stable reduction of foliated surfacesbuonerba/stable (1).pdfthis deep relation between algebraic...

STABLE REDUCTION OF FOLIATED SURFACES

FEDERICO BUONERBA

Abstract We study one-dimensional families of foliated algebraic surfaces of

general type we analyze the foliation-invariant subscheme along which the foliated

canonical bundle fails to be ample prove the existence of canonical models and

analyze their structure in detail This leads to a compactification of the moduli

space of foliated surfaces

Contents

I Introduction 2

II Preliminaries 8

III Operations on Deligne-Mumford stacks 8

IIII Width of embedded parabolic champs 10

IIIII Gorenstein foliation singularities 11

IIIV Foliated adjunction 13

IIV Canonical models of foliated surfaces with canonical singularities 14

IIVI Canonical models of foliated surfaces with log-canonical singularities 16

IIVII Set-up 23

III Invariant curves and singularities local description 25

IIII sing(F ) not isolated 25

IIIII sing(F ) isolated 28

IIIIII Local consequences 30

IV Invariant curves along which KF vanishes 33

IVI KF -nil curves 33

IVII Invariant curves inside sing(F ) 40

V Configurations 42

VI Configurations of rigid invariant curves intersecting KF trivially 431

2 FEDERICO BUONERBA

VII Configurations of non-rigid invariant curves intersecting KF trivially 49

VI Contractions and proof of the Main Theorem 56

VII Moduli space of foliated surfaces of general type 59

References 62

I Introduction

Foliations are ubiquitous throughout mathematics In algebraic geometry they are

widely used as tools to investigate hyperbolicity and boundedness phenomena The

first example of this is Bogomolovrsquos ground-breaking theorem [Bo78] that on a mini-

mal surface of general type with c21 gt c2 the degree of curves can be bounded linearly

in terms of their geometric genera The proof is extremely simple the abundance

of symmetric differentials defines a birational map φ on the projectivised cotangent

bundle to our surface whose base locus is an algebraic surface carrying a canoni-

cally defined algebraic foliation Curves on our original surface lift by way of their

derivative to the projectivised cotangent bundle If such lift does not lie in the base

locus of φ the required bound on the degree follows if the lift does lie in the base

locus then the curve is invariant by the canonical foliation The conclusion follows

by two standard facts in the theory of algebraic foliations Seidenbergrsquos reduction of

foliation singularities to simple ones [Se67] and Jouanoloursquos remark that a foliated

surface with infinitely many invariant curves is fibered over a curve

This deep relation between algebraic foliations and boundedness of curves was in-

vestigated by McQuillan who in [McQ98] extended Bogomolovrsquos technique to non-

compact curves thus proving the Green-Griffiths conjecture for minimal surfaces of

general type with c21 gt c2 namely that entire curves on such surfaces cannot be

Zariski-dense The key new ingredient to the proof is the tautological inequality a

principle that can be vaguely stated as follows a foliation has simple singularities

if and only if the intersection multiplicity between any invariant disk not factor-

ing through the algebraic leaves and the so-called foliated canonical bundle can be

bounded up to a small error by the logarithm of the length of the diskrsquos boundary

Stable reduction of foliated surfaces 3

In particular a sequence of bigger and bigger invariant disks not converging in the

Gromov-Hausdorff sense to a disk with bubbles defines in the limit a closed invari-

ant measure wich intersects the foliated canonical bundle trivially

Hyperbolicity phenomena in presence of algebraic foliations can therefore be poten-

tially studied via the birational geometry of the foliated canonical bundle obstruc-

tions locate on special invariant currents along which the foliated canonical bundle

fails to be positive Such point of view was extensively developed by McQuillan and

the present work is a contribution to it The reader is invited to consult the webpage

[McQ12] for a complete discussion around this theory

As it became clear in McQuillanrsquos proof of the Green-Griffiths conjecture for c21 gt c2

the birational geometry of the foliated canonical bundle is amenable much in the

same spirit as that of the absolute canonical bundle is to a complete classification in

dimension 2 Such classification was completed in [McQ08] where an exhaustive list

of foliated surfaces with canonical Gorenstein singularities is provided Interestingly

there are some important differences between the foliated theory and the absolute

theory First abundance fails namely there exist examples of foliated canonical

bundles whose numerical Kodaira dimension and Kodaira dimension differ Second

vanishing theorems of Kodaira type fail Third foliated minimal models may have

quotient singularities while foliated canonical models may have elliptic singularities

which are never Q-Gorenstein and may even fail to be projective

The achievement of the classification in dimension 2 along with its important ap-

plications to hyperbolicity of algebraic surfaces call for a systematic study of the

birational geometry of higher-dimensional algebraic varieties foliated by curves We

refer again to [McQ12] for a discussion of the main open problems in this direction

namely existence of canonical resolution of singularities and the residue lemma in

dimensions ge 4

The present work is devoted to an in-depth study of 1-dimensional families of foliated

algebraic surfaces of general type We develop a profusion of techniques that can

be systematically used in the study of general foliated 3-folds while we settle some

of the outstanding open problems in the theory of foliated surfaces More precisely

4 FEDERICO BUONERBA

we prove the stable reduction theorem in this context and deduce the existence of

canonical compactifications of the moduli of foliated surfaces of general type

Main Theorem Let p (X F ) rarr ∆ be a one-dimensional projective semi-

stable family of foliated orbifold surfaces whose total space has Gorenstein canonical

foliation singularities such that KF is big and nef Then there exists a digram of

birational maps

(X F ) 99K (X F )canpcanminusminusrarr ∆

Where

bull The dashed arrow is a composition of foliated flops including contractions of

divisors and curves along foliation-invariant centers

bull The singularities of (X F )can not contained in the central fiber of pcan define

sections of pcan on its general fiber these singularities are elliptic featuring a

minimal resolution with exceptional divisor a foliation-invariant nodal ratio-

nal curve Along such singularities Fcan is not Q-Gorenstein and pcan might

fail to be projective

bull The singularities of (X F )can contained in the central fiber of pcan are de-

scribed as follows

(1) One-dimensional They admit a resolution with foliation-invariant ir-

reducible exceptional divisor ruled by nodal rational curves on whose

normalization the foliation is birationally isotrivial

(2) Zero-dimensional They admit a resolution with exceptional set a tree

of foliation-invariant quadric surfaces and orbifold curves such that

quadric surfaces have cohomologous rulings and the foliation restricts

to a Kronecker vector field each orbifold curve is unibranch with at

most one singular point has universal cover whose normalization has

rational moduli it is either fully contained in sing(F ) or it intersects

sing(F ) in two points

bull The birational transform KFcan of KF is numerically big and nef and sat-

isfies if KFcan middot C = 0 for some curve C sub Xcan then C is not foliation-

invariant


The output (X F )can is called foliated canonical model of (X F )

The core of the previous statement is that there exists an algorithmic birational

transformation (X F ) 99K (X F )can which contracts all the foliation-invariant

curves which intersect KF trivially One might wonder about the origin of the

notion of foliated canonical model In the absolute theory a canonical model of a

variety of general type is one where the canonical bundle is as positive as possible

In the foliated theory we only care about positivity against invariant measures - on

a philosophical level this is due to the fact that obstructions to hyperbolicity lie

on such measures Moreover we can provide an example which shows clearly that

this is the good notion of foliated canonical model Consider the algebraic foliation

defined by the natural projection

π M g1 rarrM g

The foliated canonical bundle coincides with the relative canonical bundle ωπ which

is big and nef but not semi-ample in characteristic zero as shown by Keel in [Ke99]

Its base locus coincides with the nodal locus of π and it is not even known what

type of analytic structure if any can be supported by the topological contraction

of such nodal locus From a purely algebraic perspective π cannot be improved to

a better foliated canonical model Fortunately this agrees with our definition since

the nodal locus of π is certainly not a foliation-invariant subvariety

We can now discuss the structure of the paper and of the proof

In Section II we first review the basic definitions and constructions to be used sys-

tematically in the proof These include some operations of general character on

Deligne-Mumford stacks such as building roots and Vistoli covers as well as net

completion we then turn to the basic definitions of birational foliation theory namely

the notion of (log)canonical Gorenstein singularities and the adjunction formula for

invariant curves we continue by recalling McQuillanrsquos classification of foliated sur-

faces with canonical Gorenstein singularities we conclude by describing out how to

extend McQuillanrsquos 2-dimensional theory to the more general case of foliated surfaces

with log-canonical Gorenstein singularities - in particular we classify the singularities

of the underlying surface construct minimal amp canonical models and describe the

6 FEDERICO BUONERBA

corresponding exceptional sets The situation differs considerably from the canoni-

cal world indeed surfaces supporting log-canonical foliation singularities are cones

over smooth curves of arbitrary high genus and embedding dimension and even on

smooth surfaces invariant curves through a log-canonical singularity might acquire

cusps of arbitrary high multiplicity It is worth pointing out at this stage that one

of the main technical tools we use all time is the existence of Jordan decomposition

of a vector field in a formal neighborhood of a singular point This allows us to

decompose a formal vector field part as a sum partS + partN of commuting formal vector

fields where partS is semi-simple with respect to a choice of regular parameters and

partN is formally nilpotent

In Section III we compute the local structure of a foliation by curves tangent to a

semi-stable morphism of relative dimension 2 In particular we focus on the local

configurations of foliation-invariant curves through a foliation singularity We achieve

this by distinguishing all possible 18 combinations of number of eigenvalues of partS

at the singularity whether the singularity is 0 or 1-dimensional number of local

branches of the central fiber of the semi-stable morphism through the singularity

This classification is the first step towards the proof of the Main Theorem

In Section IV we study the geometry of foliation-invariant curves along which KF

vanishes this is technically the most important chapter of the paper To understand

the problem observe that the locus of points in ∆ over which the foliation has a

log-canonical singularity can be non-discrete even dense in its closure If we happen

to find a rigid curve in a smooth fiber of p that we wish to contract then the only

possibility is that the rigid curve is smooth and rational In particular we have to

prove that rigid cuspidal rational curves dotting KF trivially cannot appear in the

smooth locus of p even though log-canonical singularities certainly do The first

major result is Proposition IVI6 that indeed invariant curves in the smooth locus

of p that intersect KF trivially are rational with at most one node and do move in a

family flat over the base ∆ The main ingredient of the proof is the existence of Jor-

dan decomposition uniformly in a formal neighborhood of our curve this provides us

with an extremely useful linear relation equation 22 between the eigenvalues of the

Jordan semi-simple fields around the foliation singularities located along the curve -


linear relation which depends uniquely on the weights of the normal bundle to the

curve This allows to easily show that the width of the curve must be infinite Having

obtained the best possible result for invariant curves in the general fiber we switch

our attention to curves located in the central fiber of p Also in this case we have

good news indeed such curves can be flopped and can be described as a complete

intersection of two formal divisors which are eigenfunctions for the global Jordan

semi-simple field - this is the content of Proposition IVI8 We conclude this im-

portant chapter by describing in Proposition IVII3 foliation-invariant curves fully

contained in the foliation singularity which turn out to be smooth and rational The

proof requires a simple but non-trivial trick and provides a drastic simplification of

the combinatorics to be dealt with in the next chapter

In Section V we globalize the informations gathered in the previous two chap-

ters namely we describe all possible configurations of invariant curves dotting KF

trivially These can be split into two groups configurations all of whose sub-

configurations are rigid and configurations of movable curves The first group is

analyzed in Proposition VI10 and it turns out that the dual graph of such con-

figurations contains no cycles - essentially the presence of cycles would force some

sub-curve to move either filling an irreducible component of the central fiber or in

the general fiber transversely to p The second group is the most tricky to study

however the result is optimal Certainly there are chains and cycles of ruled surfaces

on which p restricts to a flat morphism The structure of irreducible components of

the central fiber which are filled by movable invariant curves dotting KF trivially

is remakably poor and is summarized in Corollary VII9 there are quadric surfaces

with cohomologous rulings and carrying a Kronecker vector field and there are sur-

faces ruled by nodal rational curves on which the foliation is birationally isotrivial

Moreover the latter components are very sporadic and isolated from other curves of

interest indeed curves in the first group can only intersect quadric surfaces which

themselves can be thought of as rigid curves if one is prepared to lose projectivity

of the total space of p As such the contribution coming from movable curves is

concentrated on the general fiber of p and is a well solved 2-dimensional problem

8 FEDERICO BUONERBA

In Section VI we prove the Main Theorem we only need to work in a formal neigh-

borhood of the curve we wish to contract which by the previous chapter is a tree

of unibranch foliation-invariant rational curves The existence of a contraction is

established once we produce an effective divisor which is anti-ample along the tree

The construction of such divisor is a rather straightforward process which profits

critically from the tree structure of the curve

In Section VII we investigate the existence of compact moduli of canonical models

of foliated surfaces of general type The main issue here is the existence of a rep-

resentable functor indeed Artinrsquos results tend to use Grothendieckrsquos existence in

a rather crucial way which indeed relies on some projectivity assumption - a lux-

ury that we do not have in the foliated context Regardless it is possible to define

a functor parametrizing deformations of foliated canonical models together with a

suitably defined unique projective resolution of singularities This is enough to push

Artinrsquos method through and establish the existence of a separated algebraic space

representing this functor Its properness is the content of our Main Theorem

II Preliminaries

This section is mostly a summary of known results about holomorphic foliations by

curves By this we mean a Deligne-Mumford stack X over a field k of characteristic

zero endowed with a torsion-free quotient Ω1X k rarr Qrarr 0 generically of rank 1 We

will discuss the construction of Vistoli covers roots of divisors and net completions

in the generality of Deligne-Mumford stacks a notion of singularities well adapted to

the machinery of birational geometry a foliated version of the adjunction formula

McQuillanrsquos classification of canonical Gorenstein foliations on algebraic surfaces

a classification of log-canonical Gorenstein foliation singularities on surfaces along

with the existence of (numerical) canonical models the behavior of singularities on

a family of Gorenstein foliated surfaces

III Operations on Deligne-Mumford stacks In this subsection we describe

some canonical operations that can be performed on DM stacks over a base field

k We follow the treatment of [McQ05][IaIe] closely Proofs can also be found in


[Bu] A DM stack X is always assumed to be separated and generically scheme-like

ie without generic stabilizer A DM stack is smooth if it admits an etale atlas

U rarr X by smooth k-schemes in which case it can also be referred to as orbifold

By [KM97][13] every DM stack admits a moduli space which is an algebraic space

By [Vis89][28] every algebraic space with tame quotient singularities is the moduli

of a canonical smooth DM stack referred to as Vistoli cover It is useful to keep in

mind the following Vistoli correspondence

Fact III1 [McQ05 Ia3] Let X rarr X be the moduli of a normal DM stack and

let U rarrX be an etale atlas The groupoid R = normalization of U timesX U rArr U has

classifying space [UR] equivalent to X

Next we turn to extraction of roots of Q-Cartier divisors This is rather straight-

forward locally and can hardly be globalised on algebraic spaces It can however

be globalized on DM stacks

Fact III2 (Cartification) [McQ05 Ia8] Let L be a Q-cartier divisor on a normal

DM stack X Then there exists a finite morphism f XL rarrX from a normal DM

stack such that f lowastL is Cartier Moreover there exists f which is universal for this

property called Cartification of L

Similarly one can extract global n-th roots of effective Cartier divisors

Fact III3 (Extraction of roots) [McQ05 Ia9] Let D subX be an effective Cartier

divisor and n a positive integer invertible on X Then there exists a finite proper

morphism f X ( nradic

D)rarrX an effective Cartier divisor nradic

D subX ( nradic

D) such that

f lowastD = n nradic

D Moreover there exists f which is universal for this property called

n-th root of D which is a degree n cyclic cover etale outside D

In a different vein we proceed to discuss the notion of net completion This is a

mild generalization of formal completion in the sense that it is performed along a

local embedding rather than a global embedding Let f Y rarr X be a net morphism

ie a local embedding of algebraic spaces For every closed point y isin Y there is

a Zariski-open neighborhood y isin U sub Y such that f|U is a closed embedding In

10 FEDERICO BUONERBA

particular there exists an ideal sheaf I sub fminus1OX and an exact sequence of coherent

sheaves

(1) 0rarr I rarr fminus1OX rarr OY rarr 0

For every integer n define Ofn = fminus1OXI n By [McQ05][Ie1] the ringed space

Yn = (YOfn) is a scheme and the direct limit Y = limYn in the category of formal

schemes is the net completion along f More generally let f Y rarr X be a net

morphism ie etale-locally a closed embedding of DM stacks Let Y = [RrArr U ] be

a sufficiently fine presentation then we can define as above thickenings Un Rn along

f and moreover the induced relation [Rn rArr Un] is etale and defines thickenings

fn Yn rarrX

Definition III4 (Net completion) [McQ05 ie5] Notation as above we define the

net completion as Y = lim Yn It defines a factorization Y rarr Y rarr X where the

leftmost arrow is a closed embedding and the rightmost is net

IIII Width of embedded parabolic champs In this subsection we recall the

basic geometric properties of three-dimensional formal neighborhoods of smooth

champs with rational moduli We follow [Re83] and [McQ05rdquo] As such let X

be a three-dimensional smooth formal scheme with trace a smooth rational curve C

Our main concern is

(2) NCXsimminusrarr O(minusn)oplus O(minusm) n gt 0 m ge 0

In case m gt 0 the knowledge of (nm) determines X uniquely In fact there exists

by [Ar70] a contraction X rarrX0 of C to a point and an isomorphism everywhere

else In particular C is a complete intersection in X and everything can be made

explicit by way of embedding coordinates for X0 This is explained in the proof of

Proposition IVI8 On the other hand the case m = 0 is far more complicated

Definition IIII1 [Re83] The width width(C) of C is the maximal integer k

such that there exists an inclusion C times Spec(C[ε]εk) sub Xk the latter being the

infinitesimal neighborhood of order k


Clearly width(C) = 1 if and only if the normal bundle to C is anti-ample

width(C) ge 2 whenever such bundle is O oplus O(minusn) n gt 0 and width(C) = infinif and only if C moves in a scroll As explained in opcit width can be understood

in terms of blow ups as follows Let q1 X1 rarr X be the blow up in C Its excep-

tional divisor E1 is a Hirzebruch surface which is not a product hence q1 has two

natural sections when restricted to E1 Let C1 the negative section There are two

possibilities for its normal bundle in X1

bull it is a direct sum of strictly negative line bundles In this case width(C) = 2

bull It is a direct sum of a strictly negative line bundle and the trivial one

In the second case we can repeat the construction by blowing up C1 more generally

we can inductively construct Ck sub Ek sub Xk k isin N following the same pattern as

long as NCkminus1

simminusrarr O oplus O(minusnkminus1) Therefore width(C) is the minimal k such that

Ckminus1 has normal bundle splitting as a sum of strictly negative line bundles No-

tice that the exceptional divisor Ekminus1 neither is one of the formal surfaces defining

the complete intersection structure of Ck nor it is everywhere transverse to either

ie it has a tangency point with both This is clear by the description Reidrsquos

Pagoda [Re83] Observe that the pair (nwidth(C)) determines X uniquely there

exists a contraction X rarr X0 collapsing C to a point and an isomorphism every-

where else In particular X0 can be explicitly constructed as a ramified covering of

degree=width(C) of the contraction of a curve with anti-ample normal bundle

The notion of width can also be understood in terms of lifting sections of line bun-

dles along infinitesimal neighborhoods of C As shown in [McQ05rdquo II43] we have

assume NCp

simminusrarr O oplus O(minusnp) then the natural inclusion OC(n) rarr N orCX can be

lifted to a section OXp+2(n)rarr OXp+2

IIIII Gorenstein foliation singularities In this subsection we define certain

properties of foliation singularities which are well-suited for both local and global

considerations From now on we assume X is normal and give some definitions

taken from [McQ05] and [MP13] If Qoror is a Q-line bundle we say that F is Q-

foliated Gorenstein or simply Q-Gorenstein In this case we denote by KF = Qoror

and call it the canonical bundle of the foliation In the Gorenstein case there exists a


codimension ge 2 subscheme Z subX such that Q = KF middot IZ Such Z will be referred

to as the singular locus of F We remark that Gorenstein means that the foliation

is locally defined by a saturated vector field

Next we define the notion of discrepancy of a divisorial valuation in this context let

(UF ) be a local germ of a Q-Gorenstein foliation and v a divisorial valuation on

k(U) there exists a birational morphism p U rarr U with exceptional divisor E such

that OU E is the valuation ring of v Certainly F lifts to a Q-Gorenstein foliation

F on U and we have

(3) KF = plowastKF + aF (v)E

Where aF (v) is said discrepancy For D an irreducible Weil divisor define ε(D) = 0

if D if F -invariant and ε(D) = 1 if not We are now ready to define

Definition IIIII1 The local germ (UF ) is said

bull Terminal if aF (v) gt ε(v)

bull Canonical if aF (v) ge ε(v)

bull Log-terminal if aF (v) gt 0

bull Log-canonical if aF (v) ge 0

For every divisorial valuation v on k(U)

These classes of singularities admit a rather clear local description If part denotes a

singular derivation of the local k-algebra O there is a natural k-linear linearization

(4) part mm2 rarr mm2

As such we have the following statements

Fact IIIII2 [MP13 Iii4 IIIi1 IIIi3] A Gorenstein foliation is

bull log-canonical if and only if it is smooth or its linearization is non-nilpotent

bull terminal if and only if it is log-terminal if and only if it is smooth and gener-

ically transverse to its singular locus

bull log-canonical but not canonical if and only if it is a radial foliation


Where a derivation on a complete local ring O is termed radial if there ex-

ist x1 middot middot middotxn isin m independent mod m2 and positive integers ni such that part =sumi nixi

partpartxi

In this case the singular locus is the center of a divisorial valuation with

zero discrepancy and non-invariant exceptional divisor

A very useful tool which is emplyed in the analysis of local properties of foliation

singularities is the Jordan decompositon [McQ08 I23] Notation as above the

linearization part admits a Jordan decomposition partS + partN into commuting semi-simple

and nilpotent part It is easy to see inductively that such decomposition lifts canon-

ically over successive infinitesimal neighborhoods Omn+1 rarr Omn and in the limit

we obtain a Jordan decomposition for the linear action of part on the whole complete

ring O

IIIV Foliated adjunction In this subsection we provide an adjunction formula

for foliation-invariant curves Let (X F ) be a Gorenstein foliation and f L rarrX the normalization of a F -invariant one-dimensional stack not contained in the

singular locus Z Denote by χ(L ) its topological Euler characteristic and by sZ(f)

the multiplicity of the ideal sheaf fminus1IZ We have

Fact IIIV1 [McQ05 IId4]

(5) KF middotL = minusχ(L )minus Ramf +sZ(f)

Which is a simple consequence of the isomorphism f lowastKF middotfminus1IZsimminusrarr Ω1

L (minusRamf )

The local contribution of sZ(f)minusRamf computed for a branch of f around a point

p in the support of fminus1IZ is np|Gp|minus1 with np a positive integer and Gp the local

monodromy of L at p Assume now that L has no generic stabilizer and let |L |denote its moduli The previous formula can be rewritten more usefully

Fact IIIV2

(6) KF middotL = 2g(|L |)minus 2 + fminus1IZ +sumf(p)isinZ

(np minus 1)|Gp|minus1 +sumf(p)isinZ

(1minus |Gp|minus1)

This can be easily deduced via a comparison between χ(L ) and χ(|L |) The

adjunction estimate 6 gives a complete description of invariant curves which are not


contained in the singular locus and intersect the canonical KF non-positively A

complete analysis of the structure of KF -negative curves and much more is done

in [McQ05]

Definition IIIV3 A KF -nil curve is an invariant one-dimensional orbifold f

C rarr X such that KF middotf C = 0 and f does not factor through the singular locus

Z of the foliation

By adjunction 6 we have

Proposition IIIV4 The following is a complete list of possibilities for KF -nil

curves f

bull C is an elliptic curve without non-schematic points and f misses the singular

locus

bull |C| is a rational curve f hits the singular locus in two points with np = 1

there are no non-schematic points off the singular locus

bull |C| is a rational curve f hits the singular locus in one point with np = 1 there

are two non-schematic points off the singular locus with local monodromy

Z2Z

bull |C| is a rational curve f hits the singular locus in one point p there is at

most one non-schematic point q off the singular locus we have the identity

(np minus 1)|Gp|minus1 = |Gq|minus1

As shown in [McQ08] all these can happen In the sequel we will always assume

that a KF -nil curve is simply connected We remark that an invariant curve can have

rather bad singularities where it intersects the foliation singularities First it could

fail to be unibranch moreover each branch could acquire a cusp if going through

a radial singularity This phenomenon of deep ramification appears naturally in

presence of log-canonical singularities

IIV Canonical models of foliated surfaces with canonical singularities In

this subsection we provide a summary of the birational classification of Gorenstein

foliations with canonical singularities on algebraic surfaces obtained in [McQ08] Let


X be a two-dimensional smooth DM stack with projective moduli and F a foliation

with canonical singularities Since X is smooth certainly F is Q-Gorenstein If

KF is not pseudo-effective then foliated bend and break [BM16 Main Theorem]

shows that F is birationally a fibration by rational curves If KF is pseudo-effective

its Zariski decomposition has negative part a finite collection of invariant chains of

rational curves which can be contracted to a smooth DM stack with projective

moduli on which KF is nef At this point those foliations such that the Kodaira

dimension k(KF ) le 1 can be completely classified

Fact IIV1 [McQ08 IV3] Foliations with canonical singularities and Kodaira di-

mension zero are up to a ramified cover and birational transformations defined by

a global vector field The minimal models belong the following list

bull A Kronecker vector field on an abelian surface

bull The suspension of a representation π1(E)rarr PGL2(C) for E an elliptic curve

bull A Kronecker vector field on P1 timesP1

bull An isotrivial elliptic fibration


mension one are classified by their Kodaira fibration The linear system |KF | defines

a fibration onto a curve and the minimal models belong to the following list

bull The foliation and the fibration coincide so then the fibration is non-isotrivial

elliptic

bull The foliation is transverse to a projective bundle (Riccati)

bull The foliation is everywhere smooth and transverse to an isotrivial elliptic

fibration (turbolent)

bull The foliation is parallel to an isotrivial fibration in hyperbolic curves

On the other hand for foliations of general type the new phenomenon is that

global generation fails The problem is the appearence of elliptic Gorenstein leaves

these are cycles possibly irreducible of invariant rational curves around which KF

is numerically trivial but might fail to be torsion Assume that KF is big and nef


and consider morphisms

(7) X rarrXe rarrXc

Where the composite is the contraction of all the KF -nil curves and the rightmost

is the minimal resolution of elliptic Gorenstein singularities

Fact IIV3 [McQ08 IV21 IV22] Xe is projective there exists an ample divisor

A and an effective divisor E supported on minimal elliptic Gorenstein leaves such

that KFe = A+E On the other hand Xc might fail to be projective and Fc is never

Q-Gorenstein

We refer to [McQ08 III32] for the descriptions of possible dual graphs of config-

urations of invariant KF -negative or nil curves

IIVI Canonical models of foliated surfaces with log-canonical singulari-

ties In this subsection we study Gorenstein foliations with log-canonical singulari-

ties on algebraic surfaces In particular we will classify the singularities appearing

on the underlying surface prove the existence of minimal and canonical models

describe the exceptional curves appearing in the contraction to the canonical model

Proposition IIVI1 Let (UF ) be a germ of Gorenstein log-canonical foliation

singularity Then U is a cone over a subvariety Y of a weighted projective space

whose weights are determined by the eigenvalues of F Moreover F is defined by

the rulings of the cone

Proof By Fact IIIII2 there exists a closed embedding i U rarr M - with M a

smooth local formal scheme - a regular system of parameters x1 middot middot middot xk on M and

positive integers n1 middot middot middotnk such that F is the restriction of the foliation defined by

part =sumnixi

partpartxi

to U Let I sub OM denote the ideal defining i(U) subM then part(I) sube I

We are going to prove that I is homogeneous where each xi has weight ni Let f isin I

and write f =sum

dge1 fd where fd is homogeneous of degree d in x1 middot middot middot xk - ie fd is

a k-linear combination of monomials xa11 xakk with d =

sumi aini For every N isin N

let FN = (xa11 xakk

sumi aini ge N) This collection of ideals defines a natural

filtration on OM For a = max ai we have mN sube FN sube mNa ie the FN -filtration


is equivalent to the one by powers of the maximal ideal and therefore OM is also

complete with respect to the FN -filtration

We will prove that if f isin I then fd isin I for every d

Let IN = (I + FN)FN Since OM is complete with respect to the FN -filtration

I = limlarrminus IN Therefore it is enough to show

Claim IIVI2 For every N gt 0 we have fd isin IN sub OMFN for all d lt N

Proof For every g isin IN denote by n(g) the unique integer such that g isin Fn(g) Fn(g)+1 (ie the minimal index d for which fd 6= 0) and by N(g) = N minus n(g)

We will prove the statement of the Claim by induction on N(f) If N(f) = 1 then

f = fNminus1 holds in OMFN and the statement is true If N(f) gt 1 we have part(f) =sumd d middot fd hence consider f1 = D(f)minus n(f)f =

sumdgtn(f)(dminus n(f))fd Tautologically

we have N(f1) lt N(f) and therefore fd isin IN for every d gt n(f) so then fn(f) =

f minussum

dgtn(f) fd isin IN as well

This implies that I is a homogeneous ideal and hence U is the germ of a cone over

a variety in the weighted projective space P(n1 nk)

Corollary IIVI3 If the germ U is normal then Y is normal If U is normal

of dimension 2 the weighted blow-up p U rarr U at the vertex of the cone has only

quotient singularities the exceptional divisor E(simminusrarr Y ) is smooth and everywhere

transverse to the induced foliation Moreover we have

(8) plowastKF = KF + E

Certainly the converse to Proposition IIVI1 fails even on surfaces indeed let

(X F ) be a Gorenstein foliated surface and let C sub X be a generic curve so

in particular smooth and not F -invariant We can assume perhaps after a finite

sequence of simple blow-ups along C that both X and F are smooth in a neigh-

borhood of C C and F are everywhere transverse and C2 lt 0

Proposition IIVI4 The formal contraction q X rarr X0 of C is isomorphic to

the cone over C the projected foliation F0 coincides with that by rulings on the cone

F0 is Q-Gorenstein if C rational or elliptic but not in general


Proof F0 is Q-Gorenstein if and only if the bundle KF (C) is trivial on the formal

completion of X along C The obstructions to lift the isomorphism OC(KF +C)simminusrarr

OC over infinitesimal neighborhoods of C lie in H1(C (1 minus n)C + KF ) for every

n isin N These groups vanish for every n isin N if 2g(C) minus 2 + C2 lt 0 (which is

always true for rational or elliptic curves) but do provide non-trivial obstructions in

general

We focus on the minimal model program for Gorenstein log-canonical foliations

on algebraic surfaces Let X be a DM stack of dimension 2 with projective moduli

and F a Gorenstein foliation with log-canonical singularities Denote by LC(F )

the set of points where F is log-canonical and not canonical and by Z the singular

sub-scheme of F We assume LC(F ) 6= empty since the empty case has been completely

settled Moreover we can assume that X is smooth outside LC(F ) Let f C rarrX

be a morphism from a 1-dimensional stack with trivial generic stabilizer such that

fminus1 LC(F ) 6= empty Since we will need them several times we formulate some technical

results

Lemma IIVI5 Let p X rarr X be the resolution of the log-canonical foliation

singularity intersecting C with exceptional divisor E Then

(9) KF middot C minus C middot E = KF middot C

Proof We have

(10) plowastC = C minus (C middot EE2)E

Intersecting this equation with equation 8 we obtain the result

This formula is important because it shows that passing from foliations with log-

canonical singularities to their canonical resolution increases the negativity of inter-

sections between invariant curves and the canonical bundle In fact the log-canonical

theory reduces to the canonical one after resolving the log-canonical singularities

Further we list some strong constraints given by invariant curves along which the

foliation is smooth


Fact IIVI6 [BB72 Baum-Bott] Let g C rarrX be an F -invariant curve missing

the foliation singularities Then C2 = NF middotg C = 0

Corollary IIVI7 Let g C rarr X be an F -invariant curve missing the foliation

singularities and such that KF middotg C lt 0 Then F is birationally a fibration by

rational curves

Proof By assumption KF middot C = minusχ(C) This together with Baum-Bott IIVI6

imply that KX middot C = minusχ(C) and from usual adjunction we get C2 = 0 Riemann-

Roch gives h0(X mC) ge 2 for m gtgt 0 so then |C| defines a fibration by rational

curves tangent to F

The rest of this subsection is devoted to the construction of minimal and canonical

models in presence of log-canonical singularities The only technique we use is

resolve the log-canonical singularities in order to reduce to the canonical case and

keep track of the exceptional divisor

We are now ready to handle the existence of minimal models

Theorem IIVI8 (Minimal models) Let X be a DM stack of dimension 2 with pro-

jective moduli and F a Gorenstein foliation with log-canonical singularities Then

either

bull F is birational to a fibration by rational curves or

bull There exist a birational contraction q X rarr X0 such that KF0 is nef

Moreover the exceptional curves of q donrsquot intersect LC(F )

Proof Let f be as above ie KF -negative and intersecting LC(F ) If f is not

F -invariant then KF middotC +C2 ge 0 so C moves and KF is not pseudo-effective We

conclude by foliated bend and break [BM16]

If f is F -invariant then foliated adjunction 6 implies that C is rational it intersects

the singular locus of F in exactly one point By Lemma IIVI5 after resolving the

log-canonical singularity KF middot C lt 0 and F is smooth along C We conlude by

Corollary IIVI7


From now on we add the assumption that KF is nef and we want to analyze

those curves f satisfying KF middotf C = 0 By adjunction formula 6 we know that C

has rational moduli and fminus1Z is supported on two points

Remark IIVI9 If fminus1Z = fminus1 LC(F ) then F is birationally a fibration by ratio-

nal curves

Proof After resolving the log-canonical singularities of F along the image of C we

have KF middot C lt 0 and F smooth along C We conclude by Corollary IIVI7

Therefore we can assume that f satisfies the following condition

Definition IIVI10 An invariant curve f C rarr X is called log-contractible if

KF middot C = 0 and it intersects Z in two points one canonical and one log-canonical

not canonical

It follows that f(C) is everywhere unibranch but it might have a cusp in LC(F )

Definition IIVI11 A log-chain is a chain of invariant rational orbifolds C =

cupni=0Ci scheme-like outside the foliation singularities with KF middot C = 0 and such

that

bull C0 is log-contractible with lowast(C ) = C0 cap LC(F ) called the marking of C

bull Ci is disjoint from LC(F ) if i ge 1

There is a unique point t(C ) satisfying t(C ) isin Z cap (Cn Cnminus1) called the tail of

the log-chain

Proposition IIVI12 Let C1C2 be two log-chains intersecting non-trivially Then

lowast(C1) = lowast(C2) and C1 cap C2 = lowast(C1)

Proof It is enough to prove that the two log-chains cannot intersect in their tails

Replace X be the resolution of the at most two markings of the log-chains and Ci

by their proper transforms C1cupC2 = cupmi=0Ri is a connected chain of rational curves

intersecting Z outisde LC(F ) Moreover each of the two extremal curves R0 and

Rm intersects Z in one point only In particular KF middot Ri = 0 for i = 1 middot middot middot m minus 1

and KF middot Ri lt 0 for i = 0m Since we assumed F is not birational to a fibration


by rational curves there exists a contraction p X rarr X0 with exceptional curve

cupmminus1i=0 Ri However F0 is smooth along plowastRm so by adjunction formula KF0 middotplowastRm lt

0 and we find a contradiction by Corollary IIVI7

Corollary IIVI13 Let C be a maximal connected curve such that KF middotC = 0 and

C cap LC(F ) 6= empty Then C is a union of finitely many log-chains intersecting each

other in their common marking

The following shows a configuration of log-chains sharing the same marking col-

ored in black

We are now ready to state and prove the canonical model theorem in presence of

log-canonical singularities

Theorem IIVI14 (Canonical models) Let X be a proper DM stack of dimension

2 and F a foliation with isolated singularities Moreover assume that KF is nef

that every KF -nil curve is contained in a log-chain and that F is Gorenstein and

log-canonical in a neighborhood of every KF -nil curve Then either

bull There exists a birational contraction p X rarr X0 whose exceptional curves

are unions of log-chains If all the log-contractible curves contained in the log-

chains are smooth then F0 is Gorenstein and log-canonical If furthermore

X has projective moduli then X0 has projective moduli In any case KF0

is numerically ample

bull There exists a fibration p X rarr B over a curve whose fibers intersect

KF trivially Each F -invariant fiber of p intersecting LC(F ) is union of

mutually compatible log-chains

Proof If all the KF -nil curves can be contracted by a birational morphism we are in

the first case of the Theorem Else let C be a maximal connected curve withKF middotC =

0 and which is not contractible It is a union of pairwise compatible log-chains


with marking lowast Replace (X F ) by the resolution of the log-canonical singularity

at lowast with exceptional divisor E Therefore F is Gorenstein and canonical in a

neighborhood of C cupE By equation 9 we have KF middotC = minusC middotE lt 0 and therefore

C can be contracted to a canonical foliation singularity Replace (X F ) by the

contraction of C We have a smooth curve E around which F is smooth and

nowhere tangent to E Moreover

(11) KF middot E = minusE2

And we distinguish cases

bull E2 gt 0 so KF is not pseudo-effective a contradiction

bull E2 lt 0 so the initial curve C is contractible a contradiction

bull E2 = 0 which we proceed to analyze

The following is what we need to handle the third case

Proposition IIVI15 Let (X F ) be a foliation with canonical singularities with

X a smooth 2-dimensional DM stack with projective moduli and KF nef If there

exists a smooth curve E everywhere transverse to F such that KF middot E = E2 = 0

then E moves in a pencil and defines a fibration p X rarr B onto a curve whose

fibers intersect KF trivially

Proof If KF has Kodaira dimension one then its linear system contains an effective

divisor and by Hodge index KF = cE as divisors for some c isin Q+ Hence there is

a fibration onto a curve with fiber E and F is transverse to it by assumption If

KF has Kodaira dimension zero we use the classification IIV1 in all cases but the

suspension over an elliptic curve E defines a product structure on the underlying

surface in the case of a suspension every section of the underlying projective bundle

is clearly F -invariant Therefore E is a fiber of the bundle

This concludes the proof of the Theorem

We remark that the second alternative in the above theorem is rather natural

let (X F ) be a foliation which is transverse to a fibration by rational or elliptic

curves ie a suspension over elliptic curve turbolent Riccati or isotrivial Pick a


general fiber E of the fibration and blow up any number of points on it The proper

transform E is certainly contractible By Proposition IIVI4 the contraction of E is

Gorenstein and log-canonical Moreover the fiber E has been replaced by a union of

compatible log-chains Note in particular that the number of log-chains appearing

is arbitrary

Corollary IIVI16 Let X be a proper DM stack of dimension 2 and F a Goren-

stein foliation with log-canonical singularities Assume that KF is big and pseudo-

effective Then there exists a birational morphism p X rarr X0 such that C is

exceptional if and only if KF middot C le 0 In particular KF0 is numerically ample

For future applications we spell out

Corollary IIVI17 Hypothesis as in Theorem IIVI14 assume further that X

is everywhere smooth Then in the second alternative of the theorem the fibration

p X rarr B is by rational curves

Proof Indeed in this case the curve E which appears in the proof above as the

exceptional locus for the resolution of the log-canonical singularity at lowast must be

rational By the previous Proposition p is defined by a pencil in |E|

IIVII Set-up In this subsection we prepare the set-up we will deal with in the

rest of the manuscript Briefly we want to handle the existence of canonical models

for a family of foliated surfaces parametrized by a one-dimensional base To this

end we will discuss the modular behavior of singularities of foliated surfaces with

Gorenstein singularities recall McQuillan-Panazzolorsquos resolution of foliation singu-

larities building on the existence of minimal models [McQ05] prepare the set-up

for the sequel

Let us begin with the modular behavior of Gorenstein singularities

Proposition IIVII1 Let (X F ) be a smooth DM stack with a foliation by

curves Assume there exists a flat proper morphism p X rarr C onto an algebraic

curve such that F restricts to a foliation on every fiber of p Then the set

(12) s isin C Fs is log-canonical


is Zariski-open while the set

(13) s isin C Fs is canonical

is the complement to a countable subset of C lying on a real-analytic curve

Proof By Fact IIIII2 being log-canonical is equivalent to the linearized endomor-

phism part IZI2Z rarr IZI

2Z being non-nilpotent where Z is the singular sub-scheme

of the foliation Of course being non-nilpotent is a Zariski open condition The

second statement also follows from opcit since a log-canonical singularity which is

not canonical has linearization with all its eigenvalues positive and rational The

statement is optimal as shown by the vector field on A3xys

(14) part(s) =part

partx+ s

part

party

Next we recall the existence of a functorial canonical resolution of foliation singu-

larities for foliations by curves on 3-dimensional stacks

Fact IIVII2 [MP13] Let X = (XDF ) be a 3-dimensional DM stack with bound-

ary foliated by curves Then there exists a proper birational morphism Xprime rarr X such

that X prime is a smooth DM stack and F prime has only canonical foliation singularities

The original opcit contains a more refined statement including a detailed de-

scription of the claimed birational morphism

In order for our set-up to make sense we mention one last very important result

Fact IIVII3 [McQ05] Let X be a smooth DM stack with projective moduli and F

a foliation by curves with (log)canonical singularities Then there exists a birational

map X 99K Xm obtained as a composition of flips such that Xm is a smooth

DM stack with projective moduli the birational transform Fm has (log)canonical

singularities and one of the following happens

bull KFm is nef or


bull There exists a fibration Xm rarr B over a smooth base whose fibers are

weighted projective spaces such that Fm is tangent to the fibers and restricts

to a radial foliation on those

Finally we can organize our notation

Set-up IIVII4 (X F ) is a smooth 3-dimensional DM stack with projective mod-

uli F a foliation by curves with canonical singularities such that KF is big and nef

There exists a semi-stable morphism p X rarr ∆ whose moduli is projective onto

the (formal) unit disk F restricts to a foliation on every fiber of p it has canon-

ical singularities on the very general fiber of p and log-canonical singularities on

every fiber of p Our task from now on is to analyze and contract the maximal

foliation-invariant sub-scheme of X along which KF is not ample

III Invariant curves and singularities local description

In this section we give a complete decription of germs of invariant curves that

appear in the formal completion of any point in the set-up IIVII4 Let us replace

X by its formal completion in a point lowast where the foliation is singular F is defined

by a vector field part with Jordan decomposition partS + partN see the end of subsection

IIIII Since [partS partN ] = 0 partN preserves the eigenspaces of partS Moreover the fact that

F is tangent to p translates into

(15) partS(p) = partN(p) = 0

The description of invariant germs of curves through lowast will be achieved via case-by-

case analysis organized by computing what happens for all possible combinations

of the following informations whether the singular locus Z is isolated at lowast the

number of eigenvalues of partS the number of components of the fiber through p

IIII sing(F ) not isolated

ni (1) partS has one eigenvalue

bull We have partS = x partpartx

and partN = f partparty

+ g partpartz

for some functions f g isin (y z)

Moreover partN is non-trivial


bull Z = (x = f = g = 0) is not isolated

(a) The fiber has one component y = 0 Then 0 = partN(y) = f hence

Z = (x = g = 0) The restriction on the fiber is part|(y=0) = x partpartx

+g|(y=0)partpartz

Z has a one-dimensional component contained in the fiber iff g isin (y) in

which case such component is smooth equal to (x = 0) moreover the

foliation on the fiber saturates to a smooth one with invariant curves

z = 0 transverse to Z

Otherwise the foliation is saturated on the fiber and has a saddle-node

singularity

(b) The fiber has two components yz = 0 Then 0 = partN(yz) = zf + yg

hence f = yh g = minuszh for some h and Z = (x = h = 0) The restriction

on a fiber component say y = 0 is part|(y=0) = x partpartxminus zh|(y=0)

partpartz

Z has an

irreducible component contained in y = 0 iff h isin (y) in which case such

component is smooth equal to (x = 0) moreover the foliation on the

fiber saturates to a smooth one with invariant curves z = 0 transverse

to Z


singularity

(c) The fiber has three components xyz = 0 0 = partS(xyz) = xyz a contra-

diction

Summary with one eigenvalue there are either one or two components The

foliation might have a one-dimensional singularity when restricted to a fiber

component in this case such singularity is a smooth curve and the foliation

saturates to a smooth one with invariant curves transverse to the singular-

ity If the singularity is isolated on a fiber component then it is a canonical

saddle-node with at most two invariant branches


ni (2) partS has two eigenvalues

We have partS = x partpartx

+ λy partparty

and partN(x y z) = k partpartx

+ f partparty

+ g partpartz

for some func-

tions k f g

(a) The fiber has one component z = 0 Then g = 0 k f isin (x y) and

necessarily Z = (x = y = 0) is transverse to the fiber The restriction on

the fiber is saturated and non-degenerate It has canonical singularities

if either partN 6= 0 or λ isin Q+ otherwise it is log-canonical

(b) The fiber has two components xy = 0 Then 0 = partS(xy) hence λ = minus1

Also 0 = partN(xy) hence k = xh f = minusyh g isin (xy) and necessarily

Z = (x = y = 0) coincides with the intersection of the two fiber compo-

nents which is smooth the foliation saturates to a smooth one on each

component with invariant curves transverse to Z

(c) The fiber has three components xyz = 0 hence λ = minus1 Also k = xklowast

f = yf lowast g = zglowast with klowast + f lowast + glowast = 0 and glowast isin (xy) The singularity

Z = (x = y = 0) coincides with the intersection of two of the fiber

components The restriction to either such components saturates to a

smooth foliation with invariant curves transverse to Z the restriction

to z = 0 has a canonical non-degenerate singularity all the invariant

curves are those in xy = 0 transverse to Z

Summary with 2 eigenvalues there can be any number of components With

one component the foliation restricts to a saturated one with a (log)canonical

singularity which moves transversely to p With two components the folia-

tion is not saturated on either and saturates to a smooth one with invariant

curves transverse to the singularity With three components the foliation

is not saturated on two of them and saturates to a smooth one on both

with invariant curves transverse to the singularity the foliation is saturated

and has a canonical non-degenerate singularity on the third component In


the last two cases the singularity is fully contained in an intersection of two

components

ni (3) partS has three eigenvalues But then the singularity is isolated a contradiction

IIIII sing(F ) isolated

i (1) partS has one eigenvalue


and partN = xk partpartx

+ f partparty

+ g partpartz

for some functions

k f g isin (y z) Moreover partN is non-trivial

bull Z = (x = f = g = 0) is isolated

(a) The fiber has one component y = 0 Then 0 = partN(y) = f and Z cannot

be isolated a contradiction

(b) The fiber has two components yz = 0 Then 0 = partN(yz) = zf + yg so

f = yh g = minuszh with h non-constant Hence Z = (x = h = 0) is not

isolated a contradiction

(c) The fiber has three components xyz = 0 Then 0 = partS(xyz) = xyz a

contradiction

Summary the 1 eigenvalue case does not exist

i (2) partS has two eigenvalues


+ λy partparty

partN = xk partpartx

+ f partparty

+ g partpartz

for some functions

k f g

bull g is non-trivial and g isin (x y)

bull Z = (x = y = g = 0) is isolated

(a) The fiber has one component z = 0 Then g = 0 a contradiction


(b) The fiber has two components xy = 0 hence λ = minus1 and g isin (xy z)

the restriction to y = 0 is part|(y=0) = x partpartx

+g|(y=0)partpartz

and necessarily g|(y=0)

is non-vanishing Therefore the restriction to either components is satu-

rated and the singularity is a canonical saddle-node on each of them

The invariant curves are among the axes x = y = 0 x = z = 0 y = z =

0

(c) The fiber has three components xyz = 0 hence λ = minus1 Necessarily

k = xklowast f = yf lowast g = zglowast with klowast + f lowast + glowast = 0 The restriction to

y = 0 is part|(y=0) = x(1 + klowast|(y=0))partpartx

+ zglowast|(y=0)partpartz

since g isin (x y) we have

glowast|(y=0) non-trivial Therefore the restriction to either x = 0 and y = 0

is saturated and the singularity is a canonical saddle-node on each of

them

The restriction to z = 0 is part|(z=0) = x(1 + klowast|(z=0))partpartx

+ y(minus1 + f lowast|(z=0))partparty

therefore the restriction is saturated and has a canonical non-degenerate

singularity


0

Summary with 2 eigenvalues there are either two or three components the

foliation restricts to a canonical saddle-node on two of them and to a non-

degenerate singularity on the third one if it exists All the invariant curves

are in the axes

i (3) partS has three eigenvalues We have partS = x partpartx

+ λy partparty

+ microz partpartz

(a) The fiber has one component y = 0 hence λ = 0 a contradiction

(b) The fiber has two components xy = 0 hence λ = minus1 The restriction to

y = 0 is part|(y=0) = x partpartx

+ microz partpartz

+ partN|(y=0) The singularity is log-canonical


iff micro isin Qgt0 and partN|(y=0)= 0 and in this case the singularity on x = 0 is

canonical non-degenerate Invariant curves are those in y = 0 and the

axis x = z = 0

If the restriction to both fiber components is canonical then it non-

degenerate on both and the invariant curves are contained in the axes

x = y = 0 x = z = 0 y = z = 0

The fiber could be yz = 0 hence λ+micro = 0 It can be analyzed as in the

next item

(c) The fiber has three components xyz = 0 hence 1 + λ + micro = 0 In par-

ticular if λ = cmicro c isin Q+ then λ micro isin Q The restriction to y = 0

is part|(y=0) = x partpartx

+ microz partpartz

+ partN|(y=0) This singularity is log-canonical iff

micro isin Qgt0 and partN|(y=0)= 0 in which case the singularity on both x = 0

and z = 0 is canonical non-degenerate In particular if the singularity

is log-canonical on some fiber component then it is so on exactly one

component and all eigenvalues are rational Invariant curves are those

in y = 0 and the axis x = z = 0

If the restriction to all fiber components is canonical then it non-degenerate

on each of them and the invariant curves are contained in the axes

x = y = 0 x = z = 0 y = z = 0

Summary with 3 eigenvalues there are either two or three fiber components

the foliation restricts to a singular foliation on each of them and it has a

log-canonical singularity on at most one of them in this case all eigenvalues

of the semi-simple field are rational

IIIIII Local consequences We list some corollaries for future reference As al-

ways in this section the statements are local so everything happens in a formal

neighborhood of our point lowast First we state those corollaries we will need to describe

configurations of KF -nil curves


Definition IIIIII1 An LC-center Π sub X is an irreducible F -invariant formal

hypersurface such that the local vector field generating the foliation when restricted

to Π has semi-simple part with log-canonical singularity or a one-dimensional sin-

gular locus An LC-center with a log-canonical singularity is necessarily an algebraic

component of the singular fiber of p

Corollary IIIIII2 There is at most one LC-center with a log-canonical singularity

through lowast If it exists then all eigenvalues of the semi-simple field at lowast are rational

If there are at least two LC-centers through lowast then there are exactly two and the

foliation singularity is a smooth curve on each of them

Proof Possible cases in which an LC-center with log-canonical singularities appears

ni (2)a i (3)b i (3)c

Corollary IIIIII3 If there is no LC-center thorugh a point lowast then there are at

most three invariant curves through lowast all smooth Any number of them spans a sub-

space of the tangent space at lowast of maximal possible dimension

If there is an LC-center through lowast then there is at most one invariant curve through

lowast and not contained in the LC-center

If there is an LC-center with non-isolated singularity then the singularity is smooth

and all invariant curves contained in the LC-center intersect the singularity trans-

versely

Proof Invariant curves not in an LC-centers with log-canonical singularities are con-

tained in the three axes x = z = 0 x = y = 0 y = z = 0 mentioned in the above

classification The other statements are clear

Corollary IIIIII4 If p is smooth at lowast then Z is one dimensional around lowast

Proof Possible casesni (1)a ni (2)a

In particular if there is no LC-center the situation is the familiary

xz


While if there is an LC-center (pink) with log-canonical singularity it will look like

y

LC-center

While if there is an LC-center with non-isolated singularity (red) it will look like

(the black arrows being invariant curves)

y

LC-center

The next corollary describes one-dimensional components of the foliation singularity

Corollary IIIIII5 If Z0 is a one-dimensional irreducible component of the folia-

tion singularity then it is at worst nodal The foliation when restricted to any fiber

component containing Z0 saturates to a smooth foliation in a neighborhood of Z0

and all the invariant curves are transverse to Z0

If Z0 is contained in exactly one fiber component then the semi-simple field has one

eigenvalue everywhere along Z0

If Z0 lies at the intersection of two fiber components then the semi-simple field has

two constant eigenvalues everywhere along Z0

Moreover if there exist a sequence of invariant curves converging in the compact-

open topology to Z0 then there is generically one eigenvalue along Z0

Proof Possible casesni (1)a ni (1)b ni (2)b ni (2)c Concerning the one-

eigenvalue case invariant curves converging to Z0 can be found in the formal hyper-

surface x = 0


IV Invariant curves along which KF vanishes

In this section we study in great detail a formal neighborhood of invariant curves

where KF is numerically trivial In the first subsection we deal with KF -nil curves

namely those whose generic point is not contained in the foliation singularity In this

case the normal bundle to the curve determines its formal neighborhood uniquely

and this has plenty of amazing consequences In the second subsection we study

those invariant curves which are contained in the foliation singularity In this case

we are not able to control the full formal neighborhood but important informations

can still be obtained

IVI KF -nil curves In this subsection we study in detail the structure of KF -

nil curves and their formal neighborhoods appearing in the set-up IIVII4 In

particular we will describe a simple process to replace cuspidal KF -nil curves by

net ones prove that the foliation is in the net completion around a net KF -nil curve

defined by a global vector field which admits a global Jordan decomposition deduce

that eigenfunctions for the semi-simple component of such field are also globally

defined on a formal neighborhood

The first important remark is of technical nature and explains how to deal with non-

net KF -nil curves By Proposition IIIV4 this is indeed possible and the images of

such curves will acquire cusp-like singularities It is hard to analyze the geometry

of F around such cuspidal curves and therefore necessary to remove them Let us

assume that C is simply connected with rational moduli f is ramified at 0 isin C

and and denote by X the formal completion along f(C) Then formally around

f(0) the curve f(C) has defining ideal (y xa minus zb) where y = 0 is a fiber of the

fibration p X rarr ∆ containing f(C) x z isin H0(X OX ) restrict to suitable local

coordinates on such fiber and a b ge 2 are coprime integers

Claim IVI1 Let r X ( aradicz = 0) rarr X be the a-th root of z = 0 III3 Then

(rlowastC)norm =∐a

1 C and the induced map rlowastf (rlowastC)norm rarrX ( aradicz = 0) is net

Proof Since r is etale away from z = 0 also rlowastC rarr C is etale away from 0 isin C

Since C 0 is simply connected we have rlowastC rminus1(0) =∐a

1 C 0 This proves


the first statement Let ζ be a function on X ( aradicz = 0) such that ζa = z so then

rlowast(xa minus zb) =prod

ηa=1(x minus ηζb) Every branch is net in 0 and we obtain the second

statement as well

As such upon taking roots we can replace non-net KF -nil invariant curves with

rational moduli by net ones

We now proceed to study net KF -nil curves We will show that the foliation is

defined by a global vector field around a net KF -nil curve with rational moduli and

that such vector field admits a global Jordan decomposition

Proposition IVI2 Notation as in the set-up IIVII4 let f C rarrX be a KF -nil

curve Assume that f is net and let j C rarr C be the net completion along f Then

there exists an isomorphism

(16) OC(KF )simminusrarr OC

In particular there exists a global vector field partf on C that generates F

Proof Since C has rational moduli and is simply connected it has infinite-cyclic

Picard group and we have an isomorphism


Moreover we have a non-canonical splitting of the normal bundle Njsimminusrarr O(minusn) oplus

O(minusm) and we claim that both nm ge 0 Indeed consider the induced semi-stable

fibration C rarr ∆ and let F be the normalization of an irreducible component of a

fiber such that C sub F Consider the exact sequence

(18) 0rarr NCF rarr Nj rarr NFC|C rarr 0

Since KF is nef NCF is not ample since F is a fiber of a fibration also NFC|C is

not ample and this is enough to conclude even though nm may be different from

the degrees of such normal bundles

At this point we can lift the isomorphism 17 over infinitesimal neighborhoods of j

Indeed for every k ge 0 we have an exact sequence

(19) 0rarr Symk ICI2C rarr OCk+1

(KF )rarr OCk(KF )rarr 0


which shows that the obstruction to lifting the isomorphism lies in

(20) H1(C Symk ICI2C) = H1(C Symk(O(n)oplus O(m))) = 0

From this we can deduce

Proposition IVI3 The vector field partf admits a Jordan decomposition uniformly

along C

Proof The cohomology sequence attached to 19 is

(21) 0rarr H0(C Symk ICI2C)rarr H0(COCk+1

)rarr H0(COCk)rarr 0

Let partk+1 denote the image of partf in End(H0(COCk+1)) Looking at

H0(COCk+1) minusminusminusrarr H0(COCk

)

partk+1

y partk

yH0(COCk+1

) minusminusminusrarr H0(COCk)

we see the canonical Jordan deomposition of partk+1 restricts to that of partk

The above result describes a very useful interplay between normal bundles to

KF -nil curves and the structure of the semi-simple field It turns out that these

informations determine formal neighborhoods uniquely For simplicity let C be

an irreducible component of C without orbifold points Let us assume that part|C is

singular in 0infin Then there exist formal functions x y z at 0 isin C and ξ η ζ at

infin isin C such that x ξ are coordinates along C y η are local sections of OC(minusn)

so η = xny mod(I2C) z ζ are local sections of OC(minusm) so ζ = xmz mod(I2C)

Normalizing so that xminus1partS(x) = ε isin 0 1 we obtain

(22) part0S = εxpart

partx+ λy

part

party+ microz

part

partz partinfinS = minusεξ part

partξ+ (λ+ εn)η

part

partη+ (micro+ εm)ζ

part

partζ

More generally ie C = P1(d e) we have glueing rules x = te ξ = tminusd η = tny and

ζ = tmz mod(I2C) Normalizing so that tminus1partS(t) = ε isin 0 1 we obtain

(23) part0S = eεxpart

partx+ λy

part

party+ microz

part

partz partinfinS = minusdεξ part

partξ+ (λ+ εn)η

part


part

partζ


It is worth remarking that the above relations hold between eigenvalues at the two

singular point of singular ie nodal or cuspidal KF -nil curves as well this is

easily seen by taking a root followed by net completion and keeping track of the

eigenvalues

Definition IVI4 Let C be a KF -nil curve The foliation singular points along C

are called ends

As well as the existence of Hilbert scheme in the following form

Fact IVI5 Let W sub Xlowast be a curve and D a divisor in the formal neighborhood

of W such that W = D capXlowast scheme-theoretically Then W moves in D

In what follows our main concern is to understand formal neighborhoods of KF -

nil curves Observe that the fibration p when restricted to the formal neighborhood

of such curves provides at least one eigenfunction of the semi-simple field Hence

in order to compute the width IIII we must study the behavior of the other eigen-

function on infinitesimal neighborhoods We first describe the situation inside the

smooth locus of p

Proposition IVI6 (Smooth fiber) Let f C rarrX be an irreducible KF -nil curve

which is contained in the smooth locus of a fiber Xlowast of p Then the end(s) on C are

flat over ∆ ni (1)b and ni (1)a with z2|g(y z) and C moves in a family flat over

∆

Proof Let us assume that f is net Then upon passing to net completion we can

actually assume it is embedded From our local classification of singularities we case

in case ni (1)a and ni (2)a We want to show that both ends have components

that move transversely to p hence we have to exclude case ni (1)a with g(y z) = ya

By way of contradiction say we have g(y z) = ya in the notation of opcit around

an end lowast isin C We proceed to compute what happens at the other end we find

formal coordinates ξ η ζ which by equation 23 satisfy

(24) ξ = xminus1 η = y ζ = xmz mod(I2C)


As such we compute part(ζ) = mζ + ξminusmηa mod(I2C) which is holomorphic if and only

if m le 0 so then C moves in Xlowast contradicting that the foliation has general type

Hence we are in cases ni (1)a ni (2)a with g = microyazb b ge 1

(25) part0 = xpart

partx+ microyazb

part

partz partinfin = minusξ part

partξ+ ζ(m+ microξm(bminus1)ηaζbminus1)

part

partζ(mod(I2C))

Following notation from subsection IIII observe that C1 is the section of E1C which

is disjoint from the proper transform of the fiber of p In coordinates over 0

(26) x1 = x y1 = y z1 = zyminus1

And similarly over infin Using the previous equation and 25 on C1 we find

(27)

part01 = x1part

partx1+microya+bminus11 zb1

part

partz1 partinfin1 = minusξ1

part

partξ1+ζ1(m+microξ

m(bminus1)1 ηa+bminus11 ζbminus11 )

part

partζ1(mod(I2C1

))

In particular the transform of the nilpotent field is still nilpotent after blowing up in

C It follows that the eigenvalues of the semi-simple field at the two ends of C and

C1 are the same Using equation 23 we deduce that the normal bundle to C1X1 to

isomorphic to that of CX in particular it contains a trivial direct summand This

proves that width(C) =infin

Next we show that f must be net Consider the induced flat projection pD D rarr ∆

its general fiber cannot be a cuspidal curve since along such cusp the foliation sin-

gularity would be log-canonical but not canonical Hence if C has a cusp necessarily

the general fiber of pD is a nodal curve Let X rarr X denote the blow up of the

nodal locus of pD with exceptional divisor E ruled over ∆ In a neighborhood of E

we have sing(F ) = D cap E which projects to ∆ as a double cover ramified over lowastObserve that the foliation on Xlowast is again log-canonical and not canonical hence its

local structure is ni (2)a In that case however p is unramified along the foliation

singularity Contradiction

We state a general criterion for movability of KF -nil curves

Proposition IVI7 Let C be a connected KF -nil curve such that p(C ) = lowast isin ∆

Assume its dual graph is a chain that none of its irreducible components is double


VI3 and that its two extremal ends are flat over ∆ Then C moves in a family flat

over ∆ and each of its irreducible components is embedded

Proof We work in a formal neighborhood of C After taking roots we can assume

C is net and since its dual graph is a chain it is actually embedded We proceed by

induction on its length l where the case l = 1 is Proposition IVI6 For l gt 1 let C1

be a component of C which is extremal in its dual graph If C1 moves transversely

to p then both its ends do and hence we can apply the induction hypothesis to

C C1 If C1 is rigid then it can be flopped It is straightforward to check that we

can apply the induction hypothesis to the transform C + C+1 and we are done

Now we deal with rigid curves inside the singular fiber

Proposition IVI8 (Singular fiber) Let f C rarr X be an embedded net KF -

nil curve with anti-ample normal bundle Then there exists a birational map φ

(X F ) 99K (X +F+) and a KF+-nil curve C+ such that

bull φ is an isomorphism away from C+

bull In a formal neighborhood of at least one among C and C+ there exist two

invariant hypersurfaces D1D2 locally defined by the vanishing of eigenfunc-

tions of the semi-simple field such that C (resp C+)= D1 capD2

bull If F middot C lt 0 for some hypersurface F then F+ middot C+ gt 0 In particular X +

is projective

Such φ is called flop

Proof Our assumption is that Nfsimminusrarr O(minusn)oplusO(minusm) is anti-ample namely nm gt

0 If two invariant hypersurfaces as in our statement do exist around C the flop is

easily constructed as a weighted blow-up followed by a weighted blow-down As such

let us assume that such hypersurfaces do not exist around C In particular C is not

double VI3 By [Ar70] there exists a formal contraction c X rarrX0 collapsing C

to a point and an isomorphism everywhere else By Proposition IVI3 the foliation

F0 is Gorenstein around the node of X0 For ease of exposition let us assume

n = m = 1 the general case is handled the same way only the notation becomes more

cumbersome due to the presence of weights The node has embedding dimension 4


and in suitable local coordinates x0 y0 w0 z0 can be defined by x0z0minusy0w0 = 0 The

morphism c can be identified with the blow up of the embedded surface (x0 = y0 = 0)

Explicitly in the chart y = y0 x = x0yminus10 it is defined by

(28) y0 = y x0 = xy z0 = zprime w0 = xzprime

Here x is a coordinate along C while y = 0 is the pull-back of (x0 = y0 = 0) and

zprime some function in the normal bundle to C By the previous Proposition IVI6

we know C cannot be contained in the smooth locus of p Since C is not double

we can assume that in this local chart p is defined by (x y zprime) rarr xy - the case

(x y zprime) rarr xzprime being isomorphic as it is isomorphic to blow-up (z0 = w0 = 0)

instead of (x0 = y0 = 0) The issue is as emphasized by the notation zprime 6= z that

zprime may fail to be an eigenfunction for partS if it were we would be done since zprime = 0

is the local equation for the global pullback of (z0 = w0 = 0) However we have

a b isin C such that

(29) partS0(y0) = minusy0 partS0(x0) = 0 partS0(z0) = ay0 + bz0 partS0(w0) = ax0 + (b+ 1)w0

Observe that the embedded surfaces D01 = (y0 = z0 = 0) and D0

2 = (x0 = w0 = 0)

are both foliation invariant As such let b X + rarr X0 denote the blow-up of X0

along D01 - which is isomorphic to that along D0

2 - and let Di = blowastD0i i = 1 2

Finally X + is projective by Nakai-Moishezon Indeed every C with anti-ample

normal bundle is contained in an irreducible component F of X0 such that F middotC lt 0

Then F+ middot C+ gt 0 and we can find an ample divisor of the form H+ + εF+ where

H+ is the proper transform of an ample on X

Remark IVI9 In the previous proof a 6= b+ 1 Indeed such identity would create

a non-trivial Jordan block of partS0 in the (x0 w0)-plane thus contradicting the semi-

simplicity of partS0 In particular by a straightforward computation it is impossible to

find eigenfunctions of the form wlowast0 = w0 + αx0 zlowast0 = z0 + βy0 with x0z

lowast0 minus y0wlowast0 = 0

unless a = 0

We can conclude by pointing out an important point understood in this section

A painful aspect of the theory is that log-canonical singularities which appear natu-

rally as analytic degenerations of canonical ones come along with cuspidal invariant


curves There is in particular no a-priori obstruction to the existence of rigid

cuspidal KF -nil curves in a smooth fiber of our family of foliated surfaces p Quite

generally a rigid curve of positive arithmetic genus in a smooth family of surfaces can

never be contracted to a point Therefore this would have been a counter-example

to any reasonable foliated canonical model theory one may wish to achieve Fortu-

nately such phenomenon does not happen and this establishes the Main Theorem I

modulo finitely many rigid curves confined in the singular fiber X0

IVII Invariant curves inside sing(F ) In this subsection we study curves Z0 subeXlowast cap sing(F ) which are inside the foliation singularity and invariant By this we

mean that there exists a sequence fn Cn rarrX of not necessarily algebraic invariant

curves converging in the compact-open topology to Z0 In other words the map

(30) NorF rarr Ω1Z0

is trivial By Corollary IIIIII5 we know that Z0 is at worst nodal it is smooth

wherever p is smooth for every point z isin Z0 the semi-simple part of the gener-

ating vector field has exactly one eigenvalue The problem here is that the Jordan

decomposition need not extend uniformly along Z0 not even along an irreducible

component In case ni (1)a the functions x and g may develop essential singulari-

ties arbitrary close to z Nonetheless the function x is unique and this says that x

does extend along Z0

Proposition IVII1 Let Z0 be as above Then in a formal neighborhood of Z0

there exists a unique smooth invariant hypersurface D0 containing Z0 which restricts

around any point in Z0 to the unique eigenfunction for the semi-simple field whose

eigenvalue is non-zero

Proof The local structure of the foliation around a point z isin Z0 is described by

cases ni (1)a and ni (1)b In both cases partS has one non-zero eigenvalue with

eigenfunction x Such eigenfunction x is a-priori only defined in the normal bundle to

Z0 but it is clear that the equation partx = x has a unique up to constants solution in

the formal completion around z As such there exists a unique smooth hypersurface


D0 in the formal completion around Z0 which restricts to the eigenfunction x around

z

Observe that the previous argument goes through since we are assuming Z0 subX0

Keeping notations from case ni (1)a if say g(y z) = yz then in the hypersurface

y equiv q isin Qgt0 the foliation has a log-canonical singularity and the eigenfunction x

extends highly non-uniquely to a formal neighborhood of the point (x y z) = (0 q 0)

In particular the existence of D0 as in the previous Proposition might fail along

the component (x = z = 0) of the foliation singularity This phenomenon is called

the beast

Anyway uniqueness in the previous proposition yields

Corollary IVII2 Let Z prime be a connected invariant curve inside sing(F ) Then in

a formal neighborhood of Z prime there exists a unique smooth invariant hypersurface D prime

containing Z prime which restricts around any point in Z prime to the hypersurface constructed

in the previous proposition

Next we turn to the intrinsic structure of invariant singular curves

Proposition IVII3 Let Z0 be an irreducible invariant curve inside sing(F )

Then Z0 is smooth and rational

Proof We work in a formal neighborhood of Z0 Let F0 denote the unique component

of X0 containing Z0 For any other fiber component F intersecting Z0 non-trivially

blow-up the invariant curve F cap F0 with exceptional divisor EF A straightforward

computation using the local formula ni (1)b shows that the fiber of the natural

projection EF rarr F cap F0 intersecting Z0 is also contained in sing(F ) For Z prime the

maximal connected singular invariant curve containing Z0 let G denote the saturated

foliation induced on D prime It is canonical with reduced singularities and it satisfies

KG = KF |Dprime minus Zprime By adjunction

(31) 2g(Z0)minus 2 + sZG(Z0) = minusZ prime middot Z0

Observe that thanks to our initial sequence of blow-ups the singularities of G along

Z0 coincide with the intersection of Z0 and Z prime Z0 In particular since D prime is formally


flat over ∆ we deduce Z prime middot Z0 = 0 and the previous formula becomes

(32) 2g(Z0)minus 2 + sZG(Z0) = 0

This implies that Z0 is rational or elliptic If Z0 were elliptic then sZG(Z0) = 0 and

D0 defines an invariant elliptic curve in the general fiber of p By Baum-Bott this

contradicts that KF is big and nef Now we show that Z0 is smooth If it has a node

then D0 moves it in a family of nodal rational curves flat over ∆ In particular D0

itself has a node in codimension 1 hence Xlowast is smooth around the node of Z0 In its

formal neighborhood the foliation must be as in case ni (1)a which implies that

Z0 is smooth contradiction

We can improve as follows

Proposition IVII4 Let Z prime sube sing F capXlowast be a connected invariant curve with

KF middot Z prime = 0 Then its dual graph is a chain

Proof Assume Z prime is a cycle Then necessarily Z prime = D prime capXlowast scheme-theoretically

and by the existence of Hilbert scheme Z prime moves in the general fiber Its general

member is either a cycle of smooth rational curves or an elliptic curve Arguing as

in the previous proof we see that both options are impossible

V Configurations

In this section we analyze in great detail all possible configurations of invariant

curves along which KF vanishes These can be divided into three groups those

that do not move those that move in a one-dimensional family covering a surface

those that move in a higher-dimensional family covering a surface Curves in the first

group can form rather complicated configurations while those in the second group

are more simple they either move in the general fiber or they cover an isolated

Riccati component in the singular fiber Those in the third group turn out to be

represented by the two rulings of P1timesP1 if they happen to be cohomologous in the

ambient stack First we will analyze rigid curves then those which move and finally

we will describe their configurations


VI Configurations of rigid invariant curves intersecting KF trivially In

this subsection we describe configurations of rigid invariant curves intersecting KF

trivially which arise in the situation described by set-up IIVII4

First we focus on KF -nil curves At the end of the section we deal with the simpler

case in which a curve in the configuration is fully contained in sing(F ) The general

fiber was handled in Proposition IVI6

Proposition VI1 Let C be a connected KF -nil curve contained in the smooth

locus of a fiber of p Then it is a chain or a cycle and it moves in a family flat over

∆

In what follows we let C denote a connected KF -nil curve contained in the

singular fiber X0 all of whose sub-curves are rigid in X Observe a consequence of

equation 23

Proposition VI2 Let C = cupiCi be a connected KF -nil curve Then the eigen-

values of partS are all rational at every point of sing(F )capC if and only if they are all

rational at one point

We can deduce many non-trivial properties about configurations of KF -nil curves

First we need

Definition VI3 Let C subX0 be a curve It is called double if there exists formally

around C two components F F prime of X0 such that C = F cap F prime A point lowast isin C where

three fiber components intersect is called a triple point If C lies on exactly one fiber

component it is called non-double We denote by F (C) such component

Remark VI4 Let Csimminusrarr P1(d e) be double with two triple points Then we know

e+ λ+ micro = minusd+ (λ+ n) + (micro+m) = 0 whence we obtain

(33) n+m = d+ e

Which is called triple-point formula

Next we show that essentially KF -nil curves do not form cycles unless something

deforms in X0 First we show that there are no cycles of rigid non-double curves


Proposition VI5 Let C be a connected KF -nil curve contained in the singular

fiber X0 of p Assume that every component of C is non-double and that every

sub-curve is rigid Then C contains no cycles

Proof We work in a formal neighborhood of C which we assume is a cycle Let

us first consider the case in which all components C1 middot middot middot CN are smooth with no

orbifold points and denote lowasti = Ci cap Ci+1 with CN+1 = C1 Moreover let us

assume no lowasti lies in the smooth locus of p

Since no curve is double the semi-simple field at every lowasti is as in case i (3)b in

particular the eigenvalues along the branches of Ci and Ci+1 add up to zero More-

over let minusni denote the self-intersection of Ci sub F (Ci) Normalize the eigenvalues

of partS so that at lowastn they are 1 in the C1 direction minus1 in the CN direction λ in the

F (C1)capF (CN) direction It follows by induction that at lowasti the eigenvalues are minus1

in the Ci direction 1 in the Ci+1 direction λ+ n1 + middot middot middot+ ni in the F (Ci) cap F (Ci+1)

direction After traveling around the cycle we obtain n1+middot middot middot+nN = 0 contradicting

that ni gt 0 for every i

We established the Proposition assuming every Ci is an embedded scheme and ev-

ery lowasti is in the singular locus of p The case in which Cisimminusrarr P1(di ei) has orbifold

points is analogous we end up with a combination c1n1 + middot middot middot + cNnN = 0 with ci

the eigenvalue at lowasti in the Ci+1 direction Normalizing so that c1 = e1 gt 0 again by

induction on i we see that every ci isin Qgt0 Next we show that our assumption on lowastiis always satisfied

Claim VI6 Every lowasti must belong to the singular locus of p

Proof We give two essentially equivalent proofs

bull If N = 2 3 then the previous argument goes through indeed we can normal-

ize eigenvalues at lowasti to be negative in the Ciminus1 direction and positive in the

Ci-direction and argue as above Hence we can assume N ge 4 If say lowasti is in

the smooth locus of p then F (Ci) = F (Ci+1) If also F (Ciminus1) = F (Ci) then Ci

lies in the smooth locus of p and hence it moves Therefore F (Ciminus1) 6= F (Ci)


and Ciminus1 can be flopped inside F (Ci) Since N ge 4 this preserves the smooth-

ness of F (Ci) and Ci is now is the smooth locus of p - contradiction

bull If lowasti belongs to the smooth locus of p then the end through lowasti moves We

can form a partial net completion fi Xi rarrX by normalizing C in lowasti only

Then f lowasti C is a chain with movable ends and hence moves in the general fiber

of p fi

Finally the general case in which some among the Ci have cusps can be easily

reduced to the smooth case in a couple of ways

bull Eigenvalues at the singularities of any KF -nil curve satisfy the same relations

as in the smooth case

bull Upon taking roots and net completion we can find a foliated formal 3-fold

supported on a cycle of smooth KF -nil curves which is a ramified cover of a

formal neighborhood of our cycle

Similarly we can show that there are no rigid cycles of double curves

Proposition VI7 Let C be a nodal KF -nil curve which is double Then C

deforms inside X0 In particular there cannot exist a rigid cycle of KF -nil double

curves

Proof Following the previous notation we have 0 = infin and if C = (xy = z = 0)

around 0 then we have identities

(34) x = η y = ξ z = ζ

The last one of which implies m = 0 that is C deforms Since C is double necessarily

the surface it spans must be contained in X0

The statement about cycles follows since a cycle of double curves can be transformed

after finitely many flops into a chain of smooth double curves with a nodal one

attached at one end


Later on we will show that the deformations of C cannot span an elliptic surface

but only a family of nodal rational curves with a special Riccati foliation on it

We are ready to organize a detailed description of the dual graph of C More pre-

cisely we only need to understand how double curves intersect non-double curves

Lemma VI8 There exists a sequence of flops X 99KX + such that if C + denotes

the transform of C every double curve supported on C + contains two triple points

Proof Let C sube C be an irreducible component with at most one triple point and

let X 99K X + be its flop Then its transform C+ is non-double and the number

of double curves contained in C descreases It is clear that after finitely many flops

all double curves contain exactly two triple points

From now on all double curves in C have two triple points Thanks to proposition

VI2 we know that there are two cases to consider those curves along which the

semi-simple field has not only rational eigenvalues and those along which the semi-

simple field has only rational eigenvalues The first case being strictly easier than

the second it is a good starting place to warm up

Proposition VI9 (irrational eigenvalues) Assume that for every foliation singu-

larity along C not all eigenvalues of partS are rational Then every component of C

is net with at most one node and 1-skeleton of its dual graph is at worst trivalent

Write C = C axes cup C free where

C axes is the union of double curves

C free is the union of non-double curves

Then one among C axes and C free is empty Moreover

bull If C free is non-empty then it is a chain of smooth curves with rational moduli

bull If C axes is non-empty there exists a sequence of flops X 99KX + and an ir-

reducible component of the fiber FC subX + such that X + is a smooth stack

the transform C+ of C is contained in FC Moreover the dual graph of C+

is a tree which by [McQ08 III32] becomes a chain inside the normalization

of FC


Proof The first sentence is just corollary IIIIII3

Observe that if a non-double curve intersects a double curve then the non-double

curve must be contained in an LC-center around such intersection This contradicts

that not all eigenvalues are rational and establishes that one among C axes and C free

is empty

In case C free is not empty then it is clearly a chain or a cycle the latter being

impossible by Proposition VI5

We are left to show the existence of flops in case C = C axes We remark that

every component is smooth Let C0 be such a component and S0 a component of

X0 containing it Let C1 be the collection of irreducible components of C which

intersect S0 but are not contained in it All the components of C1 are smooth hence

we can flop them in any order until the transform of C1 lies in the transform of S0

Since the number of irreducible components of C lying outside S0 decreases after

such flop by induction we deduce the existence of finitely many flops after which

the birational transform of C is fully contained in the transform of S0

Next we take care of rational eigenvalues The complication here arises from

the fact that there may be double and non-double curves intersecting each other

However the situation is best possible

Proposition VI10 (rational eigenvalues) Assume that for every foliation singu-

larity along C all eigenvalues of partS are rational Write C = C axes cup C free where



Then there exists a sequence of flops X 99K X + and an irreducible component of

the fiber FC axes subX + such that

bull X + is a smooth stack

bull The union of double curves of C axes is contained in FC axes the 1-skeleton of

its dual graph is a tree and each of its components has two triple points

bull The 1-skeleton of the dual graph of the transform C free+ is a disjoint union of

trees

bull Each connected component of C free+ intersects C axes in exactly one point


In particular the 1-skeleton of the dual graph of C is a tree

Proof Notice that for every point lowast isin C cap Z there exists one LC-center through lowastIndeed there are three rational eigenvalues and they cannot be all non-vanishing and

all with different signs at the same time Now C axes can be treated as in proposition

VI9 This proves the first two items of the statement The third item follows from

Proposition VI5 so we can focus on the fourth one The proof is based on the

following obvious

Remark VI11 let C be a double curve such that either it is KF -nil hence has

two triple points or it is fully contained in sing(F ) Let C0 sube C free intersect C

non-trivially in a point lowast then necessarily C0 sub LClowast

Now let C0 middot middot middot CN form a chain of non-double curves with C0 and CN both

intersecting double curves as in the remark Denote by lowasti = Ci cap Ci+1 Since

C1 sub LClowast necessarily C1 is transverse to LClowast1 otherwise C1 would deform in F (C1)

Inductively we see that Ci+1 sub LClowasti and is transverse to LClowasti+1 In particular CN

cannot touch a double curve This contradiction proves the fourth item

Let us add a remark about configurations featuring invariant curves inside sing(F )

Proposition VI12 Let C be an invariant curve with KF middot C = 0 such that

Z0 sube C for some component Z0 sube sing(F ) Then C = Z prime cup C where

Z prime is a connected invariant curve inside the foliation singularity with dual graph a

chain

C is the union of at most two pairwise disjoint KF -nil curves with anti-ample

normal bundle

If not empty C can only intersect Z prime at the two ends of its dual graph Moreover

perhaps after a flop the intersection of C with Z prime supports the foliation singularity

along C As such if C has two components Z prime cup C deforms in the general fiber

Proof The structure of Z prime is Proposition IVII3 while that of Co is Proposition

IVI6 Let C prime be a component of C intersecting Z0 in a point lowast If C prime and Z0 belong


to different components of X0 then by case ni (1)b the eigenvalue of partS along C prime

vanishes so supp(sing(F ) cap C prime) = lowast If C prime and Z0 belong to the same component

then we can flop C prime into a different component

Observe that the divisor D prime from op cit can be extended along C prime This implies

that C can only intersect Z prime at its ends Moreover if C touches both ends necessarily

D prime capX0 = C scheme-theoretically hence by the existence of Hilbert scheme IVI5

we conclude

Our new set-up is

Set-up VI13 Everything as in set-up IIVII4 and moreover for any connected

invariant curve C with KF middot C = 0 all of whose sub-curves are rigid we have

configurations computed in proposition VI9 VI10 and Lemma VI12

VII Configurations of non-rigid invariant curves intersecting KF trivially

In this subsection we describe configurations of surfaces which are covered by KF -nil

curves The structure of each such surface is described by classification IIV1 IIV2

and Corollary IIVI17 Our first step is to simplify the situation by contracting

chains of ruled surfaces The next step is to try and exclude the possibility that

the singular fiber of p contains fibrations by elliptic or nodal rational curves with

at least one invariant fiber This can be achieved with good success indeed there

may be at most one such Riccati foliation which is anyway rather isolated from the

remaining KF -nil curves Finally we observe that the only class of surfaces we have

not mentioned is

Definition VII1 A irreducible component F of the singular fiber of p is called

small Kronecker if

bull (FFF ) is birational to a Kronecker vector field on P1 timesP1 and

bull The two rulings of this P1 timesP1 are cohomologous inside X

The name comes from the fact that contraction of a small Kronecker yields a node

in the ambient threefold which admits a small resolution In fact in many respects

these fiber components behave very much like rigid KF -nil curves


We start by getting rid of some fiber components which are ruled by KF -nil gener-

ically smooth rational curves

Proposition VII2 Notation as in Set-up VI13 there exists a sequence of flops

and divisorial contractions c X 99KX + such that


bull let S+ sub X + be an irreducible invariant divisor that contains a positive-

dimensional family of invariant rational curves along which KF+ vanishes

Then S+ is either birationally ruled by nodal rational curves or it is a small

Kronecker

Proof We focus first on contracting components of the singular fiber of p Consider

the following algorithm If there is no birationally ruled component different from

the ones listed in the second item we are done otherwise pick one such F and let

C denote the general member of the ruling If the cohomology class [C] is inde-

composable F is minimal Otherwise [C] = [C1] + [C2] where Ci are rational and

intersect KF trivially After finitely many steps we have a decomposition as a finite

sum of indecomposables [C] =sumn

i=0 ni[Ci] where every Ci is rational intersects KF

trivially We have two cases

bull There exists k such that Ck deforms in a birationally ruled necessarily min-

imal surface Fk

bull Every Ci has anti-ample normal bundle

In the first case denote by X rarr X1 the contraction of Fk along |Ck| and relpace

X by X1 Since Fk is minimal smoothness of the ambient stack is preserved unless

Fk is a small Kronecker This is however impossible if n ge 2 if Fk were a small

Kronecker then Ck deforms away from Fkminus1 so then Fkminus1 middot Ck = 0 on the other

hand Fkminus1 intersects the other ruling of Fk positively

In the second case flop C1 middot middot middot Cn in this order The transform of F is minimal and

not a small Kronecker so it can be contracted along its ruling preserving ambient

smoothness By induction on the Picard rank of X after finitely many steps this

algorithm ends Call X the output of this algorithm


Let S subX be an invariant divisor with a generically smooth ruling Then pS S rarr∆ is flat and after flopping finitely many rational curves in S cap plowastS(0) the resulting

pS is smooth Hence we can contract S along pS After finitely many steps the

Proposition is proven

For ease of notation in the sequel we call X the output of the previous proposition

Notice that there might still be divisors carrying a pencil of nodal rational curves

both inside and outside the singular fiber of p

We can start analyzing configurations of fiber components where the restriction of

F is not of general type We need some notation whose reason to exist will be clear

shortly

Definition VII3 Let F be an irreducible component of the singular fiber of p

along which KF is not big F is called special Riccati if

bull F is minimal

bull There exists a morphism s F rarr B onto a curve whose fibers are nodal

rational curve

bull F is transverse to the general fiber of s in particular KF is numerically

trivial along s

bull At least one fiber of s is invariant

bull There exists a section σ of s which is invariant and does not coincide with

the nodal locus of F

In particular the intersections of σ with invariant fibers of s isolated singularities

of F the node of F all coincide There may be fibers of s fully contained in sing(F )

in which case they are everywhere transverse to the saturated foliation

The definition looks nasty but special Riccati components have a rather simple

origin as the following proof shows

Lemma VII4 Let F be special Riccati Up to base change by an etale double cover

Bprime rarr B there is a natural isomorphism F norm simminusrarr BprimetimesP1 and the restriction of F

to F norm is birational to the projection onto P1


Proof We work over the normalization F norm Denote by Bprime the pre-image of the

node of F which is an etale double cover of B and let f denote an invariant fiber

of s with lowast = σ cap f Observe that F norm carries a Riccati foliation with 3 invariant

curves through lowast - namely σ f and a component of Bprime Hence FF has a log-canonical

singularity in lowast On F norm perform a weighted blow-up in each log-canonical singu-

larity followed by the contraction of the transforms of the corresponding fibers Let

s+ F+ rarr B denote the output of this operation If the proper transform of Bprime is

connected replace F+ by its base change under Bprime rarr B Hence we can assume Bprime

is the disjoint union of two sections of s+ and the transform of σ is a third section

which is disjoint from Bprime Therefore F+ = P1 timesBprime and the foliation restricts to the

projection onto P1

Next we describe those surfaces which cannot appear as components of the sin-

gular fiber X0


along which KF is not big F is called illusory if

bull there exists a morphism t F rarr B onto a curve

bull KF is trivial along t

bull For any other fiber component F prime the intersection F cap F prime is either empty or

contained in a fiber of t

We give examples of illusory fiber components

Lemma VII6 The following fiber components are necessarily illusory

bull t is elliptic the foliation is either parallel to t or turbolent with at least one

invariant fiber

bull t is a nodal ruling the foliation is either parallel to t or transverse to t with

at least one invariant fiber and not birational to a special Riccati VII3

Proof Let f = F cap F prime for some fiber component F prime It is F -invariant

Assume t and F are parallel and f is transverse to both Then necessarily f subesing(F ) and KF is positive along the fibers of t contradiction


Assume t and F are generically transverse and f is transverse to t Let e be an

invariant fiber of t then ecap f isin sing(F ) hence t must be a family of nodal rational

curves and ecapf coincides with the node of e - otherwise KF would be positive along

t But this implies F is special Riccati contradiction

We proceed to show that illusory fibers do not exist

Lemma VII7 Let F be an illusory fiber component then every fiber of t is irre-

ducible If F prime is any fiber component intersecting F non-trivially then F prime is illusory

Proof Assume t has a reducible fiber f Then we can write f = f prime cup f primeprime where f prime

is irreducible and f primeprime is a disjoint union of chains of rational curves In particular

every subscheme supported on f primeprime is rigid in X f prime is rational and smooth iff f is a

cycle of rational curves hence the foliation on F is parallel to t and the choice of f prime

is not unique otherwise f prime is nodal or elliptic Let X 99KX + denote the flop of the

irreducible components of f primeprime in any order The proper transform f prime+ (below in red)

is the irreducible fiber of the induced morphism t+ F+ rarr B Since f primeprime+ intersects

f prime+ and is not contained in F+ there exists a component F prime of the fiber of p+ such

that F+ cap f prime+ 6= empty Since F+ is illusory necessarily F+ cap F prime = f prime+ Moreover F+

and F prime are the only fiber components intersecting f prime+ again because F+ is illusory

We deduce that F+ middot f prime+ = F prime middot f prime+ = 0 and therefore f prime+ is nef inside F prime It follows

that he restriction of F+ to F prime is not of general type and by the classification of

foliated surfaces [McQ08] necessarily f prime+ deforms inside F prime and defines a fibration

u F prime rarr Bprime In particular f prime+ is an irreducible fiber of u This is however impossible

since u contracts all the components of f primeprime+ contained in F prime (below in blue) and by

definition of F prime there exists at least one such component This proves that every

fiber of t is irreducible The following depicts the situation

F prime F+

f prime+

f primeprime+

t+u


Concerning the second statement we only need to check the third condition for F prime

to be illussory namely that every other fiber component F primeprime intersecting F prime does

so along the fibers of u If F primeprime intersects F prime transversely to u then F primeprime intersects F

non-trivially and transversely to t as well absurd

Corollary VII8 Illusory components do not exist

Proof By the above lemma the existence of one illusory component implies that

every irreducible component of the singular fiber of p is illusory contradicting that

KF is big

We can now focus on fiber components that do appear

Corollary VII9 Let F be an irreducible component of the singular fiber of p

containing an invariant curve C which moves and such that KF middot C = 0 Then

following the classification [McQ08] we have two cases

(1) The minimal model of F is a special Riccati component

(2) F is a small Kronecker

Proof Certainly the ones listed are the only cases where C is KF -nil We have to

exclude the possibility that C is invariant inside sing(F ) By Proposition IVII3

such C would be rational with at most one node Since we already contracted at

the beginning of the present subsection all ruled surfaces necessarily C is nodal

However by opcit C cannot intersect any other fiber component of X0 which

implies the absurdity that X0 = F

We proceed to show that special Riccati components are rather isolated in the

fiber

Lemma VII10 Let F be a special Riccati component If F prime is a fiber component

intersecting F non-trivially then it does so transversely to s Morover there is

exactly one such F prime

Proof After finitely many flops we can assume that every fiber of s is irreducible

By the non-existence of illusory components there exists a fiber component F lowast with


σprime = F cap F prime transverse to s If F primeprime is a fiber component with f = F cap F primeprime a fiber

of s then we claim f sube sing(F ) If not then since KF middot f = 0 and σprime is invariant

necessarily σprimecap f is the node of f But then the existence of F primeprime contradicts that p is

snc Hence f sube sing(F ) so then around the node of f there are 3 fiber components -

two branches coming from F one from F primeprime - and sing(F ) has 2 branches - namely f

This is impossible by our case analysis ni (1)c ni (2)c The very same reasoning

establishes the uniqueness of F prime

Lemma VII11 Let F be as in the previous Lemma If C is a connected curve

with KF middot C = 0 then it is disjoint from F

Proof After finitely many flops s is irreducible and C intersects F transversely away

from its node Moreover lowast = C cap F isin sing(F ) and we deduce that the fiber of s

through lowast is fully contained in the singular locus of F This contradicts Proposition

IVII3 since such fiber is nodal

We conclude by observing the obvious

Lemma VII12 Let F be a small Kronecker Then there exists a contraction

c X rarr X prime which is an isomorphism away from F and contracts F along one

of its rulings Necessarily X prime is not projective since every global Cartier divisor

intersects c(F ) trivially On the other hand in a formal neighborhood of c(F ) there

exist Cartier divisors intersecting it non-trivialliy

As such the contribution coming from non-rigid curves is rather poor there are

curves moving in the general fiber there are special Riccati components disjoint from

any other invariant curve intersecting KF trivially there are small Kroneckers who

can be contracted to curves We shall see in the next section that the complications

arising from the contraction of such non-rigid curves is purely psychological and

totally harmless


VI Contractions and proof of the Main Theorem

In this section we contract all invariant curves along which KF is zero thereby

establishing the existence of canonical models I First we contract non-rigid curves

as described in the previous subsection

bull Small Kroneckers along either ruling Albeit this destroys projectivity of

the ambient threefold there still exist Cartier divisors defined in a formal

neighborhood of the base curve that intersect it non-trivially

bull Families of irreducible nodal rational curves which are flat over ∆ This

creates elliptic singularities on the general fiber of p which are disjoint from

other invariant curves intersecting KF trivially

bull Special Riccati again families of irreducible nodal rational curves fully con-

tained in X0 This creates elliptic singularities disjoint from other invariant

curves intersecting KF trivially

We can summarize our situation by way of

Set-up VI1 X is a 3-dimensional DM stack with at worst elliptic singularities

F a foliation by curves with canonical singularities which is not Q-Cartier in a

neighborhood of Xsing

There exists a proper morphism p X rarr ∆ smooth off 0 isin ∆ F restricts to a

foliation on every fiber of p it has canonical singularities on the very general fiber

of p and log-canonical singularities on every fiber of p KF is big and nef on every

irreducible component of every fiber of p

Let C be the invariant sub-scheme along which KF is not big then C is a curve

fully contained in the smooth locus X0capXsmooth every sub-curve of C has anti-ample

normal bundle the 1-simplex of its dual graph is a tree and detailed configurations

of such are referenced in set-up VI13

We proceed to contract such C We assume that along C the semi-simple field has

only rational eigenvalues and moreover no component of C is inside the foliation

singularity The other two cases - irrational eigenvalues and presence of singular

component - are strictly easier and can be obtained with the very same method


The proof takes place in the formal completion around C = C free cup C axes The

former is a disjoint union of trees of non-double curves For each non-double C let

F (C) denote the unique component of X0 containing it The latter is a tree of double

curves each with two triple points all contained in the same irreducible FC axes Each

tree of non-double curves touches C axes in one point through a LC-center In order

to construct the contraction we will use Artinrsquos celebrated criterion in the following

form

Fact VI1 ([Ar70]) Let X be an irreducible algebraic space i Y rarr X a closed

subspace of pure codimension 1 and f Y rarr Y0 a proper morphism with flowastOY = OY0

Let X denote the formal completion of X along Y and assume there exists a Cartier

divisor D in X which is supported on Y and is f -anti-ample Then there exists a

proper morphism of algebraic spaces F X rarr X0 which is an isomorphism away

from Y and such that ilowastF = f

Now we use the tree structure of C to construct a suitable divisor to which apply

Artinrsquos theorem Namely consider the following inductive defintion

(35) C0 = C axes F0 = FC axes

For k ge 0 assume Ck Fk are defined

bull Let Ck+1 denote the union of irreducible components of C that intersect Fk

non-trivially but are not contained in cupilekFibull Let Fk+1 denote the reduced sum (

sumCsubeCk+1

F (C))red

This process stops with a collection C1 middot middot middotCN of curves and F1 middot middot middot FN of reduced

divisors

Lemma VI1 We have the following identities

(36)

Fk middot C lt 0 for every C sube Ck

(F0 + F1) middot C le 0 for every C sube C0

(Fkminus1 + Fk + Fk+1) middot C = 0 for every C sube Ck 1 lt k lt N

(FNminus1 + FN) middot C lt 0 for every C sube CN


Proof The first is just the fact that every component of C is rigid The remaining

three follow from X0 middot C = 0 but the last one requires further justification Indeed

if C sube CN N gt 0 then C is one end of a tree of non-double curves In particular

there exists a foliation singularity lowast isin C disjoint from any other component of

C and LClowast is transverse to C This implies LClowast middotC gt 0 which together with

(FNminus1 + FN + LClowast) middot C = 0 concludes the proof

Let ε1 middot middot middot εN isin Qgt0 and define

(37) F Nminusk = (1 + ε1 + middot middot middot+ εk)FNminusk

bull If C sube CN we have

(38) (F Nminus1 + F

N) middot C = ε1FNminus1 middot C + (FNminus1 + FN) middot C

bull If C sube CNminusk 1 lt k lt N we have

(39)

(F Nminuskminus1 + F

Nminusk + F Nminusk+1) middot C =

(1 + middot middot middot+ εk)(FNminuskminus1 + FNminusk + FNminusk+1) middot C minus εkFNminusk+1 middot C + εk+1FNminuskminus1 middot C =

minus εkFNminusk+1 middot C + εk+1FNminuskminus1 middot C

bull If C sube C0 we have

(40) (F 0 + F

1) middot C = εNF0 middot C + (Nminus1sumj=0

εj)(F0 + F1) middot C

It is clear that provided

(41) 1 gtgt ε1 gtgt middot middot middot gtgt εN

the divisor F =sumF k is negative along every component of C

This proves the existence of the contraction and concludes the proof of the Main

Theorem I


VII Moduli space of foliated surfaces of general type

In this section we study the modular variation of foliated surfaces of general type

The upshot is that it is possible to define a functor which is representable separated

and every irreducible component is proper The main issue here is that the supporting

surface of a foliated canonical model may well fail to be projective indeed KF is not

Q-Gorenstein in general Hence in order to proceed via standard methods [Ko17]

we have to parametrize our canonical models together with a suitable projective

resolution

Definition VII13 Let (SF ) be a foliated proper reduced purely 2-dimensional

DM stack Assume S has only normal crossing singularities in codimension 1 that

F has log-canonical foliation singularities and that KF is big and nef on at least

one of the irreducible components of S If whenever KF middot C = 0 for some curve C

then C moves in an irreducible component of S and no member of such deformation

is F -invariant then (SF ) is called foliated canonical model of general type

Clearly foliated canonical models are exactly the stable degenerations appearing

as the central fiber of our Main Theorem I Their lack of projectivity has two sources

namely the contraction of cycles of smooth KF -nil curves and of trees of cuspidal

ones Singularities arising from cycles behave well in families namely they always

deform into the general fiber Those arising from cuspidal curves on the other hand

tend to be rather isolated inside the degenerate fiber

Observe that the deformation theory of a foliated surface is rather simple suppose

given an infinitesimal C-algebras extension 0 rarr J rarr Ae rarr A rarr 0 and (SF )

a foliated surface flat over Spec(A) then the obstruction to lifting to Ae can be

understood in three simple steps

(1) Deformation of the underlying surface S once a deformation Se rarr Spec(Ae)

is given

(2) Deformation of the line bundle KF by the exponential sequence

(42) 0rarr J otimes OS rarr OlowastSerarr OlowastS rarr 0

obstructions lie in H2(SOS)otimes J once a deformation of KF is given


(3) Deformation of the defining section OS rarr KF otimes TS obstructions lie in

H1(SKF otimes TS)otimes J

Now we have

Definition VII14 Let (SF ) be a foliated canonical model of general type A

proper birational morphism q (SF )proj rarr (SF ) is called its projective resolution

if

bull Every irreducible component of Sproj is smooth

bull The exceptional locus of q is a disjoint union of cycles of KFproj-nil smooth

curves and trees of KFproj-nil curves each of which has a non-trivial cuspidal

singularity

bull In a neighborhood of every q-exceptional cycle not only Sproj but also its

moduli space is smooth

We need a few remarks about projective resolutions

(1) Projective resolutions are always unique Indeed the only possible ambiguity

could lie in some extra smooth irreducible components in the exceptional

locus This is forbidden by definition in the trees of cuspidal curves it is

also forbidden by way of the third item in cycles of smooth ones ie on

algebraic surfaces the minimal resolution is unique

(2) Denote by E the collection of q-exceptional cycles Then there exist ε δ isinQgt0 such that KFproj

minus εE + δKSprojis ample by Nakai-Moishezon

(3) Deformations of a projective resolution of a foliated canonical model are

canonically identified with deformations of the underlying birational mor-

phism q Sproj rarr S followed by a deformation of the foliation F

We are now ready to construct a moduli functor for foliated surfaces more precisely

for projective resolutions of foliated canonical models of general type For m dN isinN and ε δ isin Qgt0 let

(43) MmdNεδ (C-schemes)rarr (sets)


be defined by MmdNεδ(S) = Diagrams

(44) (X F )projqminusrarr (X F )

pminusrarr S

Where

bull Both p and p q are flat and proper

bull For every closed point s isin S the induced morphism qs is the projective

resolution of (X F )s which is a foliated canonical model of general type

bull Denote by Es the collection of exceptional cycles of qs By our Main Theorem

I there is a fibrewise divisor E subXproj flat over S restricting to Es for every

s isin S

bull Let j W rarr Xproj denote the maximal closed subscheme away from which

KFprojand KXprojS are locally free Then

(45) L [m] = (jlowast(KFprojotimes OXproj

(minusεE )otimesKδXprojS

)otimesm)oror

is flat over S and commutes with base-change

bull L [m] is very ample of degree d on every fiber and defines an embedding into

PNS

We are now ready for

Theorem VII15 The functor MmdNεδ is represented by a separated algebraic

space each of whose irreducible components is proper

Proof We show that our functor satisfies Artinrsquos conditions [Ar74 Corollary 54] By

definition condition 1) and 4) hold while our previous remark on foliated obstruction

theory shows that 3) is true

We proceed to show 2) namely effectivity of formal deformations for MmdNεδ

What follows owes much to Artinrsquos [A74rsquo] Let T be a complete C-algebra Tn its

n-truncation so that Tsimminusrarr limlarrminusTn and finally qn isinMmdNεδ(Spec(Tn)) a compatible

sequence For each n there is a maximal open set Un sube Xprojn where qn is an

isomorphism denote by sn Cn rarr Zn the restriction of qn to the complement of Un

Clearly sn is projective over Spec(Tn)


Denote by X C etc the formal algebraic spaces obtained as limits of Xn Cn

etc Then Xproj sub PNˆSpec(T )

hence by Grothendieckrsquos existence Theorem we find

bull a flat projective family (X F )proj rarr Spec(T ) resp a closed subscheme

C subXproj whose completion is Xproj resp C

bull a scheme Z whose completion is Z and a morphism s C rarr Z everything

being projective over Spec(T )

Further we have a formal contraction limminusrarr qn C rarr Z hence the main theorem

of [Ar70] gives algebraic contraction (X F )proj rarr (X F ) This proves the repre-

sentability of MmdNεδ The properness and separability of each irreducible com-

ponent can be checked using valutative criteria indeed properness is the content of

our Main Theorem I separatedness follows from the uniqueness of foliated canonical

models of general type and their projective resolution

References

[Ar70] M Artin Algebraization of Formal Moduli II Existence of Modifications Ann

of Math Vol 91 No 1 (Jan 1970) pp 88-135 httpwwwjstororgstable

1970602

[Ar74] M Artin Versal Deformations and Algebraic Stacks Inventiones Math 27 165- 189

(1974) httpmathuchicagoedu~drinfeldArtin_on_stackspdf

[A74rsquo] M Artin Algebraic construction of Brieskornrsquos resolutions J Alg Vol 29 Issue

2 May 1974 pp 330-348 httpwwwsciencedirectcomsciencearticlepii

0021869374901021

[BB72] P Baum R Bott Singularities of holomorphic foliations J Diff Geom 7 (1972)

279-342 httpsprojecteuclidorgdownloadpdf_1euclidjdg1214431158

[Bo77] F Bogomolov Families of curves on a surface of general type Doklady Akademii Nauk

USSR (in Russian) 236 (5) 1041-1044

[Bo78] F Bogomolov Holomorphic tensors and vector bundles on projective varieties Math

USSR Izv 13 (1978) 499-555

[BM16] F Bogomolov M McQuillan Rational Curves on Foliated Varieties In Cascini

McKernan Pereira (eds) Foliation Theory in Algebraic Geometry Simons Symposia

Springer Cham pp 21-51 (2016) httpslinkspringercomchapter101007

978-3-319-24460-0_2


[Bu] F Buonerba Functorial resolution of tame quotient singularities in positive charac-

teristic httpsarxivorgabs151100550

[Ke99] S Keel Basepoint Freeness for Nef and Big Line Bundles in Positive Characteris-

tic Annals of Math 149 No 1 (Jan 1999) pp 253-286 httpwwwjstororg

stable121025 httpsarxivorgabsmath9901149

[KM97] S Keel S Mori Quotients by groupoids Annals of Math 145 (1997) pp

193-213 httpwwwjstororgstable2951828seq=1page_scan_tab_contents

httparxivorgabsalg-geom9508012

[Ko17] J Kollar Families of varieties of general type httpswebmathprincetonedu

~kollarbookmodbook20170720pdf

[LN02] A Lins Neto Some examples for the Poincare and Painleve problems Annales sci-

entifiques de lrsquoENS 352 (2002) 231-266 httparchivenumdamorgarticle

ASENS_2002_4_35_2_231_0pdf

[McQ12] M McQuillan Foliated Mori Theory amp Hyperbolicity of Algebraic Surfaces http

wwwmatuniroma2it~mcquillahomehtml

[McQ98] M McQuillan Diophantine approximation and foliations Inst Hautes Etudes Sci

Publ Math 87 (1998) 121-174 httpwwwmatuniroma2it~mcquillafiles

ihespdf

[McQ05rsquo] M McQuillan Bloch hyperbolicity preprint httpwwwmatuniroma2it

~mcquillafilesfileshtml

[McQ05] M McQuillan Semi-stable reduction of foliations IHES pre-print IHESM0502

(2005) httpwwwmatuniroma2it~mcquillafilesfileshtml

[McQ05rdquo] M McQuillan Uniform Uniformisation IHES pre-print (2005) IHES M0503 http

wwwmatuniroma2it~mcquillafilesfileshtml

[McQ08] M McQuillan Canonical models of foliations Pure Appl Math Q 4

(2008) no 3 Special Issue In honor of Fedor Bogomolov Part 2 877-1012

httpintlpresscomsitepubfiles_fulltextjournalspamq20080004

0003PAMQ-2008-0004-0003-a009pdf

[MP13] M McQuillan D Panazzolo Almost etale resolution of foliations J Diff Geom 95

(2013) pp 279-319 httpwwwmatuniroma2it~mcquillafilesmp1bispdf

[Pe02] JV Pereira On the Poincare Problem for Foliations of General Type Mathematische

Annalen 323 no 2 pp217-226 2002 httpsarxivorgabsmath0104185

[PS16] JV Pereira R Svaldi Effective algebraic integration in bounded genus preprint

httpsarxivorgabs161206932


[Re83] M Reid Minimal models of canonical threefolds in rdquoAlgebraic Varieties and Ana-

lytic Varietiesrdquo Adv Stud Pure Math vol 1 S Iitaka ed Kinokuniya and North-

Holland 1983 pp 131-180 httpwwwmathsedacukcheltsovquotientpdf

reid2pdf

[Se67] A Seidenberg Reduction of singularities of the differential equation Ady = Bdx Amer

J Math 89 (1967) 248-269 httpwwwjstororgstable2373435

[SB83] N Shepherd-Barron Degenerations with numerically effective canonical divisor The

Birational Geometry of Degenerations (R Friedman and D Morrison eds) Progr

Math no 29 Birkhuser 1983 pp 33-84

[Vis89] A Vistoli Intersection theory on algebraic stacks and on their moduli spaces

Invent Math (1989) pp 613-670 httplinkspringercomarticle101007

2FBF01388892page-1

Courant Institute of Mathematical Sciences New York University 251 Mercer

Street New York NY 10012 USA

E-mail address buonerbacimsnyuedu

I Introduction

II Preliminaries

III Operations on Deligne-Mumford stacks

IIII Width of embedded parabolic champs

IIIII Gorenstein foliation singularities

IIIV Foliated adjunction

IIV Canonical models of foliated surfaces with canonical singularities

IIVI Canonical models of foliated surfaces with log-canonical singularities

IIVII Set-up


IIII `39`42`613A``45`47`603Asing(F) not isolated

IIIII `39`42`613A``45`47`603Asing(F) isolated

IIIIII Local consequences


IVI KF-nil curves

IVII Invariant curves inside `39`42`613A``45`47`603Asing(F)

V Configurations

VI Configurations of rigid invariant curves intersecting KF trivially




References

2 FEDERICO BUONERBA

VII Configurations of non-rigid invariant curves intersecting KF trivially 49

VI Contractions and proof of the Main Theorem 56

VII Moduli space of foliated surfaces of general type 59

References 62

I Introduction

Foliations are ubiquitous throughout mathematics In algebraic geometry they are

widely used as tools to investigate hyperbolicity and boundedness phenomena The

first example of this is Bogomolovrsquos ground-breaking theorem [Bo78] that on a mini-

mal surface of general type with c21 gt c2 the degree of curves can be bounded linearly

in terms of their geometric genera The proof is extremely simple the abundance

of symmetric differentials defines a birational map φ on the projectivised cotangent

bundle to our surface whose base locus is an algebraic surface carrying a canoni-

cally defined algebraic foliation Curves on our original surface lift by way of their

derivative to the projectivised cotangent bundle If such lift does not lie in the base

locus of φ the required bound on the degree follows if the lift does lie in the base

locus then the curve is invariant by the canonical foliation The conclusion follows

by two standard facts in the theory of algebraic foliations Seidenbergrsquos reduction of

foliation singularities to simple ones [Se67] and Jouanoloursquos remark that a foliated

surface with infinitely many invariant curves is fibered over a curve

This deep relation between algebraic foliations and boundedness of curves was in-

vestigated by McQuillan who in [McQ98] extended Bogomolovrsquos technique to non-

compact curves thus proving the Green-Griffiths conjecture for minimal surfaces of

general type with c21 gt c2 namely that entire curves on such surfaces cannot be

Zariski-dense The key new ingredient to the proof is the tautological inequality a

principle that can be vaguely stated as follows a foliation has simple singularities

if and only if the intersection multiplicity between any invariant disk not factor-

ing through the algebraic leaves and the so-called foliated canonical bundle can be

bounded up to a small error by the logarithm of the length of the diskrsquos boundary



























dimensions ge 4





4 FEDERICO BUONERBA






birational maps


Where









scribed as follows













invariant













π M g1 rarrM g



















6 FEDERICO BUONERBA































































8 FEDERICO BUONERBA
















II Preliminaries





































D subX ( nradic

D) such that











sheaves








fn Yn rarrX








Our main concern is





















long as NCkminus1












assume NCp

















F on U and we have






















partpartxi









ring O








L (minusRamf )




Fact IIIV2



(1minus |Gp|minus1)






in [McQ05]



Z of the foliation



curves f


locus





Z2Z

































elliptic











(7) X rarrXe rarrXc






Q-Gorenstein















part =sumnixi

partpartxi



and write f =sum




let FN = (xa11 xakk















f minussum























general









results




Proof We have








foliation is smooth






rational curves




curves tangent to F








either










Corollary IIVI7






nal curves






not canonical




that




the log-chain

















ored in black




















































is arbitrary


















for the sequel

















(14) part(s) =part

partx+ s

part

party














bull KFm is nef or





























+ g partpartz







+g|(y=0)partpartz






singularity




partpartz

Z has an




to Z


singularity


diction










+ λy partparty


+ f partparty

+ g partpartz

for some func-

tions k f g



























components






+ f partparty

+ g partpartz

for some functions









contradiction




+ λy partparty


+ f partparty

+ g partpartz

for some functions

k f g







+g|(y=0)partpartz





0








them




singularity


0




are in the axes


+ λy partparty

+ microz partpartz




+ microz partpartz





axis x = z = 0



x = y = 0 x = z = 0 y = z = 0


next item




+ microz partpartz









x = y = 0 x = z = 0 y = z = 0




















ni (2)a i (3)b i (3)c








versely







xz



y

LC-center



y

LC-center














surface x = 0
































1 C 0 This proves





statement as well































along C






)

partk+1

y partk

yH0(COCk+1












partx+ λy

part

party+ microz

part


partξ+ (λ+ εn)η

part


part

partζ




partx+ λy

part

party+ microz

part


partξ+ (λ+ εn)η

part


part

partζ





eigenvalues


are called ends









smooth locus of p




∆













(25) part0 = xpart

partx+ microyazb

part



part

partζ(mod(I2C))



(26) x1 = x y1 = y z1 = zyminus1


(27)

part01 = x1part


part


part



part

partζ1(mod(I2C1

))




































is projective



























2 = (x0 = w0 = 0)













unless a = 0


































z







the beast






















(32) 2g(Z0)minus 2 + sZG(Z0) = 0















V Configurations




















∆



equation 23





First we need







(33) n+m = d+ e






































of p fi











curves



(34) x = η y = ξ z = ζ





attached at one end





























of FC






is empty
























trees








following obvious













chain


normal bundle













we conclude

Our new set-up is













not mentioned is


small Kronecker if















Kronecker











imal surface Fk




















shortly



bull F is minimal


rational curve


trivial along s























projection onto P1


gular fiber X0










invariant fiber






























F prime F+

f prime+

f primeprime+

t+u









KF is big














fiber































totally harmless






































form













sumCsubeCk+1

F (C))red


divisors


(36)















(38) (F Nminus1 + F



(39)






(40) (F 0 + F







Theorem I









resolution


















is given







Now we have



if




singularity




















pminusrarr S

Where






s isin S





)otimesm)oror



PNS



























References



1970602





0021869374901021




USSR (in Russian) 236 (5) 1041-1044


USSR Izv 13 (1978) 499-555




978-3-319-24460-0_2














ASENS_2002_4_35_2_231_0pdf





ihespdf










0003PAMQ-2008-0004-0003-a009pdf











reid2pdf








2FBF01388892page-1




I Introduction

II Preliminaries







IIVII Set-up






IVI KF-nil curves


V Configurations





References



























dimensions ge 4





4 FEDERICO BUONERBA






birational maps


Where









scribed as follows













invariant













π M g1 rarrM g



















6 FEDERICO BUONERBA































































8 FEDERICO BUONERBA
















II Preliminaries





































D subX ( nradic

D) such that











sheaves








fn Yn rarrX








Our main concern is





















long as NCkminus1












assume NCp

















F on U and we have






















partpartxi









ring O








L (minusRamf )




Fact IIIV2



(1minus |Gp|minus1)






in [McQ05]



Z of the foliation



curves f


locus





Z2Z

































elliptic











(7) X rarrXe rarrXc






Q-Gorenstein















part =sumnixi

partpartxi



and write f =sum




let FN = (xa11 xakk















f minussum























general









results




Proof We have








foliation is smooth






rational curves




curves tangent to F








either










Corollary IIVI7






nal curves






not canonical




that




the log-chain

















ored in black




















































is arbitrary


















for the sequel

















(14) part(s) =part

partx+ s

part

party














bull KFm is nef or





























+ g partpartz







+g|(y=0)partpartz






singularity




partpartz

Z has an




to Z


singularity


diction










+ λy partparty


+ f partparty

+ g partpartz

for some func-

tions k f g



























components






+ f partparty

+ g partpartz

for some functions









contradiction




+ λy partparty


+ f partparty

+ g partpartz

for some functions

k f g







+g|(y=0)partpartz





0








them




singularity


0




are in the axes


+ λy partparty

+ microz partpartz




+ microz partpartz





axis x = z = 0



x = y = 0 x = z = 0 y = z = 0


next item




+ microz partpartz









x = y = 0 x = z = 0 y = z = 0




















ni (2)a i (3)b i (3)c








versely







xz



y

LC-center



y

LC-center














surface x = 0
































1 C 0 This proves





statement as well































along C






)

partk+1

y partk

yH0(COCk+1












partx+ λy

part

party+ microz

part


partξ+ (λ+ εn)η

part


part

partζ




partx+ λy

part

party+ microz

part


partξ+ (λ+ εn)η

part


part

partζ





eigenvalues


are called ends









smooth locus of p




∆













(25) part0 = xpart

partx+ microyazb

part



part

partζ(mod(I2C))



(26) x1 = x y1 = y z1 = zyminus1


(27)

part01 = x1part


part


part



part

partζ1(mod(I2C1

))




































is projective



























2 = (x0 = w0 = 0)













unless a = 0


































z







the beast






















(32) 2g(Z0)minus 2 + sZG(Z0) = 0















V Configurations




















∆



equation 23





First we need







(33) n+m = d+ e






































of p fi











curves



(34) x = η y = ξ z = ζ





attached at one end





























of FC






is empty
























trees








following obvious













chain


normal bundle













we conclude

Our new set-up is













not mentioned is


small Kronecker if















Kronecker











imal surface Fk




















shortly



bull F is minimal


rational curve


trivial along s























projection onto P1


gular fiber X0










invariant fiber






























F prime F+

f prime+

f primeprime+

t+u









KF is big














fiber































totally harmless






































form













sumCsubeCk+1

F (C))red


divisors


(36)















(38) (F Nminus1 + F



(39)






(40) (F 0 + F







Theorem I









resolution


















is given







Now we have



if




singularity




















pminusrarr S

Where






s isin S





)otimesm)oror



PNS



























References



1970602





0021869374901021




USSR (in Russian) 236 (5) 1041-1044


USSR Izv 13 (1978) 499-555




978-3-319-24460-0_2














ASENS_2002_4_35_2_231_0pdf





ihespdf










0003PAMQ-2008-0004-0003-a009pdf











reid2pdf








2FBF01388892page-1




I Introduction

II Preliminaries







IIVII Set-up






IVI KF-nil curves


V Configurations





References

4 FEDERICO BUONERBA






birational maps


Where









scribed as follows













invariant













π M g1 rarrM g



















6 FEDERICO BUONERBA































































8 FEDERICO BUONERBA
















II Preliminaries





































D subX ( nradic

D) such that











sheaves








fn Yn rarrX








Our main concern is





















long as NCkminus1












assume NCp

















F on U and we have






















partpartxi









ring O








L (minusRamf )




Fact IIIV2



(1minus |Gp|minus1)






in [McQ05]



Z of the foliation



curves f


locus





Z2Z

































elliptic











(7) X rarrXe rarrXc






Q-Gorenstein















part =sumnixi

partpartxi



and write f =sum




let FN = (xa11 xakk















f minussum























general









results




Proof We have








foliation is smooth






rational curves




curves tangent to F








either










Corollary IIVI7






nal curves






not canonical




that




the log-chain

















ored in black




















































is arbitrary


















for the sequel

















(14) part(s) =part

partx+ s

part

party














bull KFm is nef or





























+ g partpartz







+g|(y=0)partpartz






singularity




partpartz

Z has an




to Z


singularity


diction










+ λy partparty


+ f partparty

+ g partpartz

for some func-

tions k f g



























components






+ f partparty

+ g partpartz

for some functions









contradiction




+ λy partparty


+ f partparty

+ g partpartz

for some functions

k f g







+g|(y=0)partpartz





0








them




singularity


0




are in the axes


+ λy partparty

+ microz partpartz




+ microz partpartz





axis x = z = 0



x = y = 0 x = z = 0 y = z = 0


next item




+ microz partpartz









x = y = 0 x = z = 0 y = z = 0




















ni (2)a i (3)b i (3)c








versely







xz



y

LC-center



y

LC-center














surface x = 0
































1 C 0 This proves





statement as well































along C






)

partk+1

y partk

yH0(COCk+1












partx+ λy

part

party+ microz

part


partξ+ (λ+ εn)η

part


part

partζ




partx+ λy

part

party+ microz

part


partξ+ (λ+ εn)η

part


part

partζ





eigenvalues


are called ends









smooth locus of p




∆













(25) part0 = xpart

partx+ microyazb

part



part

partζ(mod(I2C))



(26) x1 = x y1 = y z1 = zyminus1


(27)

part01 = x1part


part


part



part

partζ1(mod(I2C1

))




































is projective



























2 = (x0 = w0 = 0)













unless a = 0


































z







the beast






















(32) 2g(Z0)minus 2 + sZG(Z0) = 0















V Configurations




















∆



equation 23





First we need







(33) n+m = d+ e






































of p fi











curves



(34) x = η y = ξ z = ζ





attached at one end





























of FC






is empty
























trees








following obvious













chain


normal bundle













we conclude

Our new set-up is













not mentioned is


small Kronecker if















Kronecker











imal surface Fk




















shortly



bull F is minimal


rational curve


trivial along s























projection onto P1


gular fiber X0










invariant fiber






























F prime F+

f prime+

f primeprime+

t+u









KF is big














fiber































totally harmless






































form













sumCsubeCk+1

F (C))red


divisors


(36)















(38) (F Nminus1 + F



(39)






(40) (F 0 + F







Theorem I









resolution


















is given







Now we have



if




singularity




















pminusrarr S

Where






s isin S





)otimesm)oror



PNS



























References



1970602





0021869374901021




USSR (in Russian) 236 (5) 1041-1044


USSR Izv 13 (1978) 499-555




978-3-319-24460-0_2














ASENS_2002_4_35_2_231_0pdf





ihespdf










0003PAMQ-2008-0004-0003-a009pdf











reid2pdf








2FBF01388892page-1




I Introduction

II Preliminaries







IIVII Set-up






IVI KF-nil curves


V Configurations





References













π M g1 rarrM g



















6 FEDERICO BUONERBA































































8 FEDERICO BUONERBA
















II Preliminaries





































D subX ( nradic

D) such that











sheaves








fn Yn rarrX








Our main concern is





















long as NCkminus1












assume NCp

















F on U and we have






















partpartxi









ring O








L (minusRamf )




Fact IIIV2



(1minus |Gp|minus1)






in [McQ05]



Z of the foliation



curves f


locus





Z2Z

































elliptic











(7) X rarrXe rarrXc






Q-Gorenstein















part =sumnixi

partpartxi



and write f =sum




let FN = (xa11 xakk















f minussum























general









results




Proof We have








foliation is smooth






rational curves




curves tangent to F








either










Corollary IIVI7






nal curves






not canonical




that




the log-chain

















ored in black




















































is arbitrary


















for the sequel

















(14) part(s) =part

partx+ s

part

party














bull KFm is nef or





























+ g partpartz







+g|(y=0)partpartz






singularity




partpartz

Z has an




to Z


singularity


diction










+ λy partparty


+ f partparty

+ g partpartz

for some func-

tions k f g



























components






+ f partparty

+ g partpartz

for some functions









contradiction




+ λy partparty


+ f partparty

+ g partpartz

for some functions

k f g







+g|(y=0)partpartz





0








them




singularity


0




are in the axes


+ λy partparty

+ microz partpartz




+ microz partpartz





axis x = z = 0



x = y = 0 x = z = 0 y = z = 0


next item




+ microz partpartz









x = y = 0 x = z = 0 y = z = 0




















ni (2)a i (3)b i (3)c








versely







xz



y

LC-center



y

LC-center














surface x = 0
































1 C 0 This proves





statement as well































along C






)

partk+1

y partk

yH0(COCk+1












partx+ λy

part

party+ microz

part


partξ+ (λ+ εn)η

part


part

partζ




partx+ λy

part

party+ microz

part


partξ+ (λ+ εn)η

part


part

partζ





eigenvalues


are called ends









smooth locus of p




∆













(25) part0 = xpart

partx+ microyazb

part



part

partζ(mod(I2C))



(26) x1 = x y1 = y z1 = zyminus1


(27)

part01 = x1part


part


part



part

partζ1(mod(I2C1

))




































is projective



























2 = (x0 = w0 = 0)













unless a = 0


































z







the beast






















(32) 2g(Z0)minus 2 + sZG(Z0) = 0















V Configurations




















∆



equation 23





First we need







(33) n+m = d+ e






































of p fi











curves



(34) x = η y = ξ z = ζ





attached at one end





























of FC






is empty
























trees








following obvious













chain


normal bundle













we conclude

Our new set-up is













not mentioned is


small Kronecker if















Kronecker











imal surface Fk




















shortly



bull F is minimal


rational curve


trivial along s























projection onto P1


gular fiber X0










invariant fiber






























F prime F+

f prime+

f primeprime+

t+u









KF is big














fiber































totally harmless






































form













sumCsubeCk+1

F (C))red


divisors


(36)















(38) (F Nminus1 + F



(39)






(40) (F 0 + F







Theorem I









resolution


















is given







Now we have



if




singularity




















pminusrarr S

Where






s isin S





)otimesm)oror



PNS



























References



1970602





0021869374901021




USSR (in Russian) 236 (5) 1041-1044


USSR Izv 13 (1978) 499-555




978-3-319-24460-0_2














ASENS_2002_4_35_2_231_0pdf





ihespdf










0003PAMQ-2008-0004-0003-a009pdf











reid2pdf








2FBF01388892page-1




I Introduction

II Preliminaries







IIVII Set-up






IVI KF-nil curves


V Configurations





References

6 FEDERICO BUONERBA































































8 FEDERICO BUONERBA
















II Preliminaries





































D subX ( nradic

D) such that











sheaves








fn Yn rarrX








Our main concern is





















long as NCkminus1












assume NCp

















F on U and we have






















partpartxi









ring O








L (minusRamf )




Fact IIIV2



(1minus |Gp|minus1)






in [McQ05]



Z of the foliation



curves f


locus





Z2Z

































elliptic











(7) X rarrXe rarrXc






Q-Gorenstein















part =sumnixi

partpartxi



and write f =sum




let FN = (xa11 xakk















f minussum























general









results




Proof We have








foliation is smooth






rational curves




curves tangent to F








either










Corollary IIVI7






nal curves






not canonical




that




the log-chain

















ored in black




















































is arbitrary


















for the sequel

















(14) part(s) =part

partx+ s

part

party














bull KFm is nef or





























+ g partpartz







+g|(y=0)partpartz






singularity




partpartz

Z has an




to Z


singularity


diction










+ λy partparty


+ f partparty

+ g partpartz

for some func-

tions k f g



























components






+ f partparty

+ g partpartz

for some functions









contradiction




+ λy partparty


+ f partparty

+ g partpartz

for some functions

k f g







+g|(y=0)partpartz





0








them




singularity


0




are in the axes


+ λy partparty

+ microz partpartz




+ microz partpartz





axis x = z = 0



x = y = 0 x = z = 0 y = z = 0


next item




+ microz partpartz









x = y = 0 x = z = 0 y = z = 0




















ni (2)a i (3)b i (3)c








versely







xz



y

LC-center



y

LC-center














surface x = 0
































1 C 0 This proves





statement as well































along C






)

partk+1

y partk

yH0(COCk+1












partx+ λy

part

party+ microz

part


partξ+ (λ+ εn)η

part


part

partζ




partx+ λy

part

party+ microz

part


partξ+ (λ+ εn)η

part


part

partζ





eigenvalues


are called ends









smooth locus of p




∆













(25) part0 = xpart

partx+ microyazb

part



part

partζ(mod(I2C))



(26) x1 = x y1 = y z1 = zyminus1


(27)

part01 = x1part


part


part



part

partζ1(mod(I2C1

))




































is projective



























2 = (x0 = w0 = 0)













unless a = 0


































z







the beast






















(32) 2g(Z0)minus 2 + sZG(Z0) = 0















V Configurations




















∆



equation 23





First we need







(33) n+m = d+ e






































of p fi











curves



(34) x = η y = ξ z = ζ





attached at one end





























of FC






is empty
























trees








following obvious













chain


normal bundle













we conclude

Our new set-up is













not mentioned is


small Kronecker if















Kronecker











imal surface Fk




















shortly



bull F is minimal


rational curve


trivial along s























projection onto P1


gular fiber X0










invariant fiber






























F prime F+

f prime+

f primeprime+

t+u









KF is big














fiber































totally harmless






































form













sumCsubeCk+1

F (C))red


divisors


(36)















(38) (F Nminus1 + F



(39)






(40) (F 0 + F







Theorem I









resolution


















is given







Now we have



if




singularity




















pminusrarr S

Where






s isin S





)otimesm)oror



PNS



























References



1970602





0021869374901021




USSR (in Russian) 236 (5) 1041-1044


USSR Izv 13 (1978) 499-555




978-3-319-24460-0_2














ASENS_2002_4_35_2_231_0pdf





ihespdf










0003PAMQ-2008-0004-0003-a009pdf











reid2pdf








2FBF01388892page-1




I Introduction

II Preliminaries







IIVII Set-up






IVI KF-nil curves


V Configurations





References
































8 FEDERICO BUONERBA
















II Preliminaries





































D subX ( nradic

D) such that











sheaves








fn Yn rarrX








Our main concern is





















long as NCkminus1












assume NCp

















F on U and we have






















partpartxi









ring O








L (minusRamf )




Fact IIIV2



(1minus |Gp|minus1)






in [McQ05]



Z of the foliation



curves f


locus





Z2Z

































elliptic











(7) X rarrXe rarrXc






Q-Gorenstein















part =sumnixi

partpartxi



and write f =sum




let FN = (xa11 xakk















f minussum























general









results




Proof We have








foliation is smooth






rational curves




curves tangent to F








either










Corollary IIVI7






nal curves






not canonical




that




the log-chain

















ored in black




















































is arbitrary


















for the sequel

















(14) part(s) =part

partx+ s

part

party














bull KFm is nef or





























+ g partpartz







+g|(y=0)partpartz






singularity




partpartz

Z has an




to Z


singularity


diction










+ λy partparty


+ f partparty

+ g partpartz

for some func-

tions k f g



























components






+ f partparty

+ g partpartz

for some functions









contradiction




+ λy partparty


+ f partparty

+ g partpartz

for some functions

k f g







+g|(y=0)partpartz





0








them




singularity


0




are in the axes


+ λy partparty

+ microz partpartz




+ microz partpartz





axis x = z = 0



x = y = 0 x = z = 0 y = z = 0


next item




+ microz partpartz









x = y = 0 x = z = 0 y = z = 0




















ni (2)a i (3)b i (3)c








versely







xz



y

LC-center



y

LC-center














surface x = 0
































1 C 0 This proves





statement as well































along C






)

partk+1

y partk

yH0(COCk+1












partx+ λy

part

party+ microz

part


partξ+ (λ+ εn)η

part


part

partζ




partx+ λy

part

party+ microz

part


partξ+ (λ+ εn)η

part


part

partζ





eigenvalues


are called ends









smooth locus of p




∆













(25) part0 = xpart

partx+ microyazb

part



part

partζ(mod(I2C))



(26) x1 = x y1 = y z1 = zyminus1


(27)

part01 = x1part


part


part



part

partζ1(mod(I2C1

))




































is projective



























2 = (x0 = w0 = 0)













unless a = 0


































z







the beast






















(32) 2g(Z0)minus 2 + sZG(Z0) = 0















V Configurations




















∆



equation 23





First we need







(33) n+m = d+ e






































of p fi











curves



(34) x = η y = ξ z = ζ





attached at one end





























of FC






is empty
























trees








following obvious













chain


normal bundle













we conclude

Our new set-up is













not mentioned is


small Kronecker if















Kronecker











imal surface Fk




















shortly



bull F is minimal


rational curve


trivial along s























projection onto P1


gular fiber X0










invariant fiber






























F prime F+

f prime+

f primeprime+

t+u









KF is big














fiber































totally harmless






































form













sumCsubeCk+1

F (C))red


divisors


(36)















(38) (F Nminus1 + F



(39)






(40) (F 0 + F







Theorem I









resolution


















is given







Now we have



if




singularity




















pminusrarr S

Where






s isin S





)otimesm)oror



PNS



























References



1970602





0021869374901021




USSR (in Russian) 236 (5) 1041-1044


USSR Izv 13 (1978) 499-555




978-3-319-24460-0_2














ASENS_2002_4_35_2_231_0pdf





ihespdf










0003PAMQ-2008-0004-0003-a009pdf











reid2pdf








2FBF01388892page-1




I Introduction

II Preliminaries







IIVII Set-up






IVI KF-nil curves


V Configurations





References
























D subX ( nradic

D) such that











sheaves








fn Yn rarrX








Our main concern is





















long as NCkminus1












assume NCp

















F on U and we have






















partpartxi









ring O








L (minusRamf )




Fact IIIV2



(1minus |Gp|minus1)






in [McQ05]



Z of the foliation



curves f


locus





Z2Z

































elliptic











(7) X rarrXe rarrXc






Q-Gorenstein















part =sumnixi

partpartxi



and write f =sum




let FN = (xa11 xakk















f minussum























general









results




Proof We have








foliation is smooth






rational curves




curves tangent to F








either










Corollary IIVI7






nal curves






not canonical




that




the log-chain

















ored in black




















































is arbitrary


















for the sequel

















(14) part(s) =part

partx+ s

part

party














bull KFm is nef or





























+ g partpartz







+g|(y=0)partpartz






singularity




partpartz

Z has an




to Z


singularity


diction










+ λy partparty


+ f partparty

+ g partpartz

for some func-

tions k f g



























components






+ f partparty

+ g partpartz

for some functions









contradiction




+ λy partparty


+ f partparty

+ g partpartz

for some functions

k f g







+g|(y=0)partpartz





0








them




singularity


0




are in the axes


+ λy partparty

+ microz partpartz




+ microz partpartz





axis x = z = 0



x = y = 0 x = z = 0 y = z = 0


next item




+ microz partpartz









x = y = 0 x = z = 0 y = z = 0




















ni (2)a i (3)b i (3)c








versely







xz



y

LC-center



y

LC-center














surface x = 0
































1 C 0 This proves





statement as well































along C






)

partk+1

y partk

yH0(COCk+1












partx+ λy

part

party+ microz

part


partξ+ (λ+ εn)η

part


part

partζ




partx+ λy

part

party+ microz

part


partξ+ (λ+ εn)η

part


part

partζ





eigenvalues


are called ends









smooth locus of p




∆













(25) part0 = xpart

partx+ microyazb

part



part

partζ(mod(I2C))



(26) x1 = x y1 = y z1 = zyminus1


(27)

part01 = x1part


part


part



part

partζ1(mod(I2C1

))




































is projective



























2 = (x0 = w0 = 0)













unless a = 0


































z







the beast






















(32) 2g(Z0)minus 2 + sZG(Z0) = 0















V Configurations




















∆



equation 23





First we need







(33) n+m = d+ e






































of p fi











curves



(34) x = η y = ξ z = ζ





attached at one end





























of FC






is empty
























trees








following obvious













chain


normal bundle













we conclude

Our new set-up is













not mentioned is


small Kronecker if















Kronecker











imal surface Fk




















shortly



bull F is minimal


rational curve


trivial along s























projection onto P1


gular fiber X0










invariant fiber






























F prime F+

f prime+

f primeprime+

t+u









KF is big














fiber































totally harmless






































form













sumCsubeCk+1

F (C))red


divisors


(36)















(38) (F Nminus1 + F



(39)






(40) (F 0 + F







Theorem I









resolution


















is given







Now we have



if




singularity




















pminusrarr S

Where






s isin S





)otimesm)oror



PNS



























References



1970602





0021869374901021




USSR (in Russian) 236 (5) 1041-1044


USSR Izv 13 (1978) 499-555




978-3-319-24460-0_2














ASENS_2002_4_35_2_231_0pdf





ihespdf










0003PAMQ-2008-0004-0003-a009pdf











reid2pdf








2FBF01388892page-1




I Introduction

II Preliminaries







IIVII Set-up






IVI KF-nil curves


V Configurations





References



sheaves








fn Yn rarrX








Our main concern is





















long as NCkminus1












assume NCp

















F on U and we have






















partpartxi









ring O








L (minusRamf )




Fact IIIV2



(1minus |Gp|minus1)






in [McQ05]



Z of the foliation



curves f


locus





Z2Z

































elliptic











(7) X rarrXe rarrXc






Q-Gorenstein















part =sumnixi

partpartxi



and write f =sum




let FN = (xa11 xakk















f minussum























general









results




Proof We have








foliation is smooth






rational curves




curves tangent to F








either










Corollary IIVI7






nal curves






not canonical




that




the log-chain

















ored in black




















































is arbitrary


















for the sequel

















(14) part(s) =part

partx+ s

part

party














bull KFm is nef or





























+ g partpartz







+g|(y=0)partpartz






singularity




partpartz

Z has an




to Z


singularity


diction










+ λy partparty


+ f partparty

+ g partpartz

for some func-

tions k f g



























components






+ f partparty

+ g partpartz

for some functions









contradiction




+ λy partparty


+ f partparty

+ g partpartz

for some functions

k f g







+g|(y=0)partpartz





0








them




singularity


0




are in the axes


+ λy partparty

+ microz partpartz




+ microz partpartz





axis x = z = 0



x = y = 0 x = z = 0 y = z = 0


next item




+ microz partpartz









x = y = 0 x = z = 0 y = z = 0




















ni (2)a i (3)b i (3)c








versely







xz



y

LC-center



y

LC-center














surface x = 0
































1 C 0 This proves





statement as well































along C






)

partk+1

y partk

yH0(COCk+1












partx+ λy

part

party+ microz

part


partξ+ (λ+ εn)η

part


part

partζ




partx+ λy

part

party+ microz

part


partξ+ (λ+ εn)η

part


part

partζ





eigenvalues


are called ends









smooth locus of p




∆













(25) part0 = xpart

partx+ microyazb

part



part

partζ(mod(I2C))



(26) x1 = x y1 = y z1 = zyminus1


(27)

part01 = x1part


part


part



part

partζ1(mod(I2C1

))




































is projective



























2 = (x0 = w0 = 0)













unless a = 0


































z







the beast






















(32) 2g(Z0)minus 2 + sZG(Z0) = 0















V Configurations




















∆



equation 23





First we need







(33) n+m = d+ e






































of p fi











curves



(34) x = η y = ξ z = ζ





attached at one end





























of FC






is empty
























trees








following obvious













chain


normal bundle













we conclude

Our new set-up is













not mentioned is


small Kronecker if















Kronecker











imal surface Fk




















shortly



bull F is minimal


rational curve


trivial along s























projection onto P1


gular fiber X0










invariant fiber






























F prime F+

f prime+

f primeprime+

t+u









KF is big














fiber































totally harmless






































form













sumCsubeCk+1

F (C))red


divisors


(36)















(38) (F Nminus1 + F



(39)






(40) (F 0 + F







Theorem I









resolution


















is given







Now we have



if




singularity




















pminusrarr S

Where






s isin S





)otimesm)oror



PNS



























References



1970602





0021869374901021




USSR (in Russian) 236 (5) 1041-1044


USSR Izv 13 (1978) 499-555




978-3-319-24460-0_2














ASENS_2002_4_35_2_231_0pdf





ihespdf










0003PAMQ-2008-0004-0003-a009pdf











reid2pdf








2FBF01388892page-1




I Introduction

II Preliminaries







IIVII Set-up






IVI KF-nil curves


V Configurations





References












long as NCkminus1












assume NCp

















F on U and we have






















partpartxi









ring O








L (minusRamf )




Fact IIIV2



(1minus |Gp|minus1)






in [McQ05]



Z of the foliation



curves f


locus





Z2Z

































elliptic











(7) X rarrXe rarrXc






Q-Gorenstein















part =sumnixi

partpartxi



and write f =sum




let FN = (xa11 xakk















f minussum























general









results




Proof We have








foliation is smooth






rational curves




curves tangent to F








either










Corollary IIVI7






nal curves






not canonical




that




the log-chain

















ored in black




















































is arbitrary


















for the sequel

















(14) part(s) =part

partx+ s

part

party














bull KFm is nef or





























+ g partpartz







+g|(y=0)partpartz






singularity




partpartz

Z has an




to Z


singularity


diction










+ λy partparty


+ f partparty

+ g partpartz

for some func-

tions k f g



























components






+ f partparty

+ g partpartz

for some functions









contradiction




+ λy partparty


+ f partparty

+ g partpartz

for some functions

k f g







+g|(y=0)partpartz





0








them




singularity


0




are in the axes


+ λy partparty

+ microz partpartz




+ microz partpartz





axis x = z = 0



x = y = 0 x = z = 0 y = z = 0


next item




+ microz partpartz









x = y = 0 x = z = 0 y = z = 0




















ni (2)a i (3)b i (3)c








versely







xz



y

LC-center



y

LC-center














surface x = 0
































1 C 0 This proves





statement as well































along C






)

partk+1

y partk

yH0(COCk+1












partx+ λy

part

party+ microz

part


partξ+ (λ+ εn)η

part


part

partζ




partx+ λy

part

party+ microz

part


partξ+ (λ+ εn)η

part


part

partζ





eigenvalues


are called ends









smooth locus of p




∆













(25) part0 = xpart

partx+ microyazb

part



part

partζ(mod(I2C))



(26) x1 = x y1 = y z1 = zyminus1


(27)

part01 = x1part


part


part



part

partζ1(mod(I2C1

))




































is projective



























2 = (x0 = w0 = 0)













unless a = 0


































z







the beast






















(32) 2g(Z0)minus 2 + sZG(Z0) = 0















V Configurations




















∆



equation 23





First we need







(33) n+m = d+ e






































of p fi











curves



(34) x = η y = ξ z = ζ





attached at one end





























of FC






is empty
























trees








following obvious













chain


normal bundle













we conclude

Our new set-up is













not mentioned is


small Kronecker if















Kronecker











imal surface Fk




















shortly



bull F is minimal


rational curve


trivial along s























projection onto P1


gular fiber X0










invariant fiber






























F prime F+

f prime+

f primeprime+

t+u









KF is big














fiber































totally harmless






































form













sumCsubeCk+1

F (C))red


divisors


(36)















(38) (F Nminus1 + F



(39)






(40) (F 0 + F







Theorem I









resolution


















is given







Now we have



if




singularity




















pminusrarr S

Where






s isin S





)otimesm)oror



PNS



























References



1970602





0021869374901021




USSR (in Russian) 236 (5) 1041-1044


USSR Izv 13 (1978) 499-555




978-3-319-24460-0_2














ASENS_2002_4_35_2_231_0pdf





ihespdf










0003PAMQ-2008-0004-0003-a009pdf











reid2pdf








2FBF01388892page-1




I Introduction

II Preliminaries







IIVII Set-up






IVI KF-nil curves


V Configurations





References









F on U and we have






















partpartxi









ring O








L (minusRamf )




Fact IIIV2



(1minus |Gp|minus1)






in [McQ05]



Z of the foliation



curves f


locus





Z2Z

































elliptic











(7) X rarrXe rarrXc






Q-Gorenstein















part =sumnixi

partpartxi



and write f =sum




let FN = (xa11 xakk















f minussum























general









results




Proof We have








foliation is smooth






rational curves




curves tangent to F








either










Corollary IIVI7






nal curves






not canonical




that




the log-chain

















ored in black




















































is arbitrary


















for the sequel

















(14) part(s) =part

partx+ s

part

party














bull KFm is nef or





























+ g partpartz







+g|(y=0)partpartz






singularity




partpartz

Z has an




to Z


singularity


diction










+ λy partparty


+ f partparty

+ g partpartz

for some func-

tions k f g



























components






+ f partparty

+ g partpartz

for some functions









contradiction




+ λy partparty


+ f partparty

+ g partpartz

for some functions

k f g







+g|(y=0)partpartz





0








them




singularity


0




are in the axes


+ λy partparty

+ microz partpartz




+ microz partpartz





axis x = z = 0



x = y = 0 x = z = 0 y = z = 0


next item




+ microz partpartz









x = y = 0 x = z = 0 y = z = 0




















ni (2)a i (3)b i (3)c








versely







xz



y

LC-center



y

LC-center














surface x = 0
































1 C 0 This proves





statement as well































along C






)

partk+1

y partk

yH0(COCk+1












partx+ λy

part

party+ microz

part


partξ+ (λ+ εn)η

part


part

partζ




partx+ λy

part

party+ microz

part


partξ+ (λ+ εn)η

part


part

partζ





eigenvalues


are called ends









smooth locus of p




∆













(25) part0 = xpart

partx+ microyazb

part



part

partζ(mod(I2C))



(26) x1 = x y1 = y z1 = zyminus1


(27)

part01 = x1part


part


part



part

partζ1(mod(I2C1

))




































is projective



























2 = (x0 = w0 = 0)













unless a = 0


































z







the beast






















(32) 2g(Z0)minus 2 + sZG(Z0) = 0















V Configurations




















∆



equation 23





First we need







(33) n+m = d+ e






































of p fi











curves



(34) x = η y = ξ z = ζ





attached at one end





























of FC






is empty
























trees








following obvious













chain


normal bundle













we conclude

Our new set-up is













not mentioned is


small Kronecker if















Kronecker











imal surface Fk




















shortly



bull F is minimal


rational curve


trivial along s























projection onto P1


gular fiber X0










invariant fiber






























F prime F+

f prime+

f primeprime+

t+u









KF is big














fiber































totally harmless






































form













sumCsubeCk+1

F (C))red


divisors


(36)















(38) (F Nminus1 + F



(39)






(40) (F 0 + F







Theorem I









resolution


















is given







Now we have



if




singularity




















pminusrarr S

Where






s isin S





)otimesm)oror



PNS



























References



1970602





0021869374901021




USSR (in Russian) 236 (5) 1041-1044


USSR Izv 13 (1978) 499-555




978-3-319-24460-0_2














ASENS_2002_4_35_2_231_0pdf





ihespdf










0003PAMQ-2008-0004-0003-a009pdf











reid2pdf








2FBF01388892page-1




I Introduction

II Preliminaries







IIVII Set-up






IVI KF-nil curves


V Configurations





References




partpartxi









ring O








L (minusRamf )




Fact IIIV2



(1minus |Gp|minus1)






in [McQ05]



Z of the foliation



curves f


locus





Z2Z

































elliptic











(7) X rarrXe rarrXc






Q-Gorenstein















part =sumnixi

partpartxi



and write f =sum




let FN = (xa11 xakk















f minussum























general









results




Proof We have








foliation is smooth






rational curves




curves tangent to F








either










Corollary IIVI7






nal curves






not canonical




that




the log-chain

















ored in black




















































is arbitrary


















for the sequel

















(14) part(s) =part

partx+ s

part

party














bull KFm is nef or





























+ g partpartz







+g|(y=0)partpartz






singularity




partpartz

Z has an




to Z


singularity


diction










+ λy partparty


+ f partparty

+ g partpartz

for some func-

tions k f g



























components






+ f partparty

+ g partpartz

for some functions









contradiction




+ λy partparty


+ f partparty

+ g partpartz

for some functions

k f g







+g|(y=0)partpartz





0








them




singularity


0




are in the axes


+ λy partparty

+ microz partpartz




+ microz partpartz





axis x = z = 0



x = y = 0 x = z = 0 y = z = 0


next item




+ microz partpartz









x = y = 0 x = z = 0 y = z = 0




















ni (2)a i (3)b i (3)c








versely







xz



y

LC-center



y

LC-center














surface x = 0
































1 C 0 This proves





statement as well































along C






)

partk+1

y partk

yH0(COCk+1












partx+ λy

part

party+ microz

part


partξ+ (λ+ εn)η

part


part

partζ




partx+ λy

part

party+ microz

part


partξ+ (λ+ εn)η

part


part

partζ





eigenvalues


are called ends









smooth locus of p




∆













(25) part0 = xpart

partx+ microyazb

part



part

partζ(mod(I2C))



(26) x1 = x y1 = y z1 = zyminus1


(27)

part01 = x1part


part


part



part

partζ1(mod(I2C1

))




































is projective



























2 = (x0 = w0 = 0)













unless a = 0


































z







the beast






















(32) 2g(Z0)minus 2 + sZG(Z0) = 0















V Configurations




















∆



equation 23





First we need







(33) n+m = d+ e






































of p fi











curves



(34) x = η y = ξ z = ζ





attached at one end





























of FC






is empty
























trees








following obvious













chain


normal bundle













we conclude

Our new set-up is













not mentioned is


small Kronecker if















Kronecker











imal surface Fk




















shortly



bull F is minimal


rational curve


trivial along s























projection onto P1


gular fiber X0










invariant fiber






























F prime F+

f prime+

f primeprime+

t+u









KF is big














fiber































totally harmless






































form













sumCsubeCk+1

F (C))red


divisors


(36)















(38) (F Nminus1 + F



(39)






(40) (F 0 + F







Theorem I









resolution


















is given







Now we have



if




singularity




















pminusrarr S

Where






s isin S





)otimesm)oror



PNS



























References



1970602





0021869374901021




USSR (in Russian) 236 (5) 1041-1044


USSR Izv 13 (1978) 499-555




978-3-319-24460-0_2














ASENS_2002_4_35_2_231_0pdf





ihespdf










0003PAMQ-2008-0004-0003-a009pdf











reid2pdf








2FBF01388892page-1




I Introduction

II Preliminaries







IIVII Set-up






IVI KF-nil curves


V Configurations





References




in [McQ05]



Z of the foliation



curves f


locus





Z2Z

































elliptic











(7) X rarrXe rarrXc






Q-Gorenstein















part =sumnixi

partpartxi



and write f =sum




let FN = (xa11 xakk















f minussum























general









results




Proof We have








foliation is smooth






rational curves




curves tangent to F








either










Corollary IIVI7






nal curves






not canonical




that




the log-chain

















ored in black




















































is arbitrary


















for the sequel

















(14) part(s) =part

partx+ s

part

party














bull KFm is nef or





























+ g partpartz







+g|(y=0)partpartz






singularity




partpartz

Z has an




to Z


singularity


diction










+ λy partparty


+ f partparty

+ g partpartz

for some func-

tions k f g



























components






+ f partparty

+ g partpartz

for some functions









contradiction




+ λy partparty


+ f partparty

+ g partpartz

for some functions

k f g







+g|(y=0)partpartz





0








them




singularity


0




are in the axes


+ λy partparty

+ microz partpartz




+ microz partpartz





axis x = z = 0



x = y = 0 x = z = 0 y = z = 0


next item




+ microz partpartz









x = y = 0 x = z = 0 y = z = 0




















ni (2)a i (3)b i (3)c








versely







xz



y

LC-center



y

LC-center














surface x = 0
































1 C 0 This proves





statement as well































along C






)

partk+1

y partk

yH0(COCk+1












partx+ λy

part

party+ microz

part


partξ+ (λ+ εn)η

part


part

partζ




partx+ λy

part

party+ microz

part


partξ+ (λ+ εn)η

part


part

partζ





eigenvalues


are called ends









smooth locus of p




∆













(25) part0 = xpart

partx+ microyazb

part



part

partζ(mod(I2C))



(26) x1 = x y1 = y z1 = zyminus1


(27)

part01 = x1part


part


part



part

partζ1(mod(I2C1

))




































is projective



























2 = (x0 = w0 = 0)













unless a = 0


































z







the beast






















(32) 2g(Z0)minus 2 + sZG(Z0) = 0















V Configurations




















∆



equation 23





First we need







(33) n+m = d+ e






































of p fi











curves



(34) x = η y = ξ z = ζ





attached at one end





























of FC






is empty
























trees








following obvious













chain


normal bundle













we conclude

Our new set-up is













not mentioned is


small Kronecker if















Kronecker











imal surface Fk




















shortly



bull F is minimal


rational curve


trivial along s























projection onto P1


gular fiber X0










invariant fiber






























F prime F+

f prime+

f primeprime+

t+u









KF is big














fiber































totally harmless






































form













sumCsubeCk+1

F (C))red


divisors


(36)















(38) (F Nminus1 + F



(39)






(40) (F 0 + F







Theorem I









resolution


















is given







Now we have



if




singularity




















pminusrarr S

Where






s isin S





)otimesm)oror



PNS



























References



1970602





0021869374901021




USSR (in Russian) 236 (5) 1041-1044


USSR Izv 13 (1978) 499-555




978-3-319-24460-0_2














ASENS_2002_4_35_2_231_0pdf





ihespdf










0003PAMQ-2008-0004-0003-a009pdf











reid2pdf








2FBF01388892page-1




I Introduction

II Preliminaries







IIVII Set-up






IVI KF-nil curves


V Configurations





References





















elliptic











(7) X rarrXe rarrXc






Q-Gorenstein















part =sumnixi

partpartxi



and write f =sum




let FN = (xa11 xakk















f minussum























general









results




Proof We have








foliation is smooth






rational curves




curves tangent to F








either










Corollary IIVI7






nal curves






not canonical




that




the log-chain

















ored in black




















































is arbitrary


















for the sequel

















(14) part(s) =part

partx+ s

part

party














bull KFm is nef or





























+ g partpartz







+g|(y=0)partpartz






singularity




partpartz

Z has an




to Z


singularity


diction










+ λy partparty


+ f partparty

+ g partpartz

for some func-

tions k f g



























components






+ f partparty

+ g partpartz

for some functions









contradiction




+ λy partparty


+ f partparty

+ g partpartz

for some functions

k f g







+g|(y=0)partpartz





0








them




singularity


0




are in the axes


+ λy partparty

+ microz partpartz




+ microz partpartz





axis x = z = 0



x = y = 0 x = z = 0 y = z = 0


next item




+ microz partpartz









x = y = 0 x = z = 0 y = z = 0




















ni (2)a i (3)b i (3)c








versely







xz



y

LC-center



y

LC-center














surface x = 0
































1 C 0 This proves





statement as well































along C






)

partk+1

y partk

yH0(COCk+1












partx+ λy

part

party+ microz

part


partξ+ (λ+ εn)η

part


part

partζ




partx+ λy

part

party+ microz

part


partξ+ (λ+ εn)η

part


part

partζ





eigenvalues


are called ends









smooth locus of p




∆













(25) part0 = xpart

partx+ microyazb

part



part

partζ(mod(I2C))



(26) x1 = x y1 = y z1 = zyminus1


(27)

part01 = x1part


part


part



part

partζ1(mod(I2C1

))




































is projective



























2 = (x0 = w0 = 0)













unless a = 0


































z







the beast






















(32) 2g(Z0)minus 2 + sZG(Z0) = 0















V Configurations




















∆



equation 23





First we need







(33) n+m = d+ e






































of p fi











curves



(34) x = η y = ξ z = ζ





attached at one end





























of FC






is empty
























trees








following obvious













chain


normal bundle













we conclude

Our new set-up is













not mentioned is


small Kronecker if















Kronecker











imal surface Fk




















shortly



bull F is minimal


rational curve


trivial along s























projection onto P1


gular fiber X0










invariant fiber






























F prime F+

f prime+

f primeprime+

t+u









KF is big














fiber































totally harmless






































form













sumCsubeCk+1

F (C))red


divisors


(36)















(38) (F Nminus1 + F



(39)






(40) (F 0 + F







Theorem I









resolution


















is given







Now we have



if




singularity




















pminusrarr S

Where






s isin S





)otimesm)oror



PNS



























References



1970602





0021869374901021




USSR (in Russian) 236 (5) 1041-1044


USSR Izv 13 (1978) 499-555




978-3-319-24460-0_2














ASENS_2002_4_35_2_231_0pdf





ihespdf










0003PAMQ-2008-0004-0003-a009pdf











reid2pdf








2FBF01388892page-1




I Introduction

II Preliminaries







IIVII Set-up






IVI KF-nil curves


V Configurations





References



(7) X rarrXe rarrXc






Q-Gorenstein















part =sumnixi

partpartxi



and write f =sum




let FN = (xa11 xakk















f minussum























general









results




Proof We have








foliation is smooth






rational curves




curves tangent to F








either










Corollary IIVI7






nal curves






not canonical




that




the log-chain

















ored in black




















































is arbitrary


















for the sequel

















(14) part(s) =part

partx+ s

part

party














bull KFm is nef or





























+ g partpartz







+g|(y=0)partpartz






singularity




partpartz

Z has an




to Z


singularity


diction










+ λy partparty


+ f partparty

+ g partpartz

for some func-

tions k f g



























components






+ f partparty

+ g partpartz

for some functions









contradiction




+ λy partparty


+ f partparty

+ g partpartz

for some functions

k f g







+g|(y=0)partpartz





0








them




singularity


0




are in the axes


+ λy partparty

+ microz partpartz




+ microz partpartz





axis x = z = 0



x = y = 0 x = z = 0 y = z = 0


next item




+ microz partpartz









x = y = 0 x = z = 0 y = z = 0




















ni (2)a i (3)b i (3)c








versely







xz



y

LC-center



y

LC-center














surface x = 0
































1 C 0 This proves





statement as well































along C






)

partk+1

y partk

yH0(COCk+1












partx+ λy

part

party+ microz

part


partξ+ (λ+ εn)η

part


part

partζ




partx+ λy

part

party+ microz

part


partξ+ (λ+ εn)η

part


part

partζ





eigenvalues


are called ends









smooth locus of p




∆













(25) part0 = xpart

partx+ microyazb

part



part

partζ(mod(I2C))



(26) x1 = x y1 = y z1 = zyminus1


(27)

part01 = x1part


part


part



part

partζ1(mod(I2C1

))




































is projective



























2 = (x0 = w0 = 0)













unless a = 0


































z







the beast






















(32) 2g(Z0)minus 2 + sZG(Z0) = 0















V Configurations




















∆



equation 23





First we need







(33) n+m = d+ e






































of p fi











curves



(34) x = η y = ξ z = ζ





attached at one end





























of FC






is empty
























trees








following obvious













chain


normal bundle













we conclude

Our new set-up is













not mentioned is


small Kronecker if















Kronecker











imal surface Fk




















shortly



bull F is minimal


rational curve


trivial along s























projection onto P1


gular fiber X0










invariant fiber






























F prime F+

f prime+

f primeprime+

t+u









KF is big














fiber































totally harmless






































form













sumCsubeCk+1

F (C))red


divisors


(36)















(38) (F Nminus1 + F



(39)






(40) (F 0 + F







Theorem I









resolution


















is given







Now we have



if




singularity




















pminusrarr S

Where






s isin S





)otimesm)oror



PNS



























References



1970602





0021869374901021




USSR (in Russian) 236 (5) 1041-1044


USSR Izv 13 (1978) 499-555




978-3-319-24460-0_2














ASENS_2002_4_35_2_231_0pdf





ihespdf










0003PAMQ-2008-0004-0003-a009pdf











reid2pdf








2FBF01388892page-1




I Introduction

II Preliminaries







IIVII Set-up






IVI KF-nil curves


V Configurations





References













f minussum























general









results




Proof We have








foliation is smooth






rational curves




curves tangent to F








either










Corollary IIVI7






nal curves






not canonical




that




the log-chain

















ored in black




















































is arbitrary


















for the sequel

















(14) part(s) =part

partx+ s

part

party














bull KFm is nef or





























+ g partpartz







+g|(y=0)partpartz






singularity




partpartz

Z has an




to Z


singularity


diction










+ λy partparty


+ f partparty

+ g partpartz

for some func-

tions k f g



























components






+ f partparty

+ g partpartz

for some functions









contradiction




+ λy partparty


+ f partparty

+ g partpartz

for some functions

k f g







+g|(y=0)partpartz





0








them




singularity


0




are in the axes


+ λy partparty

+ microz partpartz




+ microz partpartz





axis x = z = 0



x = y = 0 x = z = 0 y = z = 0


next item




+ microz partpartz









x = y = 0 x = z = 0 y = z = 0




















ni (2)a i (3)b i (3)c








versely







xz



y

LC-center



y

LC-center














surface x = 0
































1 C 0 This proves





statement as well































along C






)

partk+1

y partk

yH0(COCk+1












partx+ λy

part

party+ microz

part


partξ+ (λ+ εn)η

part


part

partζ




partx+ λy

part

party+ microz

part


partξ+ (λ+ εn)η

part


part

partζ





eigenvalues


are called ends









smooth locus of p




∆













(25) part0 = xpart

partx+ microyazb

part



part

partζ(mod(I2C))



(26) x1 = x y1 = y z1 = zyminus1


(27)

part01 = x1part


part


part



part

partζ1(mod(I2C1

))




































is projective



























2 = (x0 = w0 = 0)













unless a = 0


































z







the beast






















(32) 2g(Z0)minus 2 + sZG(Z0) = 0















V Configurations




















∆



equation 23





First we need







(33) n+m = d+ e






































of p fi











curves



(34) x = η y = ξ z = ζ





attached at one end





























of FC






is empty
























trees








following obvious













chain


normal bundle













we conclude

Our new set-up is













not mentioned is


small Kronecker if















Kronecker











imal surface Fk




















shortly



bull F is minimal


rational curve


trivial along s























projection onto P1


gular fiber X0










invariant fiber






























F prime F+

f prime+

f primeprime+

t+u









KF is big














fiber































totally harmless






































form













sumCsubeCk+1

F (C))red


divisors


(36)















(38) (F Nminus1 + F



(39)






(40) (F 0 + F







Theorem I









resolution


















is given







Now we have



if




singularity




















pminusrarr S

Where






s isin S





)otimesm)oror



PNS



























References



1970602





0021869374901021




USSR (in Russian) 236 (5) 1041-1044


USSR Izv 13 (1978) 499-555




978-3-319-24460-0_2














ASENS_2002_4_35_2_231_0pdf





ihespdf










0003PAMQ-2008-0004-0003-a009pdf











reid2pdf








2FBF01388892page-1




I Introduction

II Preliminaries







IIVII Set-up






IVI KF-nil curves


V Configurations





References







general









results




Proof We have








foliation is smooth






rational curves




curves tangent to F








either










Corollary IIVI7






nal curves






not canonical




that




the log-chain

















ored in black




















































is arbitrary


















for the sequel

















(14) part(s) =part

partx+ s

part

party














bull KFm is nef or





























+ g partpartz







+g|(y=0)partpartz






singularity




partpartz

Z has an




to Z


singularity


diction










+ λy partparty


+ f partparty

+ g partpartz

for some func-

tions k f g



























components






+ f partparty

+ g partpartz

for some functions









contradiction




+ λy partparty


+ f partparty

+ g partpartz

for some functions

k f g







+g|(y=0)partpartz





0








them




singularity


0




are in the axes


+ λy partparty

+ microz partpartz




+ microz partpartz





axis x = z = 0



x = y = 0 x = z = 0 y = z = 0


next item




+ microz partpartz









x = y = 0 x = z = 0 y = z = 0




















ni (2)a i (3)b i (3)c








versely







xz



y

LC-center



y

LC-center














surface x = 0
































1 C 0 This proves





statement as well































along C






)

partk+1

y partk

yH0(COCk+1












partx+ λy

part

party+ microz

part


partξ+ (λ+ εn)η

part


part

partζ




partx+ λy

part

party+ microz

part


partξ+ (λ+ εn)η

part


part

partζ





eigenvalues


are called ends









smooth locus of p




∆













(25) part0 = xpart

partx+ microyazb

part



part

partζ(mod(I2C))



(26) x1 = x y1 = y z1 = zyminus1


(27)

part01 = x1part


part


part



part

partζ1(mod(I2C1

))




































is projective



























2 = (x0 = w0 = 0)













unless a = 0


































z







the beast






















(32) 2g(Z0)minus 2 + sZG(Z0) = 0















V Configurations




















∆



equation 23





First we need







(33) n+m = d+ e






































of p fi











curves



(34) x = η y = ξ z = ζ





attached at one end





























of FC






is empty
























trees








following obvious













chain


normal bundle













we conclude

Our new set-up is













not mentioned is


small Kronecker if















Kronecker











imal surface Fk




















shortly



bull F is minimal


rational curve


trivial along s























projection onto P1


gular fiber X0










invariant fiber






























F prime F+

f prime+

f primeprime+

t+u









KF is big














fiber































totally harmless






































form













sumCsubeCk+1

F (C))red


divisors


(36)















(38) (F Nminus1 + F



(39)






(40) (F 0 + F







Theorem I









resolution


















is given







Now we have



if




singularity




















pminusrarr S

Where






s isin S





)otimesm)oror



PNS



























References



1970602





0021869374901021




USSR (in Russian) 236 (5) 1041-1044


USSR Izv 13 (1978) 499-555




978-3-319-24460-0_2














ASENS_2002_4_35_2_231_0pdf





ihespdf










0003PAMQ-2008-0004-0003-a009pdf











reid2pdf








2FBF01388892page-1




I Introduction

II Preliminaries







IIVII Set-up






IVI KF-nil curves


V Configurations





References






rational curves




curves tangent to F








either










Corollary IIVI7






nal curves






not canonical




that




the log-chain

















ored in black




















































is arbitrary


















for the sequel

















(14) part(s) =part

partx+ s

part

party














bull KFm is nef or





























+ g partpartz







+g|(y=0)partpartz






singularity




partpartz

Z has an




to Z


singularity


diction










+ λy partparty


+ f partparty

+ g partpartz

for some func-

tions k f g



























components






+ f partparty

+ g partpartz

for some functions









contradiction




+ λy partparty


+ f partparty

+ g partpartz

for some functions

k f g







+g|(y=0)partpartz





0








them




singularity


0




are in the axes


+ λy partparty

+ microz partpartz




+ microz partpartz





axis x = z = 0



x = y = 0 x = z = 0 y = z = 0


next item




+ microz partpartz









x = y = 0 x = z = 0 y = z = 0




















ni (2)a i (3)b i (3)c








versely







xz



y

LC-center



y

LC-center














surface x = 0
































1 C 0 This proves





statement as well































along C






)

partk+1

y partk

yH0(COCk+1












partx+ λy

part

party+ microz

part


partξ+ (λ+ εn)η

part


part

partζ




partx+ λy

part

party+ microz

part


partξ+ (λ+ εn)η

part


part

partζ





eigenvalues


are called ends









smooth locus of p




∆













(25) part0 = xpart

partx+ microyazb

part



part

partζ(mod(I2C))



(26) x1 = x y1 = y z1 = zyminus1


(27)

part01 = x1part


part


part



part

partζ1(mod(I2C1

))




































is projective



























2 = (x0 = w0 = 0)













unless a = 0


































z







the beast






















(32) 2g(Z0)minus 2 + sZG(Z0) = 0















V Configurations




















∆



equation 23





First we need







(33) n+m = d+ e






































of p fi











curves



(34) x = η y = ξ z = ζ





attached at one end





























of FC






is empty
























trees








following obvious













chain


normal bundle













we conclude

Our new set-up is













not mentioned is


small Kronecker if















Kronecker











imal surface Fk




















shortly



bull F is minimal


rational curve


trivial along s























projection onto P1


gular fiber X0










invariant fiber






























F prime F+

f prime+

f primeprime+

t+u









KF is big














fiber































totally harmless






































form













sumCsubeCk+1

F (C))red


divisors


(36)















(38) (F Nminus1 + F



(39)






(40) (F 0 + F







Theorem I









resolution


















is given







Now we have



if




singularity




















pminusrarr S

Where






s isin S





)otimesm)oror



PNS



























References



1970602





0021869374901021




USSR (in Russian) 236 (5) 1041-1044


USSR Izv 13 (1978) 499-555




978-3-319-24460-0_2














ASENS_2002_4_35_2_231_0pdf





ihespdf










0003PAMQ-2008-0004-0003-a009pdf











reid2pdf








2FBF01388892page-1




I Introduction

II Preliminaries







IIVII Set-up






IVI KF-nil curves


V Configurations





References






nal curves






not canonical




that




the log-chain

















ored in black




















































is arbitrary


















for the sequel

















(14) part(s) =part

partx+ s

part

party














bull KFm is nef or





























+ g partpartz







+g|(y=0)partpartz






singularity




partpartz

Z has an




to Z


singularity


diction










+ λy partparty


+ f partparty

+ g partpartz

for some func-

tions k f g



























components






+ f partparty

+ g partpartz

for some functions









contradiction




+ λy partparty


+ f partparty

+ g partpartz

for some functions

k f g







+g|(y=0)partpartz





0








them




singularity


0




are in the axes


+ λy partparty

+ microz partpartz




+ microz partpartz





axis x = z = 0



x = y = 0 x = z = 0 y = z = 0


next item




+ microz partpartz









x = y = 0 x = z = 0 y = z = 0




















ni (2)a i (3)b i (3)c








versely







xz



y

LC-center



y

LC-center














surface x = 0
































1 C 0 This proves





statement as well































along C






)

partk+1

y partk

yH0(COCk+1












partx+ λy

part

party+ microz

part


partξ+ (λ+ εn)η

part


part

partζ




partx+ λy

part

party+ microz

part


partξ+ (λ+ εn)η

part


part

partζ





eigenvalues


are called ends









smooth locus of p




∆













(25) part0 = xpart

partx+ microyazb

part



part

partζ(mod(I2C))



(26) x1 = x y1 = y z1 = zyminus1


(27)

part01 = x1part


part


part



part

partζ1(mod(I2C1

))




































is projective



























2 = (x0 = w0 = 0)













unless a = 0


































z







the beast






















(32) 2g(Z0)minus 2 + sZG(Z0) = 0















V Configurations




















∆



equation 23





First we need







(33) n+m = d+ e






































of p fi











curves



(34) x = η y = ξ z = ζ





attached at one end





























of FC






is empty
























trees








following obvious













chain


normal bundle













we conclude

Our new set-up is













not mentioned is


small Kronecker if















Kronecker











imal surface Fk




















shortly



bull F is minimal


rational curve


trivial along s























projection onto P1


gular fiber X0










invariant fiber






























F prime F+

f prime+

f primeprime+

t+u









KF is big














fiber































totally harmless






































form













sumCsubeCk+1

F (C))red


divisors


(36)















(38) (F Nminus1 + F



(39)






(40) (F 0 + F







Theorem I









resolution


















is given







Now we have



if




singularity




















pminusrarr S

Where






s isin S





)otimesm)oror



PNS



























References



1970602





0021869374901021




USSR (in Russian) 236 (5) 1041-1044


USSR Izv 13 (1978) 499-555




978-3-319-24460-0_2














ASENS_2002_4_35_2_231_0pdf





ihespdf










0003PAMQ-2008-0004-0003-a009pdf











reid2pdf








2FBF01388892page-1




I Introduction

II Preliminaries







IIVII Set-up






IVI KF-nil curves


V Configurations





References









ored in black




















































is arbitrary


















for the sequel

















(14) part(s) =part

partx+ s

part

party














bull KFm is nef or





























+ g partpartz







+g|(y=0)partpartz






singularity




partpartz

Z has an




to Z


singularity


diction










+ λy partparty


+ f partparty

+ g partpartz

for some func-

tions k f g



























components






+ f partparty

+ g partpartz

for some functions









contradiction




+ λy partparty


+ f partparty

+ g partpartz

for some functions

k f g







+g|(y=0)partpartz





0








them




singularity


0




are in the axes


+ λy partparty

+ microz partpartz




+ microz partpartz





axis x = z = 0



x = y = 0 x = z = 0 y = z = 0


next item




+ microz partpartz









x = y = 0 x = z = 0 y = z = 0




















ni (2)a i (3)b i (3)c








versely







xz



y

LC-center



y

LC-center














surface x = 0
































1 C 0 This proves





statement as well































along C






)

partk+1

y partk

yH0(COCk+1












partx+ λy

part

party+ microz

part


partξ+ (λ+ εn)η

part


part

partζ




partx+ λy

part

party+ microz

part


partξ+ (λ+ εn)η

part


part

partζ





eigenvalues


are called ends









smooth locus of p




∆













(25) part0 = xpart

partx+ microyazb

part



part

partζ(mod(I2C))



(26) x1 = x y1 = y z1 = zyminus1


(27)

part01 = x1part


part


part



part

partζ1(mod(I2C1

))




































is projective



























2 = (x0 = w0 = 0)













unless a = 0


































z







the beast






















(32) 2g(Z0)minus 2 + sZG(Z0) = 0















V Configurations




















∆



equation 23





First we need







(33) n+m = d+ e






































of p fi











curves



(34) x = η y = ξ z = ζ





attached at one end





























of FC






is empty
























trees








following obvious













chain


normal bundle













we conclude

Our new set-up is













not mentioned is


small Kronecker if















Kronecker











imal surface Fk




















shortly



bull F is minimal


rational curve


trivial along s























projection onto P1


gular fiber X0










invariant fiber






























F prime F+

f prime+

f primeprime+

t+u









KF is big














fiber































totally harmless






































form













sumCsubeCk+1

F (C))red


divisors


(36)















(38) (F Nminus1 + F



(39)






(40) (F 0 + F







Theorem I









resolution


















is given







Now we have



if




singularity




















pminusrarr S

Where






s isin S





)otimesm)oror



PNS



























References



1970602





0021869374901021




USSR (in Russian) 236 (5) 1041-1044


USSR Izv 13 (1978) 499-555




978-3-319-24460-0_2














ASENS_2002_4_35_2_231_0pdf





ihespdf










0003PAMQ-2008-0004-0003-a009pdf











reid2pdf








2FBF01388892page-1




I Introduction

II Preliminaries







IIVII Set-up






IVI KF-nil curves


V Configurations





References



































is arbitrary


















for the sequel

















(14) part(s) =part

partx+ s

part

party














bull KFm is nef or





























+ g partpartz







+g|(y=0)partpartz






singularity




partpartz

Z has an




to Z


singularity


diction










+ λy partparty


+ f partparty

+ g partpartz

for some func-

tions k f g



























components






+ f partparty

+ g partpartz

for some functions









contradiction




+ λy partparty


+ f partparty

+ g partpartz

for some functions

k f g







+g|(y=0)partpartz





0








them




singularity


0




are in the axes


+ λy partparty

+ microz partpartz




+ microz partpartz





axis x = z = 0



x = y = 0 x = z = 0 y = z = 0


next item




+ microz partpartz









x = y = 0 x = z = 0 y = z = 0




















ni (2)a i (3)b i (3)c








versely







xz



y

LC-center



y

LC-center














surface x = 0
































1 C 0 This proves





statement as well































along C






)

partk+1

y partk

yH0(COCk+1












partx+ λy

part

party+ microz

part


partξ+ (λ+ εn)η

part


part

partζ




partx+ λy

part

party+ microz

part


partξ+ (λ+ εn)η

part


part

partζ





eigenvalues


are called ends









smooth locus of p




∆













(25) part0 = xpart

partx+ microyazb

part



part

partζ(mod(I2C))



(26) x1 = x y1 = y z1 = zyminus1


(27)

part01 = x1part


part


part



part

partζ1(mod(I2C1

))




































is projective



























2 = (x0 = w0 = 0)













unless a = 0


































z







the beast






















(32) 2g(Z0)minus 2 + sZG(Z0) = 0















V Configurations




















∆



equation 23





First we need







(33) n+m = d+ e






































of p fi











curves



(34) x = η y = ξ z = ζ





attached at one end





























of FC






is empty
























trees








following obvious













chain


normal bundle













we conclude

Our new set-up is













not mentioned is


small Kronecker if















Kronecker











imal surface Fk




















shortly



bull F is minimal


rational curve


trivial along s























projection onto P1


gular fiber X0










invariant fiber






























F prime F+

f prime+

f primeprime+

t+u









KF is big














fiber































totally harmless






































form













sumCsubeCk+1

F (C))red


divisors


(36)















(38) (F Nminus1 + F



(39)






(40) (F 0 + F







Theorem I









resolution


















is given







Now we have



if




singularity




















pminusrarr S

Where






s isin S





)otimesm)oror



PNS



























References



1970602





0021869374901021




USSR (in Russian) 236 (5) 1041-1044


USSR Izv 13 (1978) 499-555




978-3-319-24460-0_2














ASENS_2002_4_35_2_231_0pdf





ihespdf










0003PAMQ-2008-0004-0003-a009pdf











reid2pdf








2FBF01388892page-1




I Introduction

II Preliminaries







IIVII Set-up






IVI KF-nil curves


V Configurations





References












(14) part(s) =part

partx+ s

part

party














bull KFm is nef or





























+ g partpartz







+g|(y=0)partpartz






singularity




partpartz

Z has an




to Z


singularity


diction










+ λy partparty


+ f partparty

+ g partpartz

for some func-

tions k f g



























components






+ f partparty

+ g partpartz

for some functions









contradiction




+ λy partparty


+ f partparty

+ g partpartz

for some functions

k f g







+g|(y=0)partpartz





0








them




singularity


0




are in the axes


+ λy partparty

+ microz partpartz




+ microz partpartz





axis x = z = 0



x = y = 0 x = z = 0 y = z = 0


next item




+ microz partpartz









x = y = 0 x = z = 0 y = z = 0




















ni (2)a i (3)b i (3)c








versely







xz



y

LC-center



y

LC-center














surface x = 0
































1 C 0 This proves





statement as well































along C






)

partk+1

y partk

yH0(COCk+1












partx+ λy

part

party+ microz

part


partξ+ (λ+ εn)η

part


part

partζ




partx+ λy

part

party+ microz

part


partξ+ (λ+ εn)η

part


part

partζ





eigenvalues


are called ends









smooth locus of p




∆













(25) part0 = xpart

partx+ microyazb

part



part

partζ(mod(I2C))



(26) x1 = x y1 = y z1 = zyminus1


(27)

part01 = x1part


part


part



part

partζ1(mod(I2C1

))




































is projective



























2 = (x0 = w0 = 0)













unless a = 0


































z







the beast






















(32) 2g(Z0)minus 2 + sZG(Z0) = 0















V Configurations




















∆



equation 23





First we need







(33) n+m = d+ e






































of p fi











curves



(34) x = η y = ξ z = ζ





attached at one end





























of FC






is empty
























trees








following obvious













chain


normal bundle













we conclude

Our new set-up is













not mentioned is


small Kronecker if















Kronecker











imal surface Fk




















shortly



bull F is minimal


rational curve


trivial along s























projection onto P1


gular fiber X0










invariant fiber






























F prime F+

f prime+

f primeprime+

t+u









KF is big














fiber































totally harmless






































form













sumCsubeCk+1

F (C))red


divisors


(36)















(38) (F Nminus1 + F



(39)






(40) (F 0 + F







Theorem I









resolution


















is given







Now we have



if




singularity




















pminusrarr S

Where






s isin S





)otimesm)oror



PNS



























References



1970602





0021869374901021




USSR (in Russian) 236 (5) 1041-1044


USSR Izv 13 (1978) 499-555




978-3-319-24460-0_2














ASENS_2002_4_35_2_231_0pdf





ihespdf










0003PAMQ-2008-0004-0003-a009pdf











reid2pdf








2FBF01388892page-1




I Introduction

II Preliminaries







IIVII Set-up






IVI KF-nil curves


V Configurations





References





























+ g partpartz







+g|(y=0)partpartz






singularity




partpartz

Z has an




to Z


singularity


diction










+ λy partparty


+ f partparty

+ g partpartz

for some func-

tions k f g



























components






+ f partparty

+ g partpartz

for some functions









contradiction




+ λy partparty


+ f partparty

+ g partpartz

for some functions

k f g







+g|(y=0)partpartz





0








them




singularity


0




are in the axes


+ λy partparty

+ microz partpartz




+ microz partpartz





axis x = z = 0



x = y = 0 x = z = 0 y = z = 0


next item




+ microz partpartz









x = y = 0 x = z = 0 y = z = 0




















ni (2)a i (3)b i (3)c








versely







xz



y

LC-center



y

LC-center














surface x = 0
































1 C 0 This proves





statement as well































along C






)

partk+1

y partk

yH0(COCk+1












partx+ λy

part

party+ microz

part


partξ+ (λ+ εn)η

part


part

partζ




partx+ λy

part

party+ microz

part


partξ+ (λ+ εn)η

part


part

partζ





eigenvalues


are called ends









smooth locus of p




∆













(25) part0 = xpart

partx+ microyazb

part



part

partζ(mod(I2C))



(26) x1 = x y1 = y z1 = zyminus1


(27)

part01 = x1part


part


part



part

partζ1(mod(I2C1

))




































is projective



























2 = (x0 = w0 = 0)













unless a = 0


































z







the beast






















(32) 2g(Z0)minus 2 + sZG(Z0) = 0















V Configurations




















∆



equation 23





First we need







(33) n+m = d+ e






































of p fi











curves



(34) x = η y = ξ z = ζ





attached at one end





























of FC






is empty
























trees








following obvious













chain


normal bundle













we conclude

Our new set-up is













not mentioned is


small Kronecker if















Kronecker











imal surface Fk




















shortly



bull F is minimal


rational curve


trivial along s























projection onto P1


gular fiber X0










invariant fiber






























F prime F+

f prime+

f primeprime+

t+u









KF is big














fiber































totally harmless






































form













sumCsubeCk+1

F (C))red


divisors


(36)















(38) (F Nminus1 + F



(39)






(40) (F 0 + F







Theorem I









resolution


















is given







Now we have



if




singularity




















pminusrarr S

Where






s isin S





)otimesm)oror



PNS



























References



1970602





0021869374901021




USSR (in Russian) 236 (5) 1041-1044


USSR Izv 13 (1978) 499-555




978-3-319-24460-0_2














ASENS_2002_4_35_2_231_0pdf





ihespdf










0003PAMQ-2008-0004-0003-a009pdf











reid2pdf








2FBF01388892page-1




I Introduction

II Preliminaries







IIVII Set-up






IVI KF-nil curves


V Configurations





References





+g|(y=0)partpartz






singularity




partpartz

Z has an




to Z


singularity


diction










+ λy partparty


+ f partparty

+ g partpartz

for some func-

tions k f g



























components






+ f partparty

+ g partpartz

for some functions









contradiction




+ λy partparty


+ f partparty

+ g partpartz

for some functions

k f g







+g|(y=0)partpartz





0








them




singularity


0




are in the axes


+ λy partparty

+ microz partpartz




+ microz partpartz





axis x = z = 0



x = y = 0 x = z = 0 y = z = 0


next item




+ microz partpartz









x = y = 0 x = z = 0 y = z = 0




















ni (2)a i (3)b i (3)c








versely







xz



y

LC-center



y

LC-center














surface x = 0
































1 C 0 This proves





statement as well































along C






)

partk+1

y partk

yH0(COCk+1












partx+ λy

part

party+ microz

part


partξ+ (λ+ εn)η

part


part

partζ




partx+ λy

part

party+ microz

part


partξ+ (λ+ εn)η

part


part

partζ





eigenvalues


are called ends









smooth locus of p




∆













(25) part0 = xpart

partx+ microyazb

part



part

partζ(mod(I2C))



(26) x1 = x y1 = y z1 = zyminus1


(27)

part01 = x1part


part


part



part

partζ1(mod(I2C1

))




































is projective



























2 = (x0 = w0 = 0)













unless a = 0


































z







the beast






















(32) 2g(Z0)minus 2 + sZG(Z0) = 0















V Configurations




















∆



equation 23





First we need







(33) n+m = d+ e






































of p fi











curves



(34) x = η y = ξ z = ζ





attached at one end





























of FC






is empty
























trees








following obvious













chain


normal bundle













we conclude

Our new set-up is













not mentioned is


small Kronecker if















Kronecker











imal surface Fk




















shortly



bull F is minimal


rational curve


trivial along s























projection onto P1


gular fiber X0










invariant fiber






























F prime F+

f prime+

f primeprime+

t+u









KF is big














fiber































totally harmless






































form













sumCsubeCk+1

F (C))red


divisors


(36)















(38) (F Nminus1 + F



(39)






(40) (F 0 + F







Theorem I









resolution


















is given







Now we have



if




singularity




















pminusrarr S

Where






s isin S





)otimesm)oror



PNS



























References



1970602





0021869374901021




USSR (in Russian) 236 (5) 1041-1044


USSR Izv 13 (1978) 499-555




978-3-319-24460-0_2














ASENS_2002_4_35_2_231_0pdf





ihespdf










0003PAMQ-2008-0004-0003-a009pdf











reid2pdf








2FBF01388892page-1




I Introduction

II Preliminaries







IIVII Set-up






IVI KF-nil curves


V Configurations





References




+ λy partparty


+ f partparty

+ g partpartz

for some func-

tions k f g



























components






+ f partparty

+ g partpartz

for some functions









contradiction




+ λy partparty


+ f partparty

+ g partpartz

for some functions

k f g







+g|(y=0)partpartz





0








them




singularity


0




are in the axes


+ λy partparty

+ microz partpartz




+ microz partpartz





axis x = z = 0



x = y = 0 x = z = 0 y = z = 0


next item




+ microz partpartz









x = y = 0 x = z = 0 y = z = 0




















ni (2)a i (3)b i (3)c








versely







xz



y

LC-center



y

LC-center














surface x = 0
































1 C 0 This proves





statement as well































along C






)

partk+1

y partk

yH0(COCk+1












partx+ λy

part

party+ microz

part


partξ+ (λ+ εn)η

part


part

partζ




partx+ λy

part

party+ microz

part


partξ+ (λ+ εn)η

part


part

partζ





eigenvalues


are called ends









smooth locus of p




∆













(25) part0 = xpart

partx+ microyazb

part



part

partζ(mod(I2C))



(26) x1 = x y1 = y z1 = zyminus1


(27)

part01 = x1part


part


part



part

partζ1(mod(I2C1

))




































is projective



























2 = (x0 = w0 = 0)













unless a = 0


































z







the beast






















(32) 2g(Z0)minus 2 + sZG(Z0) = 0















V Configurations




















∆



equation 23





First we need







(33) n+m = d+ e






































of p fi











curves



(34) x = η y = ξ z = ζ





attached at one end





























of FC






is empty
























trees








following obvious













chain


normal bundle













we conclude

Our new set-up is













not mentioned is


small Kronecker if















Kronecker











imal surface Fk




















shortly



bull F is minimal


rational curve


trivial along s























projection onto P1


gular fiber X0










invariant fiber






























F prime F+

f prime+

f primeprime+

t+u









KF is big














fiber































totally harmless






































form













sumCsubeCk+1

F (C))red


divisors


(36)















(38) (F Nminus1 + F



(39)






(40) (F 0 + F







Theorem I









resolution


















is given







Now we have



if




singularity




















pminusrarr S

Where






s isin S





)otimesm)oror



PNS



























References



1970602





0021869374901021




USSR (in Russian) 236 (5) 1041-1044


USSR Izv 13 (1978) 499-555




978-3-319-24460-0_2














ASENS_2002_4_35_2_231_0pdf





ihespdf










0003PAMQ-2008-0004-0003-a009pdf











reid2pdf








2FBF01388892page-1




I Introduction

II Preliminaries







IIVII Set-up






IVI KF-nil curves


V Configurations





References



components






+ f partparty

+ g partpartz

for some functions









contradiction




+ λy partparty


+ f partparty

+ g partpartz

for some functions

k f g







+g|(y=0)partpartz





0








them




singularity


0




are in the axes


+ λy partparty

+ microz partpartz




+ microz partpartz





axis x = z = 0



x = y = 0 x = z = 0 y = z = 0


next item




+ microz partpartz









x = y = 0 x = z = 0 y = z = 0




















ni (2)a i (3)b i (3)c








versely







xz



y

LC-center



y

LC-center














surface x = 0
































1 C 0 This proves





statement as well































along C






)

partk+1

y partk

yH0(COCk+1












partx+ λy

part

party+ microz

part


partξ+ (λ+ εn)η

part


part

partζ




partx+ λy

part

party+ microz

part


partξ+ (λ+ εn)η

part


part

partζ





eigenvalues


are called ends









smooth locus of p




∆













(25) part0 = xpart

partx+ microyazb

part



part

partζ(mod(I2C))



(26) x1 = x y1 = y z1 = zyminus1


(27)

part01 = x1part


part


part



part

partζ1(mod(I2C1

))




































is projective



























2 = (x0 = w0 = 0)













unless a = 0


































z







the beast






















(32) 2g(Z0)minus 2 + sZG(Z0) = 0















V Configurations




















∆



equation 23





First we need







(33) n+m = d+ e






































of p fi











curves



(34) x = η y = ξ z = ζ





attached at one end





























of FC






is empty
























trees








following obvious













chain


normal bundle













we conclude

Our new set-up is













not mentioned is


small Kronecker if















Kronecker











imal surface Fk




















shortly



bull F is minimal


rational curve


trivial along s























projection onto P1


gular fiber X0










invariant fiber






























F prime F+

f prime+

f primeprime+

t+u









KF is big














fiber































totally harmless






































form













sumCsubeCk+1

F (C))red


divisors


(36)















(38) (F Nminus1 + F



(39)






(40) (F 0 + F







Theorem I









resolution


















is given







Now we have



if




singularity




















pminusrarr S

Where






s isin S





)otimesm)oror



PNS



























References



1970602





0021869374901021




USSR (in Russian) 236 (5) 1041-1044


USSR Izv 13 (1978) 499-555




978-3-319-24460-0_2














ASENS_2002_4_35_2_231_0pdf





ihespdf










0003PAMQ-2008-0004-0003-a009pdf











reid2pdf








2FBF01388892page-1




I Introduction

II Preliminaries







IIVII Set-up






IVI KF-nil curves


V Configurations





References




+g|(y=0)partpartz





0








them




singularity


0




are in the axes


+ λy partparty

+ microz partpartz




+ microz partpartz





axis x = z = 0



x = y = 0 x = z = 0 y = z = 0


next item




+ microz partpartz









x = y = 0 x = z = 0 y = z = 0




















ni (2)a i (3)b i (3)c








versely







xz



y

LC-center



y

LC-center














surface x = 0
































1 C 0 This proves





statement as well































along C






)

partk+1

y partk

yH0(COCk+1












partx+ λy

part

party+ microz

part


partξ+ (λ+ εn)η

part


part

partζ




partx+ λy

part

party+ microz

part


partξ+ (λ+ εn)η

part


part

partζ





eigenvalues


are called ends









smooth locus of p




∆













(25) part0 = xpart

partx+ microyazb

part



part

partζ(mod(I2C))



(26) x1 = x y1 = y z1 = zyminus1


(27)

part01 = x1part


part


part



part

partζ1(mod(I2C1

))




































is projective



























2 = (x0 = w0 = 0)













unless a = 0


































z







the beast






















(32) 2g(Z0)minus 2 + sZG(Z0) = 0















V Configurations




















∆



equation 23





First we need







(33) n+m = d+ e






































of p fi











curves



(34) x = η y = ξ z = ζ





attached at one end





























of FC






is empty
























trees








following obvious













chain


normal bundle













we conclude

Our new set-up is













not mentioned is


small Kronecker if















Kronecker











imal surface Fk




















shortly



bull F is minimal


rational curve


trivial along s























projection onto P1


gular fiber X0










invariant fiber






























F prime F+

f prime+

f primeprime+

t+u









KF is big














fiber































totally harmless






































form













sumCsubeCk+1

F (C))red


divisors


(36)















(38) (F Nminus1 + F



(39)






(40) (F 0 + F







Theorem I









resolution


















is given







Now we have



if




singularity




















pminusrarr S

Where






s isin S





)otimesm)oror



PNS



























References



1970602





0021869374901021




USSR (in Russian) 236 (5) 1041-1044


USSR Izv 13 (1978) 499-555




978-3-319-24460-0_2














ASENS_2002_4_35_2_231_0pdf





ihespdf










0003PAMQ-2008-0004-0003-a009pdf











reid2pdf








2FBF01388892page-1




I Introduction

II Preliminaries







IIVII Set-up






IVI KF-nil curves


V Configurations





References




axis x = z = 0



x = y = 0 x = z = 0 y = z = 0


next item




+ microz partpartz









x = y = 0 x = z = 0 y = z = 0




















ni (2)a i (3)b i (3)c








versely







xz



y

LC-center



y

LC-center














surface x = 0
































1 C 0 This proves





statement as well































along C






)

partk+1

y partk

yH0(COCk+1












partx+ λy

part

party+ microz

part


partξ+ (λ+ εn)η

part


part

partζ




partx+ λy

part

party+ microz

part


partξ+ (λ+ εn)η

part


part

partζ





eigenvalues


are called ends









smooth locus of p




∆













(25) part0 = xpart

partx+ microyazb

part



part

partζ(mod(I2C))



(26) x1 = x y1 = y z1 = zyminus1


(27)

part01 = x1part


part


part



part

partζ1(mod(I2C1

))




































is projective



























2 = (x0 = w0 = 0)













unless a = 0


































z







the beast






















(32) 2g(Z0)minus 2 + sZG(Z0) = 0















V Configurations




















∆



equation 23





First we need







(33) n+m = d+ e






































of p fi











curves



(34) x = η y = ξ z = ζ





attached at one end





























of FC






is empty
























trees








following obvious













chain


normal bundle













we conclude

Our new set-up is













not mentioned is


small Kronecker if















Kronecker











imal surface Fk




















shortly



bull F is minimal


rational curve


trivial along s























projection onto P1


gular fiber X0










invariant fiber






























F prime F+

f prime+

f primeprime+

t+u









KF is big














fiber































totally harmless






































form













sumCsubeCk+1

F (C))red


divisors


(36)















(38) (F Nminus1 + F



(39)






(40) (F 0 + F







Theorem I









resolution


















is given







Now we have



if




singularity




















pminusrarr S

Where






s isin S





)otimesm)oror



PNS



























References



1970602





0021869374901021




USSR (in Russian) 236 (5) 1041-1044


USSR Izv 13 (1978) 499-555




978-3-319-24460-0_2














ASENS_2002_4_35_2_231_0pdf





ihespdf










0003PAMQ-2008-0004-0003-a009pdf











reid2pdf








2FBF01388892page-1




I Introduction

II Preliminaries







IIVII Set-up






IVI KF-nil curves


V Configurations





References












ni (2)a i (3)b i (3)c








versely







xz



y

LC-center



y

LC-center














surface x = 0
































1 C 0 This proves





statement as well































along C






)

partk+1

y partk

yH0(COCk+1












partx+ λy

part

party+ microz

part


partξ+ (λ+ εn)η

part


part

partζ




partx+ λy

part

party+ microz

part


partξ+ (λ+ εn)η

part


part

partζ





eigenvalues


are called ends









smooth locus of p




∆













(25) part0 = xpart

partx+ microyazb

part



part

partζ(mod(I2C))



(26) x1 = x y1 = y z1 = zyminus1


(27)

part01 = x1part


part


part



part

partζ1(mod(I2C1

))




































is projective



























2 = (x0 = w0 = 0)













unless a = 0


































z







the beast






















(32) 2g(Z0)minus 2 + sZG(Z0) = 0















V Configurations




















∆



equation 23





First we need







(33) n+m = d+ e






































of p fi











curves



(34) x = η y = ξ z = ζ





attached at one end





























of FC






is empty
























trees








following obvious













chain


normal bundle













we conclude

Our new set-up is













not mentioned is


small Kronecker if















Kronecker











imal surface Fk




















shortly



bull F is minimal


rational curve


trivial along s























projection onto P1


gular fiber X0










invariant fiber






























F prime F+

f prime+

f primeprime+

t+u









KF is big














fiber































totally harmless






































form













sumCsubeCk+1

F (C))red


divisors


(36)















(38) (F Nminus1 + F



(39)






(40) (F 0 + F







Theorem I









resolution


















is given







Now we have



if




singularity




















pminusrarr S

Where






s isin S





)otimesm)oror



PNS



























References



1970602





0021869374901021




USSR (in Russian) 236 (5) 1041-1044


USSR Izv 13 (1978) 499-555




978-3-319-24460-0_2














ASENS_2002_4_35_2_231_0pdf





ihespdf










0003PAMQ-2008-0004-0003-a009pdf











reid2pdf








2FBF01388892page-1




I Introduction

II Preliminaries







IIVII Set-up






IVI KF-nil curves


V Configurations





References



y

LC-center



y

LC-center














surface x = 0
































1 C 0 This proves





statement as well































along C






)

partk+1

y partk

yH0(COCk+1












partx+ λy

part

party+ microz

part


partξ+ (λ+ εn)η

part


part

partζ




partx+ λy

part

party+ microz

part


partξ+ (λ+ εn)η

part


part

partζ





eigenvalues


are called ends









smooth locus of p




∆













(25) part0 = xpart

partx+ microyazb

part



part

partζ(mod(I2C))



(26) x1 = x y1 = y z1 = zyminus1


(27)

part01 = x1part


part


part



part

partζ1(mod(I2C1

))




































is projective



























2 = (x0 = w0 = 0)













unless a = 0


































z







the beast






















(32) 2g(Z0)minus 2 + sZG(Z0) = 0















V Configurations




















∆



equation 23





First we need







(33) n+m = d+ e






































of p fi











curves



(34) x = η y = ξ z = ζ





attached at one end





























of FC






is empty
























trees








following obvious













chain


normal bundle













we conclude

Our new set-up is













not mentioned is


small Kronecker if















Kronecker











imal surface Fk




















shortly



bull F is minimal


rational curve


trivial along s























projection onto P1


gular fiber X0










invariant fiber






























F prime F+

f prime+

f primeprime+

t+u









KF is big














fiber































totally harmless






































form













sumCsubeCk+1

F (C))red


divisors


(36)















(38) (F Nminus1 + F



(39)






(40) (F 0 + F







Theorem I









resolution


















is given







Now we have



if




singularity




















pminusrarr S

Where






s isin S





)otimesm)oror



PNS



























References



1970602





0021869374901021




USSR (in Russian) 236 (5) 1041-1044


USSR Izv 13 (1978) 499-555




978-3-319-24460-0_2














ASENS_2002_4_35_2_231_0pdf





ihespdf










0003PAMQ-2008-0004-0003-a009pdf











reid2pdf








2FBF01388892page-1




I Introduction

II Preliminaries







IIVII Set-up






IVI KF-nil curves


V Configurations





References
































1 C 0 This proves





statement as well































along C






)

partk+1

y partk

yH0(COCk+1












partx+ λy

part

party+ microz

part


partξ+ (λ+ εn)η

part


part

partζ




partx+ λy

part

party+ microz

part


partξ+ (λ+ εn)η

part


part

partζ





eigenvalues


are called ends









smooth locus of p




∆













(25) part0 = xpart

partx+ microyazb

part



part

partζ(mod(I2C))



(26) x1 = x y1 = y z1 = zyminus1


(27)

part01 = x1part


part


part



part

partζ1(mod(I2C1

))




































is projective



























2 = (x0 = w0 = 0)













unless a = 0


































z







the beast






















(32) 2g(Z0)minus 2 + sZG(Z0) = 0















V Configurations




















∆



equation 23





First we need







(33) n+m = d+ e






































of p fi











curves



(34) x = η y = ξ z = ζ





attached at one end





























of FC






is empty
























trees








following obvious













chain


normal bundle













we conclude

Our new set-up is













not mentioned is


small Kronecker if















Kronecker











imal surface Fk




















shortly



bull F is minimal


rational curve


trivial along s























projection onto P1


gular fiber X0










invariant fiber






























F prime F+

f prime+

f primeprime+

t+u









KF is big














fiber































totally harmless






































form













sumCsubeCk+1

F (C))red


divisors


(36)















(38) (F Nminus1 + F



(39)






(40) (F 0 + F







Theorem I









resolution


















is given







Now we have



if




singularity




















pminusrarr S

Where






s isin S





)otimesm)oror



PNS



























References



1970602





0021869374901021




USSR (in Russian) 236 (5) 1041-1044


USSR Izv 13 (1978) 499-555




978-3-319-24460-0_2














ASENS_2002_4_35_2_231_0pdf





ihespdf










0003PAMQ-2008-0004-0003-a009pdf











reid2pdf








2FBF01388892page-1




I Introduction

II Preliminaries







IIVII Set-up






IVI KF-nil curves


V Configurations





References





statement as well































along C






)

partk+1

y partk

yH0(COCk+1












partx+ λy

part

party+ microz

part


partξ+ (λ+ εn)η

part


part

partζ




partx+ λy

part

party+ microz

part


partξ+ (λ+ εn)η

part


part

partζ





eigenvalues


are called ends









smooth locus of p




∆













(25) part0 = xpart

partx+ microyazb

part



part

partζ(mod(I2C))



(26) x1 = x y1 = y z1 = zyminus1


(27)

part01 = x1part


part


part



part

partζ1(mod(I2C1

))




































is projective



























2 = (x0 = w0 = 0)













unless a = 0


































z







the beast






















(32) 2g(Z0)minus 2 + sZG(Z0) = 0















V Configurations




















∆



equation 23





First we need







(33) n+m = d+ e






































of p fi











curves



(34) x = η y = ξ z = ζ





attached at one end





























of FC






is empty
























trees








following obvious













chain


normal bundle













we conclude

Our new set-up is













not mentioned is


small Kronecker if















Kronecker











imal surface Fk




















shortly



bull F is minimal


rational curve


trivial along s























projection onto P1


gular fiber X0










invariant fiber






























F prime F+

f prime+

f primeprime+

t+u









KF is big














fiber































totally harmless






































form













sumCsubeCk+1

F (C))red


divisors


(36)















(38) (F Nminus1 + F



(39)






(40) (F 0 + F







Theorem I









resolution


















is given







Now we have



if




singularity




















pminusrarr S

Where






s isin S





)otimesm)oror



PNS



























References



1970602





0021869374901021




USSR (in Russian) 236 (5) 1041-1044


USSR Izv 13 (1978) 499-555




978-3-319-24460-0_2














ASENS_2002_4_35_2_231_0pdf





ihespdf










0003PAMQ-2008-0004-0003-a009pdf











reid2pdf








2FBF01388892page-1




I Introduction

II Preliminaries







IIVII Set-up






IVI KF-nil curves


V Configurations





References






along C






)

partk+1

y partk

yH0(COCk+1












partx+ λy

part

party+ microz

part


partξ+ (λ+ εn)η

part


part

partζ




partx+ λy

part

party+ microz

part


partξ+ (λ+ εn)η

part


part

partζ





eigenvalues


are called ends









smooth locus of p




∆













(25) part0 = xpart

partx+ microyazb

part



part

partζ(mod(I2C))



(26) x1 = x y1 = y z1 = zyminus1


(27)

part01 = x1part


part


part



part

partζ1(mod(I2C1

))




































is projective



























2 = (x0 = w0 = 0)













unless a = 0


































z







the beast






















(32) 2g(Z0)minus 2 + sZG(Z0) = 0















V Configurations




















∆



equation 23





First we need







(33) n+m = d+ e






































of p fi











curves



(34) x = η y = ξ z = ζ





attached at one end





























of FC






is empty
























trees








following obvious













chain


normal bundle













we conclude

Our new set-up is













not mentioned is


small Kronecker if















Kronecker











imal surface Fk




















shortly



bull F is minimal


rational curve


trivial along s























projection onto P1


gular fiber X0










invariant fiber






























F prime F+

f prime+

f primeprime+

t+u









KF is big














fiber































totally harmless






































form













sumCsubeCk+1

F (C))red


divisors


(36)















(38) (F Nminus1 + F



(39)






(40) (F 0 + F







Theorem I









resolution


















is given







Now we have



if




singularity




















pminusrarr S

Where






s isin S





)otimesm)oror



PNS



























References



1970602





0021869374901021




USSR (in Russian) 236 (5) 1041-1044


USSR Izv 13 (1978) 499-555




978-3-319-24460-0_2














ASENS_2002_4_35_2_231_0pdf





ihespdf










0003PAMQ-2008-0004-0003-a009pdf











reid2pdf








2FBF01388892page-1




I Introduction

II Preliminaries







IIVII Set-up






IVI KF-nil curves


V Configurations





References





eigenvalues


are called ends









smooth locus of p




∆













(25) part0 = xpart

partx+ microyazb

part



part

partζ(mod(I2C))



(26) x1 = x y1 = y z1 = zyminus1


(27)

part01 = x1part


part


part



part

partζ1(mod(I2C1

))




































is projective



























2 = (x0 = w0 = 0)













unless a = 0


































z







the beast






















(32) 2g(Z0)minus 2 + sZG(Z0) = 0















V Configurations




















∆



equation 23





First we need







(33) n+m = d+ e






































of p fi











curves



(34) x = η y = ξ z = ζ





attached at one end





























of FC






is empty
























trees








following obvious













chain


normal bundle













we conclude

Our new set-up is













not mentioned is


small Kronecker if















Kronecker











imal surface Fk




















shortly



bull F is minimal


rational curve


trivial along s























projection onto P1


gular fiber X0










invariant fiber






























F prime F+

f prime+

f primeprime+

t+u









KF is big














fiber































totally harmless






































form













sumCsubeCk+1

F (C))red


divisors


(36)















(38) (F Nminus1 + F



(39)






(40) (F 0 + F







Theorem I









resolution


















is given







Now we have



if




singularity




















pminusrarr S

Where






s isin S





)otimesm)oror



PNS



























References



1970602





0021869374901021




USSR (in Russian) 236 (5) 1041-1044


USSR Izv 13 (1978) 499-555




978-3-319-24460-0_2














ASENS_2002_4_35_2_231_0pdf





ihespdf










0003PAMQ-2008-0004-0003-a009pdf











reid2pdf








2FBF01388892page-1




I Introduction

II Preliminaries







IIVII Set-up






IVI KF-nil curves


V Configurations





References





(25) part0 = xpart

partx+ microyazb

part



part

partζ(mod(I2C))



(26) x1 = x y1 = y z1 = zyminus1


(27)

part01 = x1part


part


part



part

partζ1(mod(I2C1

))




































is projective



























2 = (x0 = w0 = 0)













unless a = 0


































z







the beast






















(32) 2g(Z0)minus 2 + sZG(Z0) = 0















V Configurations




















∆



equation 23





First we need







(33) n+m = d+ e






































of p fi











curves



(34) x = η y = ξ z = ζ





attached at one end





























of FC






is empty
























trees








following obvious













chain


normal bundle













we conclude

Our new set-up is













not mentioned is


small Kronecker if















Kronecker











imal surface Fk




















shortly



bull F is minimal


rational curve


trivial along s























projection onto P1


gular fiber X0










invariant fiber






























F prime F+

f prime+

f primeprime+

t+u









KF is big














fiber































totally harmless






































form













sumCsubeCk+1

F (C))red


divisors


(36)















(38) (F Nminus1 + F



(39)






(40) (F 0 + F







Theorem I









resolution


















is given







Now we have



if




singularity




















pminusrarr S

Where






s isin S





)otimesm)oror



PNS



























References



1970602





0021869374901021




USSR (in Russian) 236 (5) 1041-1044


USSR Izv 13 (1978) 499-555




978-3-319-24460-0_2














ASENS_2002_4_35_2_231_0pdf





ihespdf










0003PAMQ-2008-0004-0003-a009pdf











reid2pdf








2FBF01388892page-1




I Introduction

II Preliminaries







IIVII Set-up






IVI KF-nil curves


V Configurations





References




















is projective



























2 = (x0 = w0 = 0)













unless a = 0


































z







the beast






















(32) 2g(Z0)minus 2 + sZG(Z0) = 0















V Configurations




















∆



equation 23





First we need







(33) n+m = d+ e






































of p fi











curves



(34) x = η y = ξ z = ζ





attached at one end





























of FC






is empty
























trees








following obvious













chain


normal bundle













we conclude

Our new set-up is













not mentioned is


small Kronecker if















Kronecker











imal surface Fk




















shortly



bull F is minimal


rational curve


trivial along s























projection onto P1


gular fiber X0










invariant fiber






























F prime F+

f prime+

f primeprime+

t+u









KF is big














fiber































totally harmless






































form













sumCsubeCk+1

F (C))red


divisors


(36)















(38) (F Nminus1 + F



(39)






(40) (F 0 + F







Theorem I









resolution


















is given







Now we have



if




singularity




















pminusrarr S

Where






s isin S





)otimesm)oror



PNS



























References



1970602





0021869374901021




USSR (in Russian) 236 (5) 1041-1044


USSR Izv 13 (1978) 499-555




978-3-319-24460-0_2














ASENS_2002_4_35_2_231_0pdf





ihespdf










0003PAMQ-2008-0004-0003-a009pdf











reid2pdf








2FBF01388892page-1




I Introduction

II Preliminaries







IIVII Set-up






IVI KF-nil curves


V Configurations





References

















2 = (x0 = w0 = 0)













unless a = 0


































z







the beast






















(32) 2g(Z0)minus 2 + sZG(Z0) = 0















V Configurations




















∆



equation 23





First we need







(33) n+m = d+ e






































of p fi











curves



(34) x = η y = ξ z = ζ





attached at one end





























of FC






is empty
























trees








following obvious













chain


normal bundle













we conclude

Our new set-up is













not mentioned is


small Kronecker if















Kronecker











imal surface Fk




















shortly



bull F is minimal


rational curve


trivial along s























projection onto P1


gular fiber X0










invariant fiber






























F prime F+

f prime+

f primeprime+

t+u









KF is big














fiber































totally harmless






































form













sumCsubeCk+1

F (C))red


divisors


(36)















(38) (F Nminus1 + F



(39)






(40) (F 0 + F







Theorem I









resolution


















is given







Now we have



if




singularity




















pminusrarr S

Where






s isin S





)otimesm)oror



PNS



























References



1970602





0021869374901021




USSR (in Russian) 236 (5) 1041-1044


USSR Izv 13 (1978) 499-555




978-3-319-24460-0_2














ASENS_2002_4_35_2_231_0pdf





ihespdf










0003PAMQ-2008-0004-0003-a009pdf











reid2pdf








2FBF01388892page-1




I Introduction

II Preliminaries







IIVII Set-up






IVI KF-nil curves


V Configurations





References































z







the beast






















(32) 2g(Z0)minus 2 + sZG(Z0) = 0















V Configurations




















∆



equation 23





First we need







(33) n+m = d+ e






































of p fi











curves



(34) x = η y = ξ z = ζ





attached at one end





























of FC






is empty
























trees








following obvious













chain


normal bundle













we conclude

Our new set-up is













not mentioned is


small Kronecker if















Kronecker











imal surface Fk




















shortly



bull F is minimal


rational curve


trivial along s























projection onto P1


gular fiber X0










invariant fiber






























F prime F+

f prime+

f primeprime+

t+u









KF is big














fiber































totally harmless






































form













sumCsubeCk+1

F (C))red


divisors


(36)















(38) (F Nminus1 + F



(39)






(40) (F 0 + F







Theorem I









resolution


















is given







Now we have



if




singularity




















pminusrarr S

Where






s isin S





)otimesm)oror



PNS



























References



1970602





0021869374901021




USSR (in Russian) 236 (5) 1041-1044


USSR Izv 13 (1978) 499-555




978-3-319-24460-0_2














ASENS_2002_4_35_2_231_0pdf





ihespdf










0003PAMQ-2008-0004-0003-a009pdf











reid2pdf








2FBF01388892page-1




I Introduction

II Preliminaries







IIVII Set-up






IVI KF-nil curves


V Configurations





References



(32) 2g(Z0)minus 2 + sZG(Z0) = 0















V Configurations




















∆



equation 23





First we need







(33) n+m = d+ e






































of p fi











curves



(34) x = η y = ξ z = ζ





attached at one end





























of FC






is empty
























trees








following obvious













chain


normal bundle













we conclude

Our new set-up is













not mentioned is


small Kronecker if















Kronecker











imal surface Fk




















shortly



bull F is minimal


rational curve


trivial along s























projection onto P1


gular fiber X0










invariant fiber






























F prime F+

f prime+

f primeprime+

t+u









KF is big














fiber































totally harmless






































form













sumCsubeCk+1

F (C))red


divisors


(36)















(38) (F Nminus1 + F



(39)






(40) (F 0 + F







Theorem I









resolution


















is given







Now we have



if




singularity




















pminusrarr S

Where






s isin S





)otimesm)oror



PNS



























References



1970602





0021869374901021




USSR (in Russian) 236 (5) 1041-1044


USSR Izv 13 (1978) 499-555




978-3-319-24460-0_2














ASENS_2002_4_35_2_231_0pdf





ihespdf










0003PAMQ-2008-0004-0003-a009pdf











reid2pdf








2FBF01388892page-1




I Introduction

II Preliminaries







IIVII Set-up






IVI KF-nil curves


V Configurations





References










∆



equation 23





First we need







(33) n+m = d+ e






































of p fi











curves



(34) x = η y = ξ z = ζ





attached at one end





























of FC






is empty
























trees








following obvious













chain


normal bundle













we conclude

Our new set-up is













not mentioned is


small Kronecker if















Kronecker











imal surface Fk




















shortly



bull F is minimal


rational curve


trivial along s























projection onto P1


gular fiber X0










invariant fiber






























F prime F+

f prime+

f primeprime+

t+u









KF is big














fiber































totally harmless






































form













sumCsubeCk+1

F (C))red


divisors


(36)















(38) (F Nminus1 + F



(39)






(40) (F 0 + F







Theorem I









resolution


















is given







Now we have



if




singularity




















pminusrarr S

Where






s isin S





)otimesm)oror



PNS



























References



1970602





0021869374901021




USSR (in Russian) 236 (5) 1041-1044


USSR Izv 13 (1978) 499-555




978-3-319-24460-0_2














ASENS_2002_4_35_2_231_0pdf





ihespdf










0003PAMQ-2008-0004-0003-a009pdf











reid2pdf








2FBF01388892page-1




I Introduction

II Preliminaries







IIVII Set-up






IVI KF-nil curves


V Configurations





References



































of p fi











curves



(34) x = η y = ξ z = ζ





attached at one end





























of FC






is empty
























trees








following obvious













chain


normal bundle













we conclude

Our new set-up is













not mentioned is


small Kronecker if















Kronecker











imal surface Fk




















shortly



bull F is minimal


rational curve


trivial along s























projection onto P1


gular fiber X0










invariant fiber






























F prime F+

f prime+

f primeprime+

t+u









KF is big














fiber































totally harmless






































form













sumCsubeCk+1

F (C))red


divisors


(36)















(38) (F Nminus1 + F



(39)






(40) (F 0 + F







Theorem I









resolution


















is given







Now we have



if




singularity




















pminusrarr S

Where






s isin S





)otimesm)oror



PNS



























References



1970602





0021869374901021




USSR (in Russian) 236 (5) 1041-1044


USSR Izv 13 (1978) 499-555




978-3-319-24460-0_2














ASENS_2002_4_35_2_231_0pdf





ihespdf










0003PAMQ-2008-0004-0003-a009pdf











reid2pdf








2FBF01388892page-1




I Introduction

II Preliminaries







IIVII Set-up






IVI KF-nil curves


V Configurations





References





























of FC






is empty
























trees








following obvious













chain


normal bundle













we conclude

Our new set-up is













not mentioned is


small Kronecker if















Kronecker











imal surface Fk




















shortly



bull F is minimal


rational curve


trivial along s























projection onto P1


gular fiber X0










invariant fiber






























F prime F+

f prime+

f primeprime+

t+u









KF is big














fiber































totally harmless






































form













sumCsubeCk+1

F (C))red


divisors


(36)















(38) (F Nminus1 + F



(39)






(40) (F 0 + F







Theorem I









resolution


















is given







Now we have



if




singularity




















pminusrarr S

Where






s isin S





)otimesm)oror



PNS



























References



1970602





0021869374901021




USSR (in Russian) 236 (5) 1041-1044


USSR Izv 13 (1978) 499-555




978-3-319-24460-0_2














ASENS_2002_4_35_2_231_0pdf





ihespdf










0003PAMQ-2008-0004-0003-a009pdf











reid2pdf








2FBF01388892page-1




I Introduction

II Preliminaries







IIVII Set-up






IVI KF-nil curves


V Configurations





References






is empty
























trees








following obvious













chain


normal bundle













we conclude

Our new set-up is













not mentioned is


small Kronecker if















Kronecker











imal surface Fk




















shortly



bull F is minimal


rational curve


trivial along s























projection onto P1


gular fiber X0










invariant fiber






























F prime F+

f prime+

f primeprime+

t+u









KF is big














fiber































totally harmless






































form













sumCsubeCk+1

F (C))red


divisors


(36)















(38) (F Nminus1 + F



(39)






(40) (F 0 + F







Theorem I









resolution


















is given







Now we have



if




singularity




















pminusrarr S

Where






s isin S





)otimesm)oror



PNS



























References



1970602





0021869374901021




USSR (in Russian) 236 (5) 1041-1044


USSR Izv 13 (1978) 499-555




978-3-319-24460-0_2














ASENS_2002_4_35_2_231_0pdf





ihespdf










0003PAMQ-2008-0004-0003-a009pdf











reid2pdf








2FBF01388892page-1




I Introduction

II Preliminaries







IIVII Set-up






IVI KF-nil curves


V Configurations





References







following obvious













chain


normal bundle













we conclude

Our new set-up is













not mentioned is


small Kronecker if















Kronecker











imal surface Fk




















shortly



bull F is minimal


rational curve


trivial along s























projection onto P1


gular fiber X0










invariant fiber






























F prime F+

f prime+

f primeprime+

t+u









KF is big














fiber































totally harmless






































form













sumCsubeCk+1

F (C))red


divisors


(36)















(38) (F Nminus1 + F



(39)






(40) (F 0 + F







Theorem I









resolution


















is given







Now we have



if




singularity




















pminusrarr S

Where






s isin S





)otimesm)oror



PNS



























References



1970602





0021869374901021




USSR (in Russian) 236 (5) 1041-1044


USSR Izv 13 (1978) 499-555




978-3-319-24460-0_2














ASENS_2002_4_35_2_231_0pdf





ihespdf










0003PAMQ-2008-0004-0003-a009pdf











reid2pdf








2FBF01388892page-1




I Introduction

II Preliminaries







IIVII Set-up






IVI KF-nil curves


V Configurations





References








we conclude

Our new set-up is













not mentioned is


small Kronecker if















Kronecker











imal surface Fk




















shortly



bull F is minimal


rational curve


trivial along s























projection onto P1


gular fiber X0










invariant fiber






























F prime F+

f prime+

f primeprime+

t+u









KF is big














fiber































totally harmless






































form













sumCsubeCk+1

F (C))red


divisors


(36)















(38) (F Nminus1 + F



(39)






(40) (F 0 + F







Theorem I









resolution


















is given







Now we have



if




singularity




















pminusrarr S

Where






s isin S





)otimesm)oror



PNS



























References



1970602





0021869374901021




USSR (in Russian) 236 (5) 1041-1044


USSR Izv 13 (1978) 499-555




978-3-319-24460-0_2














ASENS_2002_4_35_2_231_0pdf





ihespdf










0003PAMQ-2008-0004-0003-a009pdf











reid2pdf








2FBF01388892page-1




I Introduction

II Preliminaries







IIVII Set-up






IVI KF-nil curves


V Configurations





References










Kronecker











imal surface Fk




















shortly



bull F is minimal


rational curve


trivial along s























projection onto P1


gular fiber X0










invariant fiber






























F prime F+

f prime+

f primeprime+

t+u









KF is big














fiber































totally harmless






































form













sumCsubeCk+1

F (C))red


divisors


(36)















(38) (F Nminus1 + F



(39)






(40) (F 0 + F







Theorem I









resolution


















is given







Now we have



if




singularity




















pminusrarr S

Where






s isin S





)otimesm)oror



PNS



























References



1970602





0021869374901021




USSR (in Russian) 236 (5) 1041-1044


USSR Izv 13 (1978) 499-555




978-3-319-24460-0_2














ASENS_2002_4_35_2_231_0pdf





ihespdf










0003PAMQ-2008-0004-0003-a009pdf











reid2pdf








2FBF01388892page-1




I Introduction

II Preliminaries







IIVII Set-up






IVI KF-nil curves


V Configurations





References










shortly



bull F is minimal


rational curve


trivial along s























projection onto P1


gular fiber X0










invariant fiber






























F prime F+

f prime+

f primeprime+

t+u









KF is big














fiber































totally harmless






































form













sumCsubeCk+1

F (C))red


divisors


(36)















(38) (F Nminus1 + F



(39)






(40) (F 0 + F







Theorem I









resolution


















is given







Now we have



if




singularity




















pminusrarr S

Where






s isin S





)otimesm)oror



PNS



























References



1970602





0021869374901021




USSR (in Russian) 236 (5) 1041-1044


USSR Izv 13 (1978) 499-555




978-3-319-24460-0_2














ASENS_2002_4_35_2_231_0pdf





ihespdf










0003PAMQ-2008-0004-0003-a009pdf











reid2pdf








2FBF01388892page-1




I Introduction

II Preliminaries







IIVII Set-up






IVI KF-nil curves


V Configurations





References












projection onto P1


gular fiber X0










invariant fiber






























F prime F+

f prime+

f primeprime+

t+u









KF is big














fiber































totally harmless






































form













sumCsubeCk+1

F (C))red


divisors


(36)















(38) (F Nminus1 + F



(39)






(40) (F 0 + F







Theorem I









resolution


















is given







Now we have



if




singularity




















pminusrarr S

Where






s isin S





)otimesm)oror



PNS



























References



1970602





0021869374901021




USSR (in Russian) 236 (5) 1041-1044


USSR Izv 13 (1978) 499-555




978-3-319-24460-0_2














ASENS_2002_4_35_2_231_0pdf





ihespdf










0003PAMQ-2008-0004-0003-a009pdf











reid2pdf








2FBF01388892page-1




I Introduction

II Preliminaries







IIVII Set-up






IVI KF-nil curves


V Configurations





References


























F prime F+

f prime+

f primeprime+

t+u









KF is big














fiber































totally harmless






































form













sumCsubeCk+1

F (C))red


divisors


(36)















(38) (F Nminus1 + F



(39)






(40) (F 0 + F







Theorem I









resolution


















is given







Now we have



if




singularity




















pminusrarr S

Where






s isin S





)otimesm)oror



PNS



























References



1970602





0021869374901021




USSR (in Russian) 236 (5) 1041-1044


USSR Izv 13 (1978) 499-555




978-3-319-24460-0_2














ASENS_2002_4_35_2_231_0pdf





ihespdf










0003PAMQ-2008-0004-0003-a009pdf











reid2pdf








2FBF01388892page-1




I Introduction

II Preliminaries







IIVII Set-up






IVI KF-nil curves


V Configurations





References









KF is big














fiber































totally harmless






































form













sumCsubeCk+1

F (C))red


divisors


(36)















(38) (F Nminus1 + F



(39)






(40) (F 0 + F







Theorem I









resolution


















is given







Now we have



if




singularity




















pminusrarr S

Where






s isin S





)otimesm)oror



PNS



























References



1970602





0021869374901021




USSR (in Russian) 236 (5) 1041-1044


USSR Izv 13 (1978) 499-555




978-3-319-24460-0_2














ASENS_2002_4_35_2_231_0pdf





ihespdf










0003PAMQ-2008-0004-0003-a009pdf











reid2pdf








2FBF01388892page-1




I Introduction

II Preliminaries







IIVII Set-up






IVI KF-nil curves


V Configurations





References


























totally harmless






































form













sumCsubeCk+1

F (C))red


divisors


(36)















(38) (F Nminus1 + F



(39)






(40) (F 0 + F







Theorem I









resolution


















is given







Now we have



if




singularity




















pminusrarr S

Where






s isin S





)otimesm)oror



PNS



























References



1970602





0021869374901021




USSR (in Russian) 236 (5) 1041-1044


USSR Izv 13 (1978) 499-555




978-3-319-24460-0_2














ASENS_2002_4_35_2_231_0pdf





ihespdf










0003PAMQ-2008-0004-0003-a009pdf











reid2pdf








2FBF01388892page-1




I Introduction

II Preliminaries







IIVII Set-up






IVI KF-nil curves


V Configurations





References






































form













sumCsubeCk+1

F (C))red


divisors


(36)















(38) (F Nminus1 + F



(39)






(40) (F 0 + F







Theorem I









resolution


















is given







Now we have



if




singularity




















pminusrarr S

Where






s isin S





)otimesm)oror



PNS



























References



1970602





0021869374901021




USSR (in Russian) 236 (5) 1041-1044


USSR Izv 13 (1978) 499-555




978-3-319-24460-0_2














ASENS_2002_4_35_2_231_0pdf





ihespdf










0003PAMQ-2008-0004-0003-a009pdf











reid2pdf








2FBF01388892page-1




I Introduction

II Preliminaries







IIVII Set-up






IVI KF-nil curves


V Configurations





References











(38) (F Nminus1 + F



(39)






(40) (F 0 + F







Theorem I









resolution


















is given







Now we have



if




singularity




















pminusrarr S

Where






s isin S





)otimesm)oror



PNS



























References



1970602





0021869374901021




USSR (in Russian) 236 (5) 1041-1044


USSR Izv 13 (1978) 499-555




978-3-319-24460-0_2














ASENS_2002_4_35_2_231_0pdf





ihespdf










0003PAMQ-2008-0004-0003-a009pdf











reid2pdf








2FBF01388892page-1




I Introduction

II Preliminaries







IIVII Set-up






IVI KF-nil curves


V Configurations





References









resolution


















is given







Now we have



if




singularity




















pminusrarr S

Where






s isin S





)otimesm)oror



PNS



























References



1970602





0021869374901021




USSR (in Russian) 236 (5) 1041-1044


USSR Izv 13 (1978) 499-555




978-3-319-24460-0_2














ASENS_2002_4_35_2_231_0pdf





ihespdf










0003PAMQ-2008-0004-0003-a009pdf











reid2pdf








2FBF01388892page-1




I Introduction

II Preliminaries







IIVII Set-up






IVI KF-nil curves


V Configurations





References




Now we have



if




singularity




















pminusrarr S

Where






s isin S





)otimesm)oror



PNS



























References



1970602





0021869374901021




USSR (in Russian) 236 (5) 1041-1044


USSR Izv 13 (1978) 499-555




978-3-319-24460-0_2














ASENS_2002_4_35_2_231_0pdf





ihespdf










0003PAMQ-2008-0004-0003-a009pdf











reid2pdf








2FBF01388892page-1




I Introduction

II Preliminaries







IIVII Set-up






IVI KF-nil curves


V Configurations





References




pminusrarr S

Where






s isin S





)otimesm)oror



PNS



























References



1970602





0021869374901021




USSR (in Russian) 236 (5) 1041-1044


USSR Izv 13 (1978) 499-555




978-3-319-24460-0_2














ASENS_2002_4_35_2_231_0pdf





ihespdf










0003PAMQ-2008-0004-0003-a009pdf











reid2pdf








2FBF01388892page-1




I Introduction

II Preliminaries







IIVII Set-up






IVI KF-nil curves


V Configurations





References















References



1970602





0021869374901021




USSR (in Russian) 236 (5) 1041-1044


USSR Izv 13 (1978) 499-555




978-3-319-24460-0_2














ASENS_2002_4_35_2_231_0pdf





ihespdf










0003PAMQ-2008-0004-0003-a009pdf











reid2pdf








2FBF01388892page-1




I Introduction

II Preliminaries







IIVII Set-up






IVI KF-nil curves


V Configurations





References














ASENS_2002_4_35_2_231_0pdf





ihespdf










0003PAMQ-2008-0004-0003-a009pdf











reid2pdf








2FBF01388892page-1




I Introduction

II Preliminaries







IIVII Set-up






IVI KF-nil curves


V Configurations





References





reid2pdf








2FBF01388892page-1




I Introduction

II Preliminaries







IIVII Set-up






IVI KF-nil curves


V Configurations





References

stable reduction of foliated surfacesbuonerba/stable (1).pdfthis deep relation between algebraic...

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