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Tel Aviv UniversitySchool of Mathematical Sciences

Geometry of ObstructedEquisingular Families of Algebraic

Curves and Hypersurfaces

Thesis submitted for the degree“Doctor of Philosophy”

by

Anna Gourevitch

Under the Supervision ofProf. Eugenii Shustin

Submitted to the Scientific Council of theTel Aviv University

Tel Aviv, Israel

2

March 2008

This work was carried out under the supervision of

Prof. Eugenii Shustin

Introduction

Historical background

One of the main achievements of the algebraic geometry of the 20th century was torealize that it is fruitful not only to study algebraic varieties by themselves but alsoin families. That means that one should consider the space classifying all varietieswith given properties. These spaces are called moduli spaces. The advantage ofalgebraic geometry is that many times those spaces are themselves algebraic varietiesor schemes, and hence the same methods of algebraic geometry can be used to studythese moduli spaces.

One of the first moduli spaces considered in algebraic geometry were familiesof algebraic curves with given invariants and given set of singularity types. Alreadyin the end of the 19th century, the foundation was in the works of Plucker, Severi,Segre and Zariski. Later, the theory of equisingular families had found importantapplications in the singularity theory, the topology of complex algebraic curves andsurfaces, and in real algebraic geometry.

This thesis is devoted to the study of the so-called obstructed families of planecurves, and, more generally, projective hypersurfaces, of a given degree, having oneisolated singularity of prescribed type.

Let Σ be a smooth projective variety over the complex field C. Let D bean ample divisor on Σ. Denote by V = V|D|(S1, . . . , Sr) the set of hypersurfacesin the linear system |D| having r singular points of (topological or analytic) typesS1, . . . , Sr (as only singularities). One knows that V|D|(S1, . . . , Sr) can be identifiedwith a (locally closed) subscheme (”equisingular stratum”) in the Hilbert scheme ofhypersurfaces on Σ. The main questions concerning this space are

• Existence problem: Is V|D|(S1, . . . , Sr) non-empty, that is, does there exist ahypersurface F ∈ |D| with the given collection of singularities?

• Smoothness problem: If V|D|(S1, . . . , Sr) is non-empty, is it smooth?

• Dimension problem: If V|D|(S1, . . . , Sr) is non-empty, does it have an ”ex-pected” dimension (expressible via local invariants of the singularities)?

• Irreducibility problem: Is V|D|(S1, . . . , Sr) irreducible?

• Versality problem: Is the deformation of the multisingularity of a hypersur-face H ∈ V|D|(S1, . . . , Sr) induced by the linear system |D| versal (see section1.2.2)?

6

If V|D|(S1, . . . , Sr) is non-empty, smooth and has expected dimension, it is saidto be ”T-smooth”. It is known that in this case the deformation induced by |D| isversal.

The case of plane nodal curves has been settled completely. In 1920, Severigave answers to the first three questions: there exists an irreducible plane curve ofdegree d having n nodes as only singularities if and only if

0 ≤ n ≤ (d− 1)(d− 2)

2.

Furthermore, if V irrd (n·A1) is non-empty, then it is smooth of the expected dimension

d(d+3)2

− n. In 1985, Harris proved that V irrd (n · A1) is irreducible.

Already in the case of curves with nodes and cusps there is no complete answer.Segre [Seg, Tan] gave an example of equisingular stratum of such curves (having 6m2

cusps as their only singularities) which has a component of non-expected dimension.Zariski [Zar] gave the first example of a reducible equisingular stratum (of sexticcurves with 6 cusps). Finally, Wahl [Wah] gave the first example of a non-smoothequisingular stratum (of curves of degree 104 having 3636 nodes and 900 cusps).Some other examples can be found in [Lue], [GLS2], [Kei], [Mar].

So far, the main effort in the study of equisingular families has been concen-trated on obtaining criteria for the T-smoothness. In turn, the obstructed (i.e., non-T-smooth) equisingular families and non-versal deformations have not been studiedsystematically.

Main goals and motivation

The present thesis is concerned with the study of certain obstructed equisingularfamilies of plane curves, and, more generally, projective hypersurfaces with isolatedsingularities.

Geometry of equisingular families of minimal obstructness

The deformation theory leads to the following result: the variety V|D|(S1, . . . , Sr) isT-smooth at H ∈ V|D|(S1, . . . , Sr) if and only if

h1(JZes(H)/Σ(H)) = 0, respectively h1(JZea(H)/Σ(H)) = 0,

where Zea(H) and Zes(H) are certain zero-dimensional schemes and JZea(H)/Σ(H)and JZes(H)/Σ(H) are their defining ideals (see precise definition in section 1.1.7). Inour research we focus on the minimal obstructness case, i.e. h1(JZea(H)/Σ(H)) = 1.Our first goal was to begin a systematic study of the geometry of equisingular familiesof minimal obstructness. We started from an example suggested in [DPW] in orderto show sharpness of ”4d − 4” criterion (see example 1.4.6) for the T-smoothnessand versality.

7

Dependence of geometry on stabilization

In classical examples and in our results we have seen that the geometry of equi-singular families may be very different. The next question that naturally arises isthe behavior of the geometric properties of equianalytic families with respect to thestabilization of the singularities. Stabilization of the singularities is adding squaresof new variables to the space and their squares to the local equation of the hy-persurface. The adding of squares is a known method in local deformation theory,which preserves contact equivalence. Our second goal was to find stable geometricproperties of the geometry of obstructed equisingular families.

Deformation-theoretical meaning

Let H be a projective hypersurface with unique singular point z. It is importantto know which deformations of (H, z) appear in the deformation Dz,H,|H| inducedby the linear system H. If the deformation Dz,H,|H| is versal, then all possible localdeformations appear. It is a known theorem that Dz,H,|H| is versal if and only if theequisingular family of H is T-smooth. This theorem is widely used for T-smoothfamilies. However, for non-T-smooth families the question remains open.

Our third goal was to show that if the equisingular family of H has a reducedcomponent of expected dimension, then the deformation Dz,H,|H| is 1-complete (seeprecise definition in section 1.2), and hence includes all possible one-parametricdeformations of (H, z).

Main results and methods

First, we considered the plane curve C given by local equation xk+yl+λ1xd+λ2y

d =0 such that h1(JZea(C)/P2(d)) = 1. We studied the geometry of Vd,C(S), where Sis the analytic singularity type of C and d = k + l − 5. As a result, we haveobserved various geometric phenomena: for d = 5, Vd,C(S) is smooth of non-expecteddimension; for d = 6, Vd,C(S) is non-reduced with smooth reduction of expecteddimension; for d = 7, Vd,C(S) is reducible with smooth irreducible components ofexpected dimension and smooth intersection locus; and for d ≥ 8, Vd,C(S) is reducedirreducible non-smooth variety of expected dimension with smooth singular locus.

With the same methods, we have obtained similar result for quasihomogeneoushypersurface singularities in Pn. Among them was a new example of smooth equisin-gular family of non-expected dimension: V3,H(S) where H is given by local equationx3 + y3 + z3 + w3 = 0.

The next question that naturally arose was the behavior of these geomet-ric properties of equianalytic families, with respect to the stabilization of the sin-gularities. We found out that though stabilization preserves both Tjurina alge-bra and h1(JZea(H)/Pn(H)), it can change the geometry of the equianalytic fam-

8

ily radically. We have shown that for any hypersurface H of degree d satisfyingh1(JZea(H)/Pn(2d − 2)) = 0, after adding enough squares the obtained family has areduced component of expected dimension. We have also shown that the conditionh1(JZea(H)/Pn(2d− 2)) = 0 always holds for plane curves.

The next result is concerned with deformation theory. Suppose that the germof equianalytic family has a reduced irreducible component of expected dimension.In this case the family is T-smooth at every its regular point which lies in thatcomponent. It means that our singular hypersurface H has a deformation X → Tsuch that for t 6= t0 the deformation of Xt induced by the linear system |H| is versal.We show that this implies that the deformation of H induced by the linear system|H| is 1-complete (see section 1.2 for precise definition).

The methods that we use are the technique of cohomologies of ideal sheavesof zero-dimensional schemes associated with analytic types of singularities, methodsfor their calculation, and H1-vanishing theorems. We also use the algorithms ofcomputer algebra (see [GP]) as a technical tool in the proof of the theorems.

The structure of the thesis

This thesis is organized in the following way. Chapter 1 is dedicated to the formu-lation of the necessary notions and background. We do not present the proofs ofthese classical results here but rather give references to the literature.

In section 1.1 we introduce the notions of singularity theory such as analyticsingularity types, quasihomogeneous singularities, zero-dimensional schemes associ-ated with singularities and Castelnuovo function. The main theorems of this sectionare the Mather-Yau theorem 1.1.5, finite determinacy theorem 1.1.8, theorem 1.1.15on semiquasihomogeneous polynomials, theorem 1.1.23 which gives cohomologicalcriteria of T-smoothness, and lemmas 1.1.27 and 1.1.28 on Castelnuovo function.

In section 1.2 we introduce the notions of deformation theory such as completeand versal deformations. The most important theorem for us in this section istheorem 1.2.10 on versality of deformation induced by a complete linear system.

In section 1.3 we introduce notions and algorithms of computer algebra. Themost important notions for us are the notion of normal form and the RedNFBuch-berger algorithm for its computation.

In section 1.4 we present the known examples of obstructed equisingular fam-ilies.

In Chapter 2 we formulate our main results. These can be divided into threeparts. In section 2.1 we present results concerning equianalytic families of curveshypersurfaces with quasihomogeneous singularities of minimal obstructness . Section2.2 contains results on stable properties of obstructed equianalytic families. Insection 2.3 we give an application of the obtained results to the deformation theory.

9

In Chapter 3 we prove the results on families of curves hypersurfaces withquasihomogeneous singularities of minimal obstructness.

In Chapter 4 we prove the results on stable properties of obstructed equiana-lytic families.

Acknowledgements

First of all I would like to thank my supervisor, Prof. Eugenii Shustin for hishelp, his attitude and his patience during my research work.

Next I would like to thank my M.Sc. thesis supervisor Prof. Grigorii Polo-tovskii for introducing me to the magnificent world of algebraic curves.

I would also like to thank Prof. Joseph Bernstein for teaching me basics ofAlgebraic Geometry and his warm support during my studies.

Finally, I am pleased to thank my husband and colleague Dmitry Gourevitchfor helpful discussions.

This research was supported by Hermann-Minkowski-Minerva Center for Ge-ometry at the Tel-Aviv University, by the grant no. 465/04 from the Israel ScienceFoundation.

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1 Preliminaries 131.1 Notions of singularity theory . . . . . . . . . . . . . . . . . . . . . . . 13

1.1.1 Analytic types of hypersurface singularities . . . . . . . . . . . 131.1.2 Finite determinacy . . . . . . . . . . . . . . . . . . . . . . . . 141.1.3 Stable equivalence . . . . . . . . . . . . . . . . . . . . . . . . 151.1.4 Quasihomogeneous singularities . . . . . . . . . . . . . . . . . 151.1.5 Newton polytope . . . . . . . . . . . . . . . . . . . . . . . . . 161.1.6 Families of hypersurfaces and Hilbert schemes . . . . . . . . . 161.1.7 Zero-dimensional schemes associated with singularities . . . . 191.1.8 The Castelnuovo function of a zero-dimensional scheme in Pn . 20

1.2 Some notions of the local deformation theory . . . . . . . . . . . . . . 211.2.1 Deformation of singularities . . . . . . . . . . . . . . . . . . . 211.2.2 Versal and complete deformations . . . . . . . . . . . . . . . . 23

1.3 Notions of computer algebra . . . . . . . . . . . . . . . . . . . . . . . 251.3.1 Monomial orderings . . . . . . . . . . . . . . . . . . . . . . . . 251.3.2 Normal form . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271.3.3 RedNFBuchberger algorithm for computation of normal

form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271.3.4 Highest corner . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

1.4 Examples of obstructed families . . . . . . . . . . . . . . . . . . . . . 28

2 Main results 332.1 Equianalytic families of minimal obstructness . . . . . . . . . . . . . 332.2 Stable properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.3 Deformation theoretic meaning . . . . . . . . . . . . . . . . . . . . . 36

3 The proof of the main results 393.1 Affine coordinates and the stratum V U

d . . . . . . . . . . . . . . . . . 393.2 Proof of theorem 2.1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.2.1 The structure of the proof . . . . . . . . . . . . . . . . . . . . 40

11

12 CONTENTS

3.2.2 Proof of the relation h1(JZea(C,z)/P2(d)) = 1 . . . . . . . . . . . 423.2.3 Notations that will be used during the whole proof . . . . . . 433.2.4 Case 1: l < k < 2l . . . . . . . . . . . . . . . . . . . . . . . . 433.2.5 Case 2: k = l . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.2.6 Proof of statements (i), (ii) and (iii) . . . . . . . . . . . . . . . 473.2.7 Finish of the proof of statement (iv) for cases 1 and 2 . . . . . 473.2.8 Case 3: k ≥ 2l . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.3 Proof of theorem 2.1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . 523.3.1 The structure of the proof . . . . . . . . . . . . . . . . . . . . 533.3.2 Proof that h1(JZea(H,z)/Pn(d)) = 1 . . . . . . . . . . . . . . . . 533.3.3 Switch to substratum and notations . . . . . . . . . . . . . . . 553.3.4 Proof of the theorem for the case α1 < 2αn . . . . . . . . . . . 573.3.5 Proof of the theorem for the case α1 ≥ 2αn. . . . . . . . . . . 603.3.6 Proof that the sectional singularity type is A1 for all cases . . 63

3.4 Proof of theorem 2.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 643.4.1 The structure of the proof . . . . . . . . . . . . . . . . . . . . 653.4.2 Coordinate changes . . . . . . . . . . . . . . . . . . . . . . . . 663.4.3 Equations defining V U

d,W 1 and proof of statement (a) . . . . . . 663.4.4 Proof of statement (b) . . . . . . . . . . . . . . . . . . . . . . 673.4.5 Proof of statement (e) . . . . . . . . . . . . . . . . . . . . . . 683.4.6 Proof of statement (c) . . . . . . . . . . . . . . . . . . . . . . 693.4.7 Proof of statement (d) . . . . . . . . . . . . . . . . . . . . . . 703.4.8 Proof of statement (f) . . . . . . . . . . . . . . . . . . . . . . 70

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

Chapter 1

Preliminaries

1.1 Notions of singularity theory

In this section we describe the types of isolated hypersurface singularities consideredthroughout the thesis.

1.1.1 Analytic types of hypersurface singularities

Definition 1.1.1 Let Σ be an n-dimensional smooth projective variety. Two germs(F, z) ⊂ (Σ, z) and (G,w) ⊂ (Σ, w) of isolated hypersurface singularities are said tobe analytically equivalent if there exists a local analytic isomorphism (Σ, z) → (Σ, w)mapping (F, z) to (G,w). The corresponding equivalence classes are called analytictypes.

Notation 1.1.2 We denote by C{x1, . . . , xn}, or in short C{x}, the algebra ofconvergent power series in n variables.

Definition 1.1.3 Series f, g ∈ C{x} are said to be contact equivalent if there existsan automorphism φ of C{x} and a unit u ∈ C{x}∗ such that f = u ·φ(g). We denotef

c∼ g.

Note that fc∼ g if and only if the corresponding germs (f−1(0), 0) and (g−1(0), 0)

are analytically equivalent.

Definition 1.1.4 Let S be an analytic type of reduced hypersurface singularityrepresented by (H, z) ⊂ (Σ, z) and f ∈ C{x1, . . . , xn} be a local equation for (H, z).Define the jacobian of f by j(f) = 〈 ∂f

∂x1, . . . , ∂f

∂xn〉. The analytic algebras

Mf := C{x}/j(f), Tf := C{x}/〈f, j(f)〉

13

14 CHAPTER 1. PRELIMINARIES

are called the Milnor and Tjurina algebra of f , respectively, and the numbers

µ(S) = µ(H, z) := dimCMf , τ(S) = τ(H, z) := dimC Tf

are called the Milnor and Tjurina numbers of S, respectively.

The following theorem shows that the Tjurina algebra is a complete invariantof an analytic singularity type.

Theorem 1.1.5 (Mather-Yau) Let f, g ∈ m ⊂ C{x}. The following are equiva-lent:(a) f

c∼ g;(b) for all b ≥ 0, C{x}/〈f,mbj(f)〉 ∼= C{x}/〈g,mbj(g)〉 as C-algebras;(c) there is some b ≥ 0 such that C{x}/〈f,mbj(f)〉 ∼= C{x}/〈g,mbj(g)〉 as C-algebras.In particular, f

c∼ g iff Tf ∼= Tg.

Proof. See [MaY] for the case of isolated singularity and b = 0, 1 or [GLS], Theorem2.26 for the general case.

1.1.2 Finite determinacy

The aim of this section is to show that an isolated hypersurface singularity is alreadydetermined by its Taylor series expansion up to a sufficiently high order.

Definition 1.1.6 For f ∈ C{x} we define the k−jet of f by

jet(f, k) := f (k) := image of f in C{x}/mk+1.

We identify f (k) with the power series expansion of f up to (and including) order k.

Definition 1.1.7 f ∈ C{x} is called contact k-determined if for each g ∈ C{x}with f (k) = g(k) we have f

c∼ g. The minimal such k is called contact determinacyof f .

Theorem 1.1.8 (Finite determinacy theorem) Let f ∈ m ⊂ C{x}.f is contact k−determined if mk+1 ⊂ m2 · j(f) +m〈f〉.

Proof. This theorem is well known. See, for instance, [GLS], Theorem 2.23.

Corollary 1.1.9 If f ∈ m ⊂ C{x}, has an isolated singularity with Tjurina numberτ , then f is contact τ + 1−determined .

1.1. NOTIONS OF SINGULARITY THEORY 15

1.1.3 Stable equivalence

Definition 1.1.10 Let f ∈ C{x1, . . . , xl} and g ∈ C{x1, . . . , xk}. We say that f isstably contact equivalent to g if they become contact equivalent after addition withnon-degenerate quadratic forms of additional variables. In other words,

f(x1, . . . , xl) + x2l+1 + . . .+ x2

nc∼ g(x1, . . . , xk) + x2

k+1 + . . .+ x2n

Theorem 1.1.11 Polynomials of the same number of variables are stably contactequivalent if and only if they are contact equivalent.

Proof. See, for instance, [AGV], chapter II section 11.We will use a more general lemma:

Lemma 1.1.12 (1) Let f ∈ C{x1, ..., xn}. Let g = f + x2n+1(1 + h) where h ∈

m ⊂ C{x1, ..., xn+1}. Then the Tjurina algebra Tf of f is isomorphic to the Tjurinaalgebra Tg of g.(2) Let f1, f2 ∈ C{x1, ..., xn}. Let gi = fi + x2

n+1(1 + hi) , i = 1, 2 where hi ∈ m ⊂C{x1, ..., xn+1}. Then f1

c∼ f2 if and only if g1c∼ g2

Proof. (1) In Tg,

0 =∂g

∂xn+1

= xn+1(2 + 2h+ xn+1∂h

∂xn+1

).

Since 2 + 2h + xn+1∂h

∂xn+1is invertible in Tg, the latter implies xn+1 = 0. Hence Tf

is isomorphic to Tg.(2) Follows from (1) and from the Mather-Yau theorem 1.1.5. 2

1.1.4 Quasihomogeneous singularities

Definition 1.1.13 Let f =∑

α∈Zn≥0aαx

α ∈ C[x1, . . . , xn]

(i) The polynomial f is called quasihomogeneous of type

(w; d) = (w1, . . . , wn; d)

if wi, d are positive integers satisfying

〈w, α〉 = w1α1 + . . .+ wnαn = d

for each α ∈ Zn≥0 with aα 6= 0.

(ii) An isolated hypersurface singularity (H, x) ⊂ (Cn, x) is called quasihomo-geneous if there exists a quasihomogeneous polynomial f ∈ C[x1, . . . , xn] such thatOH,x

∼= C{x}/〈f〉.


Example 1.1.14 Let f =n∑i=1

xαii . Then f is a quasihomogeneous polynomial of

type (1/α1, . . . , 1/αn; 1). Such polynomials are called canonical quasihomogeneous.

A quasihomogeneous polynomial f of type (w; d) obviously satisfies the relation

d · f =n∑i=1

wixi∂f

∂xi,

that implies that f is contained in j(f), hence, for quasihomogeneous isolated hy-persurface singularities µ = τ . It is a theorem of K.Saito [Sai] that for an isolatedsingularity the converse also holds. So, let (H, x) ⊂ (Cn, x) be an isolated hyper-surface singularity and let f ∈ C{x1, . . . , xn} be an any local equation for (H, x),then

(H, x) is quasihomogeneous ⇐⇒ f ∈ j(f) ⇐⇒ µ(H, x) = τ(H, x)

We will use the following important theorem:

Theorem 1.1.15 Let f be a quasihomogeneous polynomial of weighted degree d andg be a polynomial such that the weighted degrees of all its monomials are greater thand. Let e1, e2, ..., eµ be a system of monomials that forms basis of the Milnor algebraMf of f . Then e1, e2, ..., eµ form a basis of Mf+g as well. In particular, Mf+g

∼= Mf

and µ(f + g) = µ(f).

Proof. See [AGV], chapter II section 12.

1.1.5 Newton polytope

Definition 1.1.16 Let f =∑

α∈Zn≥0aαx

α ∈ C{x}, a0 = 0. The convex hull in Rn

of the support of f ,

∆(f) := conv{α ∈ Zn≥0 | aα 6= 0},

is called the Newton polytope of f .

If f is quasihomogeneous with weight w of degree d then the Newton polytope∆(f) will lie in the hyperplane {α ∈ Zn

≥0 |∑αiwi = d}. In particular, for a

quasihomogeneous polynomial of two variables its Newton polygon is an interval.

1.1.6 Families of hypersurfaces and Hilbert schemes

Throughout the following let Σ be a smooth, connected complex projective varietyand H ⊂ Σ a reduced hypersurface.


Definition 1.1.17 Let T be a complex space, then by a family of hypersurfaces onΣ over T we mean a commutative diagram

Hj↪→ Σ× T

ϕ↘ ↙ pr

T

where ϕ is a proper and flat morphism such that for all t ∈ T the fiberHt := ϕ−1(t) isa hypersurface in Σ, j is a closed embedding and pr denotes the natural projection. Afamily with sections is a diagram as above, together with sections s1, ..., sr : T → Hof ϕ.

To a family of reduced hypersurfaces as above and a fiber H = Ht0 we can associate,in a functorial way, the deformation∐

i

(H, zi) → (T, t0)

of multigerm (H,Sing(H)) =∐

i(H, zi) over the germ (T, t0). Having a family withsections s1, ...sr, si(t0) = zi, we obtain in the same way a deformation of

∐i(H, zi)

over (T, t0) with sections.The Hilbert functor HilbΣ on the category of complex spaces defined by

HilbΣ(T ) := {H ↪→ Σ× T → T family of reduced hypersurfaces}

is well known to be representable by a complex space HilbΣ (see [Bin]). Moreover,given any M ⊂ PN × T , flat over T , the Hilbert polynomial ht of the fiber Mt,t ∈ T , defined by

ht(n) = χ(OMt(n)),

is constant on each component of T . In particular, the universal family of reducedhypersurfaces on Σ ”breaks up” via Hilbert polynomials, that is

HilbΣ =⊔

h∈C[z]

HilbhΣ

where HilbhΣ are (unions of) connected components of HilbΣ. Note that HilbhΣ rep-resents the functor HilbhΣ defined by

HilbhΣ(T ) := {H ↪→ Σ× T → T family of reduced hypersurfaces

with Hilbert polynomial h}.


The tangent space to HilbhΣ at H is H0(NH/Σ) as well as the space of first orderembedded deformations of H ⊂ Σ. Here, NH/Σ = OΣ(H)

⊗OΣOH is the normal

sheaf of H in Σ.We are interested in subfamilies of reduced hypersurfaces on Σ with fixed

Hilbert polynomial h and having fixed analytic types S1, ..., Sr of singularities. Itis a locally closed subspace of HilbhΣ, and it represents the functor of equianalyticfamilies of given types S1, ..., Sr (see [GrL]).

Definition 1.1.18 Vh(S1, ..., Sr), respectively V irrh (S1, ..., Sr) denotes the locally

closed subspace of HilbhΣ consisting of reduced, respectively reduced irreducible hy-persurfaces H ∈ Σ having precisely r singularities of the types S1, ..., Sr.

Remark 1.1.19 If the dimension of Σ is more than 2 then the space V irrh (S1, ..., Sr)

coincides with Vh(S1, ..., Sr) because hypersurfaces of dimension at least 2 havingonly isolated singularities are always irreducible.

Example 1.1.20 (a) For Σ = Pn the Hilbert polynomial is defined by the degreeof the hypersurface, hence we consider the hypersurfaces in the linear system |dH|,where H is hyperplane section, and denote the corresponding varieties by Vd =Vd(S1, ..., Sr).

(b) For Σ = P2 we consider the curves in the linear system |dL|, where L isprojective line, and denote the corresponding varieties by Vd = Vd(S1, . . . , Sr).

Remark 1.1.21 Let H ⊂ Pn be a hypersurface of degree d with isolated singular-ities. We denote by Vd,H the germ at H of the equianalytic family of H. In otherwords, Vd,H := (Vd(S1, ..., Sr), H), where S1, . . . , Sr are the analytic types of thesingularities of H.

Classically, instead of HilbhΣ, a complete linear system |H| = P(H0(OΣ(H)))on Σ, respectively, the open subset U ⊂ |H| corresponding to reduced hypersurfaces,is considered. Note that the Hilbert polynomial h = h(H ′) of a hypersurface H ⊂ Σis constant for all (reduced) hypersurfaces H ′ ∈ |H| and, since |H| is reduced, itfollows that the incidence variety

{(x,H ′) ∈ Σ× U |x ∈ H ′} ⊂ Σ× U

is flat over U. Hence there exists a natural (injective) morphism U → HilbhΣ, whichon the tangent level, corresponds to

φ : H0(OΣ(H))/H0(OΣ) ↪→ H0(NH/Σ).

Thus, we may consider U as a subscheme of HilbhΣ. The morphism φ comes fromthe exact sequence

0 → OΣ → OΣ(H) → OH(H) → 0.


Hence if Σ is regular, that is, if h1(OΣ) = 0, then φ is an isomorphism and we canidentify U with HilbhΣ.

1.1.7 Zero-dimensional schemes associated with singulari-ties

Definition 1.1.22 Let Σ be a smooth projective variety and H ⊂ Σ a reducedhypersurface with singular locus Sing(H) = {z1, ..., zr}. We define Zea(H) ⊂ Σto be the zero-dimensional scheme, concentrated at Sing(H), given by the Tjurinaideals

JZea(H)/Σ,zi= 〈fi,

∂fi∂x1

, . . . ,∂fi∂xn

〉 ⊂ OΣ,zi,

where fi(x1, . . . , xn) is a local equation of H in a neighborhood of zi. We denote byJZea(H)/Σ ⊂ OΣ the corresponding ideal sheaf.

The degree of Zea(H) is

degZea(H) =∑

zi∈Sing(H)

dimCOΣ,zi/JZea(H)/Σ,zi

.

Theorem 1.1.23 Let H ⊂ Pn be a reduced hypersurface of degree d with preciselyr singularities z1, ..., zr of analytic types S1, ..., Sr.(a) H0(JZea(H)/Pn(d))/H0(OPn) is isomorphic to the Zariski tangent space to Vd atH. Here, H0(OPn) is embedded into H0(JZea(H)/Pn(d)) by multiplication by F .(b) h0(JZea(H)/Pn(d))− h1(JZea(H)/Pn(d))− 1 ≤ dim(Vd, H) ≤ h0(JZea(H)/Pn(d))− 1.(c) H1(JZea(H)/Pn(d)) = 0 if and only if Vd is T-smooth at H, that is, smooth of the

expected dimension(d+nn

)− 1− deg Zea(H).

Proof. See [GrK].

Lemma 1.1.24 Let H ⊂ Pn be a hypersurface and Z ⊂ H be a zero-dimensionalsubscheme. Then H1(JZ/Pn(H)) ∼= H1(JZ/H(H)).

Proof. The lemma follows from the exact sequence of sheaves

0 → JH/Pn(H) → JZ/Pn(H) → JZ/Pn(H)⊗OH → 0,

from the fact that JH/Pn(H) ∼= OPn , which implies H1(JH/Pn(H)) =H2(JH/Pn(H)) = 0, and from JZ/Pn(H)⊗OH

∼= JZ/H(H).


1.1.8 The Castelnuovo function of a zero-dimensionalscheme in Pn

Let X ⊂ Pn be a zero-dimensional scheme and JX/Pn ⊂ OPn the corresponding idealsheaf.

Definition 1.1.25 The Castelnuovo function of X is defined as

CX : Z≥0 −→ Z≥0

d 7−→ h1(JX/Pn(d− 1))− h1(JX/Pn(d)).

Remark 1.1.26 Let X ⊂ Pn be a zero-dimensional scheme and H ⊂ Pn be ageneric hyperplane not passing through the support of X. Then we have an exactreduction sequence

0 −→ JX/Pn(d− 1) −→ JX/Pn(d) −→ OH(d) −→ 0,

respectively the corresponding exact cohomology sequence

H0(JX/Pn(d− 1))πH−→ H0(OH(d)) −→ H1(JX/Pn(d− 1)) −→ H1(JX/Pn(d)) −→ 0.

In particular,CX(d) = h0(OH(d))− dimC πH(H0(JX/Pn(d))).

We associate to X the numbers

a(X) = min{d ∈ Z|h0(JX/Pn(d) > 0)}b(X) = min{d ∈ Z|P(H0(JX/Pn(d))) has no fixed component}t(X) = min{d ∈ Z|H1(JX/Pn(d)) = 0)}.

Here, a fixed component is a divisor D such that every element of the linear system|H0(JX/Pn(d))| contains D as a component. We call the maximal divisor satisfyingthis property the fixed component of |H0(JX/Pn(d))|. The following lemma containssome basic properties of the Castelnuovo function.

Lemma 1.1.27 Let X ⊂ Pn be a zero-dimensional scheme and H ⊂ Pn be ageneric hyperplane not passing through the support of X. Then(a) CX(d) ≥ 0 for all d, and CX(d) = 0 for d� 0.(b) CX(d) ≤ h0(OH(d)), with equality if and only if d < a(X).(c) a(X) ≤ b(X) ≤ t(X) + 1.(d) CX(d) = 0 if and only if d ≥ t(X) + 1.(e) If Y ⊆ X then CY (d) ≤ CX(d).(f) CX(0) + CX(1) + · · ·+ CX(d) = degX − h1(JX/Pn(d)).

1.2. SOME NOTIONS OF THE LOCAL DEFORMATION THEORY 21

Proof. See [Dav] for curves or [West] for hypersurfaces.

Lemma 1.1.28 Let Z = Cd ∩ Ck be the intersection of two plane curves Cd, Ck ofdegrees d and k without common components. Suppose k ≤ d. Then

CZ(i) ≤ k for i ≥ 0, and CZ(d+ k − j) = j − 1 for j = 1, . . . , k + 1.

Proof. See [GLS1].

1.2 Some notions of the local deformation theory

In this section we introduce some notions of the theory of deformations of complexspace germs. The key object of the theory is the vector space of the first orderdeformations T 1

(X,x) which in the case of isolated hyperserface singularities is justthe Tjurina algebra.

1.2.1 Deformation of singularities

Definition 1.2.1 Let (X, x) and (S, s) be complex space germs. A deformationof (X, x) over (S, s) consists of a flat morphism φ : (X , x) → (S, s) of complexspace germs together with an isomorphism from (X, x) to the fiber of φ, (X, x) →(Xs, x) := (φ−1(s), x).

(X , x) is called the total space, (S, s) the base space, and (Xs, x) ∼= (X, x) thespecial fiber of the deformation. We denote a deformation by

(i, φ) : (X, x)i↪→ (X , x)

φ→ (S, s),

or simply by (X , x)φ→ (S, s).

Definition 1.2.2 Let (i, φ) : (X, x)i↪→ (X , x)

φ→ (S, s) and (i′, φ′) : (X, x)i↪→

(X ′, x′)φ′→ (S ′, s′) be two deformations of (X, x). A morphism of deformations

from (i, φ) to (i′, φ′) consists of two morphisms (ψ, ϕ)

(X, x)

i′ ↙ ↘ i

(X ′, x′)ψ−→ (X , x)

φ′ ↓ ↓ φ(S ′, s′)

ϕ−→ (S, s)

Two deformations over the same base are isomorphic if there exists a morphism(ψ, ϕ) with ψ an isomorphism and ϕ the identity map.


Definition 1.2.3 Let (i, φ) : (X, x)i↪→ (X , x)

φ→ (S, s) be a deformation of (X, x)and ϕ : (T, t) → (S, s) be a morphism of germs. Denote by ϕ∗(X , x) the fiberproduct (X , x)×(S,s) (T, t). We call

ϕ∗(i, φ) := (ϕ∗i, ϕ∗φ) : (X, x)ϕ∗i↪→ ϕ∗(X , x)

ϕ∗φ→ (T, t)

the deformation induced from (i, φ) by ϕ, or just pull-back ; ϕ is called the basechange map.

Definition 1.2.41. Def(X,x) denotes the category of deformations of (X, x).2. Def(X,x)(S, s) denotes the category of deformations of (X, x) over (S, s).3. Def

(X,x)(S, s) denotes the set of isomorphism classes of deformations of (X, x)

over (S, s).4. The functor

Def(X,x)

: (complex germs) → Sets, (S, s) 7→ Def(X,x)

(S, s)

is called the deformation functor of (X, x) or the functor of isomorphism classes ofdeformations of (X, x).

Proposition 1.2.5 Let (X, 0) ⊂ (Cn, 0) be a closed subgerm. Then any deforma-

tion (i, φ) : (X, x)i↪→ (X , x)

φ→ (S, s) can be embedded. In other words, there existsa Cartesian diagram

(X, 0)i↪→ (X , x)

↓ ↓ J

(Cn, 0)j↪→ (Cn, 0)× (S, s)

↓ ↓ p

{s} ↪→ (S, s)

where J is a closed embedding, p is the second projection, j is the first inclusion andφ = p ◦ J .

In particular, the embedding dimension is semicontinuous under deformations,that is, edimφ−1(φ(y)), y) ≤edim(X, 0), for all y in X sufficiently close to x.

Proof. See [GLS], Corollary II.1.6.

1.2. SOME NOTIONS OF THE LOCAL DEFORMATION THEORY 23

1.2.2 Versal and complete deformations

A versal deformation of a complex space germ is a deformation which containsbasically all information about any possible deformation of this germ. More pre-cisely, we say that a deformation (i, φ) of (X, x) over (S, s) is complete if any otherdeformation over some base space (T, t) can be induced from (i, φ) by some basechange ϕ : (T, t) → (S, s). A complete deformation is called versal if for any defor-mation of (X, x) over some subgerm (T ′, t) ⊂ (T, t) induced by some base changeϕ′ : (T ′, t) → (S, s), ϕ can be chosen in such a way that it extends ϕ′. We will nowgive the formal definitions.

Definition 1.2.6 (1) A deformation (X, x)i↪→ (X , x)

φ→ (S, s) of (X, x) is called

complete if, for any deformation (j, ψ) : (X, x)j↪→ (Y , y)

ψ→ (T, t) of (X, x) thereexists a morphism ϕ : (T, t) → (S, s) such that (j, ψ) is isomorphic to the induceddeformation (ϕ∗i, ϕ∗φ).(2) A deformation (i, φ) is called versal if, for any deformation (j, ψ) as above thefollowing holds: for any closed embedding k : (T ′, t) ↪→ (T, t) of complex spacegerms and any morphism ϕ′ : (T ′, t) → (S, s) such that (k∗j, k∗ψ) is isomorphic tothe induced deformation (ϕ′∗i, ϕ′∗φ), there exists a morphism ϕ : (T, t) → (S, s)such that(i) ϕ ◦ k = ϕ′ and(ii)(j, ψ) is isomorphic to the induced deformation (ϕ∗i, ϕ∗φ).

This definition can be illustrated by the following commutative diagram:

(X, x)

k∗j ↙ ↓ j ↘ i

k∗(Y , y) ↪→ (Y , y) 99K (X , x)

k∗ψ ↓ ψ ↓ ↓ φ

(T ′, t)k↪→ (T, t)

ϕ99K (S, s).

(3) A versal deformation is called semiuniversal or miniversal if the Zariskitangent map T (ϕ) : T(T,t) → T(S,s) is uniquely defined by (i, φ) and (j, ψ).

Our definition of versality is very restrictive. The deformations that we call completeare sometimes called versal in the literature. For example, in [AGV] the authorscall our complete deformation ”versal” and our versal deformation ”infinitesimallyversal”.

Now we introduce a notion which is weaker than completeness, but still strongenough for many applications.


Definition 1.2.7 A deformation (X, x)i↪→ (X , x)

φ→ (S, s) of (X, x) is called 1-

complete if, for any one parametric deformation (j, ψ) : (X, x)j↪→ (Y , y)

ψ→ (C, 0)of (X, x) there exists a morphism ϕ : (C, 0) → (S, s) such that (j, ψ) is isomorphicto the induced deformation (ϕ∗i, ϕ∗φ).

An arbitrary complex space germ may not have a versal deformation. It isa fundamental theorem of Grauert, that for isolated singularity a semiuniversaldeformation exists.

Theorem 1.2.8 (Grauert, 1972) . Any complex space germ with isolated singu-larity has a semiuniversal deformation.

Proof. See [Gra].The following two theorems describe the connection between equisingular fam-

ilies and versal deformations.

Theorem 1.2.9 Let (X, 0) ⊂ (Cn, 0) be an isolated singularity defined by f ∈ OCn,0

and g1, . . . , gτ ∈ OCn,0 be a basis of the Tjurina algebra Tf . If we set

F (x, t) := f(x) +τ∑j=1

tjgj(x), (X , 0) := V (F ) ⊂ (Cn × Cτ , 0),

then (X, 0) ↪→ (X , 0)φ→ (Cτ , 0), where φ is the second projection, is a semiuniversal

deformation of (X, 0).

Proof. See [GLS1], corollary II.1.17.

Theorem 1.2.10 Let (H, z) be a germ of projective hypersurface with one isolatedsingularity z. Suppose h1(JZea(H)/Pn(H)) = 0. Then the linear system |H| inducesversal deformation of (H, z).

Proof. See [GrK] or [GLS1], proposition II.2.8 (d).We finish this section with the classical curve selection lemma.

Lemma 1.2.11 Let X be an algebraic variety over C. Let U ⊂ X be a Zariskiopen subset and x be a point in the closure of U . Then there exists a morphism ofanalytic germs φ : (C, 0) → (X, x) such that φ(C \ {0}) ⊂ U .

Proof. A basic lemma (see, for example, [Kem], lemma 7.2.1) says that there existsa smooth curve C and a morphism ν : C → U , such that ν−1(U) is non-empty andx is contained in the image of ν. Denote Z = ν−1(U \ U) . Then Z is a closedsubset of C and hence consists of finite number of points. Now take any pointz ∈ ν−1({x}) ⊂ C. It has a neighborhood which does not contain other points of Z.

1.3. NOTIONS OF COMPUTER ALGEBRA 25

Since C is smooth, the analytic germ (C, z) is isomorphic to (C, 0). Hence ν definesthe required morphism. 2

1.3 Notions of computer algebra

In our computations we want to use the methods of computer algebra. Here weintroduce the basic notions, that will be widely used in our proofs. A more detaileddescription of these notions can be found also in [GP].

1.3.1 Monomial orderings

Definition 1.3.1 A monomial ordering is a total (or linear) ordering > on the setof monomials Monn = {xα | α ∈ Zn

≥0} in n variables satisfying

xα > xβ ⇒ xγxα > xγxβ

for all α, β, γ ∈ Zn≥0. We say also > is a monomial ordering on A[x1, ..., xn], where

A is any ring, meaning that > is a monomial ordering on Monn.

We identifyMonn with Zn≥0, and then a monomial ordering is a total ordering on Zn

≥0,which is compatible with the semigroup structure on Zn

≥0 given by addition. Froma practical point of view, a monomial ordering > allows us to write a polynomialf ∈ K[x] in a unique ordered way as

f = aαxα + aβx

β + ...+ aγxγ,

with xα > xβ > ... > xγ, where no coefficient is zero.The most important distinction is between global and local orderings.

Definition 1.3.2 Let > be a monomial ordering on {xα | α ∈ Zn≥0}

(1) > is called a global ordering if xα > 1 for all α 6= (0, ..., 0),(2) > is called a local ordering if xα < 1 for all α 6= (0, ..., 0)

Important examples of monomial orderings are:

Example 1.3.3 (monomial orderings). In the following examples we fix an enumer-ation x1, ..., xn of the variables, any other enumeration leads to a different ordering.(1) Global orderings

(i) Lexicographical ordering >lp

xα >lp xβ :⇔ ∃1 ≤ i ≤ n : α1 = β1, ..., αi−1 = βi−1, αi > βi,


(ii) Degree lexicographical ordering >Dp

xα >Dp xβ :⇔ degxα > degxβ

or (degxα = degxβ and ∃1 ≤ i ≤ n : α1 = β1, ..., αi−1 = βi−1, αi > βi).

(iii) Weighted degree lexicographical ordering Wp(ω1, ..., ωn)Given a vector ω = (ω1, ..., ωn) of integers, we define the weighted degree of xα bydegω(x

α) := 〈ω, α〉 := ω1α1 + · · ·+ ωnαn, that is, the variable xi has degree ωi. Fora polynomial f =

∑α aαx

α, we define the weighted degree,

degω(f) := max{degω(xα) | aα 6= 0}.

Using the weighted degree in (ii), with all ωi > 0, instead of the usual degree,we obtain the weighted degree lexicographical ordering, Wp(ω1, ..., ωn).

(2) Local orderings(i) Negative lexicographical ordering >ls

xα >ls xβ :⇔ ∃1 ≤ i ≤ n : α1 = β1, ..., αi−1 = βi−1, αi < βi,

(ii) Negative degree lexicographical ordering >Ds:

xα >Ds xβ :⇔ degxα < degxβ

or (degxα = degxβ and ∃1 ≤ i ≤ n : α1 = β1, ..., αi−1 = βi−1, αi > βi).

(iii) Negative weighted degree lexicographical ordering Ws(ω1, ..., ωn) is aweighted version of the last ordering.

Definition 1.3.4 Let > be a fixed monomial ordering. Let f ∈ K[x], f 6= 0. Thenf can be written in a unique way as a sum of non-zero terms

f = aαxα + aβx

β + ...+ aγxγ, xα > xβ > ... > xγ,

and aα, aβ, ..., aγ ∈ K. We define:(1) LM(f) := xα, the leading monomial of f ,(2) LE(f) := α, the leading exponent of f ,(3) LT (f) := aαx

α, the leading term of f ,(4) LC(f) := aα, the leading coefficient of f ,(5) tail(f) := f − LT (f) = aβx

β + ...+ aγxγ, the tail of f .

Definition 1.3.5 For any monomial ordering > on Mon(x1, ..., xn), we define thering K[x]> associated to K[x] and > by

K[x]> := {fu| f, u ∈ K[x], LM(u) = 1}

Note that K[x]> = K[x] if and only if > is global and K[x]> = K[x]〈x1,...,xn〉 if andonly if > is local.

1.3. NOTIONS OF COMPUTER ALGEBRA 27

1.3.2 Normal form

Let > be a monomial ordering and let R = K[x1, ..., xn]> (see definition 1.3.5 above).For any subset G ⊂ R define the ideal

L>(G) := L(G) := 〈LM(g)|g ∈ G \ {0}〉K[x].

L(G) ⊂ K[x] is called the leading ideal of G. Note that if I is an ideal, then L(I)is the ideal generated by all leading monomials of all elements of I and not only bythe leading monomials of a given set of generators of I.

Definition 1.3.6 Let G denote the set of all finite subsets G ⊂ R. A map

NF : R× G → R, (f,G) 7→ NF (f | G),

is called a normal form on R if, for all f ∈ R and G ∈ G,(0) NF (0|G) = 0;(1) NF (f |G) 6= 0 ⇒ LM(NF (f |G)) /∈ L(G);(2) if G = {g1, ..., gs}, then r := f − NF (f |G) has a standard representation withrespect to G, that is, either r = 0, or

r =s∑i=1

aigi, ai ∈ R,

satisfying LM(f) ≥ LM(aigi) for all i such that aigi 6= 0.NF is called a reduced normal form, if, moreover, NF (f |G) is reduced with respectto G, i.e. no monomial of the power series expansion of NF (f |G) is contained inL(G).As we can see from the definition, NF (f |G) = 0 if and only if f ∈ 〈G〉.

1.3.3 RedNFBuchberger algorithm for computation of nor-mal form

Algorithm 1.3.7 (redNFBuchberger algorithm)Assume that > is a global monomial ordering.Input: f ∈ K[x], G ∈ GOutput: p ∈ K[x], a reduced normal form of f with respect to G.

1. p := 0; h := f ;

2. while (h 6= 0)

(a) while (h 6= 0 and Gh := {g ∈ G|LM(g) divides LM(h)} 6= 0){ choose any g ∈ Gh;h := h− (LT (h)/LT (g)) · g}


(b) if (h 6= 0){p := p+ LT (h);h := tail(h)};

3. return p/LC(p);

The algorithm works in the following way: the inner loop (2a) runs untilit meets an ”obstruction”, i.e. the first monomial that isn’t divisible by leadingmonomial of any member of G. When the inner loop (2a) stops, h stores a normalform of f . To make this normal form reduced, we add the leading term of h, i.e.the ”obstruction”, to p and continue working with the tail of h in the same way.

Note that any specific choice of ”any g ∈ Gh” can give a different normal formfunction. For proof of correctness of the algorithm see [GP], section 1.6 algorithms1.6.10 and 1.6.11.

1.3.4 Highest corner

Definition 1.3.8 Let > be a monomial ordering on Mon(x1, ..., xn) and let I ⊂K[x1, ..., xn]> be an ideal. A monomial m ∈ Mon(x1, ..., xn) is called the highestcorner of I (with respect to >), denoted by HC(I), if(1) m /∈ L(I);(2) m′ ∈Mon(x1, ..., xn), m

′ < m⇒ m′ ∈ L(I).

Lemma 1.3.9 Let > be a monomial ordering on Mon(x1, ..., xn) and let I ⊂K[x1, ..., xn]> be an ideal. Letm be a monomial such thatm′ < m impliesm′ ∈ L(I).Let f ∈ K[x1, ..., xn] such that LM(f) < m. Then f ∈ I.

Proof. See [GP] lemma 1.7.13.

Lemma 1.3.10 Let > be a weighted degree ordering on Mon(x1, ..., xn). Moreover,let f1, ..., fk be a set of generators of the ideal I ⊂ K[x1, ..., xn]> such that J :=〈LM(f1), ..., LM(fk)〉 has a highest corner m := HC(J) and f ∈ K[x1, ..., xn]>. IfLM(f) < HC(J) then f ∈ I.

Proof. See [GP] lemma 1.7.17.

1.4 Examples of obstructed families of curves and

hypersurfaces

Let D be a divisor on a smooth projective variety Σ. Let VD = V|D|(S1, ..., Sr)be the variety of reduced hypersurfaces in a complete linear system |D| having rsingularities of analytic types S1, ..., Sr.

The main questions concerning this space are

1.4. EXAMPLES OF OBSTRUCTED FAMILIES 29

• Is VD non-empty?

• Is VD irreducible?

• Is VD smooth?

• Has VD ”expected” dimension (expressible via local invariants of the singular-ities)?

This thesis focuses on the last two problems. If VD is smooth and has expecteddimension it is called T-smooth. The well-known classical result is that the varietyVD is T-smooth at H ∈ VD if and only if h1(JZea(H)/Σ(H)) = 0 (see theorem 1.1.23).We want to consider the equisingular families of curves for which the condition isnot satisfied, that is where T-smoothness fails. Such equisingular families are calledobstructed.

By the moment, obstructed families of curves have not been studied system-atically and are presented by several series of examples.

A classical example of B. Segre [Seg, Tan] gives a series of irreducible planecurves such that the corresponding germs of the equisingular strata are non-T-smooth:

Example 1.4.1 (Segre) Let F2m, respectively G3m be two generic homogeneouspolynomials in 3 variables of degree 2m, respectively 3m. Then the curve C6m ⊂ P2

of degree d = 6m given by

(F2m(x, y, z))3 + (G3m(x, y, z))2 = 0

is irreducible and has precisely 6m2 cusps as only singularities. Moreover, the di-mension of the family V ′ ⊂ V6m(6m2 · A2) of such curves is

2m(2m+ 3)

2+

3m(3m+ 3)

2+ 1 =

d(d+ 3)

2− 12m2 +

(m− 1)(m− 2)

2

It follows that for m ≥ 3 the variety V6m(6m2 · A2) contains a component whichhas a bigger dimension than the expected one. In particular, this component isnon-T-smooth.

On the other hand, h1(JZea(C6m)/P2(6m)) = (m−1)(m−2)2

, hence we can concludethat V6m(6m2 · A2) is smooth at C6m.

E.Shustin [Shu1, Shu2] modified these (classical) examples slightly and obtained thefollowing result.

Example 1.4.2 (Shustin) (a) Let m ≥ 3. Then the variety V irr7m−3(6m

2 · A2) con-tains a component of dimension (7m − 3)7m/2 − 12m2 + 1, which is 1 more than


the expected dimension. In particular, this component is non-T-smooth. Note thatin this example

#(cusps) =6

49d2 +O(d).

(b) Letm ≥ 9 and let S = D4 be an ordinary triple point. The variety V irr3m+1(m

2·D4)contains a (non-T-smooth) component of dimension at least m2 + 3m+ 4, which isgreater than the expected dimension m2/2 + 15m/2 + 2. Note that here

#(ordinary triple points) =1

9d2 +O(d).

J.Wahl [Wah] presented the first series of irreducible plane curves C (of degreed ≥ 104, with only nodes and cusps) such that the corresponding equisingularstrata V irr

d (S1, . . . , Sr) are non-smooth at C, for instance

Example 1.4.3 (Wahl) The (non-empty) variety V irr104 (3636 ·A1, 900 ·A2) is non-

smooth.

It was I.Luengo [Lue], who gave the first example of a curve C such that the corre-sponding equisingular stratum is of the expected dimension, but non-smooth:

Example 1.4.4 (Luengo) The homogeneous polynomial x9 + z(xz3 + y4)2 definesan irreducible projective plane curve C with exactly one singular point (at (0 : 0 : 1)),which is of type A35. Moreover, the equisingular stratum V irr

9 (1 · A35) has theexpected dimension 19, but is non-smooth at C.

Concerning reducible equisingular family, there is mainly one classical example dueto Zariski [Zar]:

Example 1.4.5 (Zariski) The variety V irr6 (6 · A2) is reducible.

More precisely, Zariski shows that there exist exactly two components, both beingT-smooth.

We focus on the minimal obstructness case, i.e. h1(JZea(C)/P2(d)) = 1. In thiscase the family is either smooth of unexpected dimension or non-smooth of expecteddimension.

One of the interesting recent examples of such families is due to du Plessis andWall [DPW]:

Example 1.4.6 (du Plessis and Wall) (a) For any d ≥ 5 the curve C ⊂ P2 givenby the equation (xd1 + x5

2xd−50 + xd2 = 0) has the unique singular point z = (0 : 0 : 1)

with Tjurina number τ(C, z) = 4d− 4, and satisfies

h1(JZea(C)/P2(d)) = 1.

(b) Denote by S the analytic type of plane curve singularity (C, z). If d ≥ 10 thenthe family Vd(S) is singular at C.

1.4. EXAMPLES OF OBSTRUCTED FAMILIES 31

Note that τ(C, z) = 4d − 4 is a boundary case, since for τ(C, z) < 4d − 4the family is T-smooth at C by a theorem due to E.Shustin [Shu3] (see also [GrL],[DPW] and [ST]).

Chapter 2

Main results

2.1 Equianalytic families of minimal obstructness

of hypersurfaces with quasihomogeneous sin-

gularities

We take the next step in the study of geometry of obstructed families of singularhypersurfaces. First, we study the geometry of the families from example 1.4.6 (thelast example from the previous section) and prove the following result

Theorem 2.1.1 Let C be the projective plane curve given by the equation td1 +t52t

d−50 + td2 = 0. Let Vd,C be the germ at C of the equianalytic family of C. Then for

any d ≥ 5, Vd,C is non-T-smooth and h1(JZea(C)/P2(d)) = 1.Furthermore:

(i) For d = 5, the germ V5,C(S) is smooth of codimension less than expectedby one.

(ii) For d = 6, the germ V6,C(S) is non-reduced. It is a double [V6,C(S)]red,and [V6,C(S)]red is smooth of expected codimension.

(iii) For d = 7, the germ V7,C(S) is reducible and decomposes into two smoothcomponents of expected codimension that intersect non-transversally with multiplicityone. The intersection locus is smooth. Moreover, the germ V7,C(S) has the sectionalsingularity type A1.

(iv) For d ≥ 8, the germ Vd,C(S) is reduced irreducible non-smooth variety ofexpected codimension which has a smooth singular locus. Moreover, the germ Vd,C(S)has the sectional singularity type A1.

Furthermore, we prove the following generalization of this theorem.

Theorem 2.1.2 Let C be the projective plane curve given by the equation tk1td−k0 +

tl2td−l0 + λ1t

d1 + λ2t

d2 = 0, where d = k + l− 5, and λi are complex numbers such that

33

34 CHAPTER 2. MAIN RESULTS

z = (1, 0, 0) is the unique singular point of C. Note that generic λi satisfies thiscondition. Suppose, for convenience, k ≥ l.

Let Vd,C be the germ at C of the equianalytic family of C. Then for any k, l ≥ 5,Vd,C is non-T-smooth and h1(JZea(C)/P2(d)) = 1.Furthermore:

(i) If d = 5 (i.e. l = k = 5), the germ V5,C is smooth of codimension less thanexpected by one.

(ii) If d = 6 (i.e. l = 5, k = 6), the germ V6,C is non-reduced. It is a double[V6,C ]red, and [V6,C ]red is smooth of expected codimension.

(iii) If d = 7 (i.e. l = 5, k = 7 or k = l = 6), the germ V7,C is reducibleand decomposes into two smooth components of expected codimension that intersectnon-transversally with multiplicity one. The intersection locus is smooth. Moreover,the germ V7,C has the sectional singularity type A1.

(iv) If d ≥ 8, the germ Vd,C is reduced irreducible non-smooth variety of ex-pected codimension which has a smooth singular locus. Moreover, the germ Vd,C hasthe sectional singularity type A1.

(For proof see section 3.2)Then we prove the following generalization to higher dimensions.

Theorem 2.1.3 Let H ⊂ Pn, n ≥ 3 be the projective hypersurface given by the

equationn∑i=1

tαii t

d−αi0 +

n∑i=1

λitdi = 0 where d =

n∑i=1

αi − (2n + 1) and λi are complex

numbers such that z = (1, 0, ..., 0) is the unique singular point of H. Note thatgeneric λi satisfy this condition.

Let Vd,H be the germ at H of the equianalytic family of H. Then for any{αi}ni=1 ≥ 2 such that d ≥ αi for all i, Vd,H is non-T-smooth and h1(JZea(H)/Pn(d)) =1.Furthermore:

(i) If n = 4 and d = 3 (i.e. α1 = α2 = α3 = α4 = 3), the germ V3,H(S) issmooth variety of non-expected codimension (one less than expected).

(ii) Otherwise, the germ Vd,H(S) is reduced irreducible non-smooth variety ofexpected codimension which has a smooth singular locus. Moreover, the germ Vd,H(S)has the sectional singularity type A1.

(For proof see section 3.3)

Remark 2.1.4 It can be shown that the exceptional cases of smooth stratum inthe last two theorems 2.1.2 and 2.1.3 can be generalized in the following way:Let H ⊂ Pn be the hypersurface given by local equation

∑ni=1 t

di = 0, d ≥ max{3, 7−

n}. Then the germ Vd,H is an orbit of PGLn+1 and hence is a smooth variety ofnon-expected codimension.

2.2. STABLE PROPERTIES 35

2.2 Stable properties of obstructed equianalytic

families

In theorem 2.1.3 we have seen several examples of equianalytic strata of minimalobstructness which are reduced, reducible or have unexpected dimension. However,these phenomena are not stable. Namely, if we add a new variable xn+1 to the spaceand x2

n+1 to the local equation of the hypersurface, the equianalytic stratum of thenew hypersurface has the same h1 and τ but is reduced irreducible of expected codi-mension. Apparently, the same is true for any singularity of minimal obstructness,though sometimes more variables and their squares need to be added.

More generally, for any hypersurface singularity with h1 > 0 but h1(2d− 2) =0, the equianalytic stratum obtains an irreducible component which is reduced ofexpected dimension after adding h1+1 squares. The condition h1(2d−2) = 0 alwaysholds for curves. For higher dimensions, h1(2d − 2) = 0 follows from the conditionh1 < d− 1.

The following theorem summarizes all mentioned above.

Theorem 2.2.1 Let H ⊂ Pn be hypersurface of degree d ≥ 3 with the unique sin-gular point z = (1 : 0 : . . . : 0). Let h1 := h1(JZea(H)/Pn(d)) and τ := degZea(H).Let F0 be the equation of H. Suppose h1 > 0.

For any m ≥ 1 define Wm ⊂ Pn+m to be the hypersurface given by equationF0 +

∑mj=1 t

2n+jt

d−20 and zm be the point (1 : 0 : . . . : 0). Denote by V U

d,Wm the germat Wm of the family of all hypersurfaces of degree d that have a unique singularpoint inside U = {(t0 : ... : tn+m, )|t0 6= 0} ⊂ Pn+m, and are analytically equivalentto (Wm, z) near the singular point. Then:

(a) h1(JZea(Wm)/Pn+m(d)) = h1 and degZea(Wm) = τ .

(b) If h1(JZea(H)/Pn(2d − 2)) = 0 then the germs V Ud,Wm for m ≥ h1 + 1 have

a reduced component of expected dimension.

(c) If H is a plane curve then already h1(JZea(H)/P2(2d − 4)) = 0 and hencefor m ≥ h1 + 1 the germs V U

d,Wm have a reduced component of expected dimension.

(d) If h1 < d − 1 then h1(JZea(H)/P2(2d − 2)) = 0 and hence for m ≥ h1 + 1the germs V U


(e) If h1 = 1 then the germs V Ud,Wm are non-smooth of expected dimension

for all m ≥ 1, reduced for m ≥ max{1, 5−d} and irreducible for m ≥ max{1, 6−d}.

(f) Suppose that F0 =∑n

i=1 tαii t

d−αi0 and h1(JZea(H)/Pn(t)) = 0 for some


d < t ≤ 2d − 2. Then for m ≥ dh1 (t−d+1)2(d−2)

e + 1 the germs V Ud,Wm have a re-

duced component of expected dimension. In particular, if h1(JZea(H)/Pn(2d− 3)) = 0

then for m ≥ dh1

2e + 1 the germs V U

d,Wm have a reduced component of expecteddimension.

(For proof see section 3.4)

Remark 2.2.2 1) Statement (c) is not always true for n ≥ 3. Consider, for example,H given by local equation

∑xdi = 0. Then h1(JZea(H)/Pn(k)) > 0 for k < n(d− 2).

2) It can be proven that if h1(JZea(H)/Pn(2d − 2)) = 0 and h1(JZea(H)/Pn(d + 1)) =h1(JZea(H)/Pn(d))− 1 then for m ≥ 2 the germs V U

d,Wm have a reduced component ofexpected dimension. Together with (f) that implies that if f0 is canonical quasiho-mogeneous and h1 < d− 1 then the germs Vd,Wm(Sm) have a reduced component of

expected dimension and for any m ≥ dh1

2e+ 1.

2.3 Deformation theoretic meaning

Theorem 2.3.1 Let H ∈ Pn be a projective hypersurface of degree d with the uniquesingular point z. Suppose that the equianalytic stratum germ Vd,H has a reducedcomponent R of expected dimension. Then the deformation of H induced by thelinear system |H| is 1-complete.

The proof of the theorem is based on the following observation: at everysmooth point H ′ of R, the stratum Vd,H is T -smooth. Hence the deformation of H ′

induced by the linear system |H| is versal, and hence any 1-parametric deformation(H, z) ↪→ (X , x) → (C, 0) of H can be induced from it by a map ψH′ : (C, 0) →(|H|, H ′). By curve selection lemma, there exists a map φ : (C, 0) → (R,H) suchthat all points except 0 are mapped to non-singular points. Now we define therequested map ϕ : (C, 0) → (|H|, H) by ϕ(t) := ψφ(t)(t) for t 6= 0 and ϕ(0) = 0.

Now we give a precise proof, which includes the description how to choose themaps ψφ(t) analytically.

Proof. Denote U = R \ Sing(Vd) where Sing(Vd) is the singular locus of Vd.Let τ be the Tjurina number of (H, z). Consider the coincidence variety

Z := {(H ′,W )|H ′ ∈ R, W is a τ− dimensional affine subspace of |H| and H ′ ∈ W}.

Let Y ⊂ Z be the open subset defined by

Y := {(H ′,W ) ∈ Z|H ′ ∈ U and W is transversal to R at H ′}.

By curve selection lemma (lemma 1.2.11), there exists a morphism of analytic germsφ : (C, 0) → (Z, (H,W )) (for some τ -dimensional subspace W ) such that φ(C \{0}) ⊂ Y .

2.3. DEFORMATION THEORETIC MEANING 37

Now let (H, z) ↪→ (X , x) → (C, 0) be a one-parametric deformation of (H, z).Let TH denote the Tjurina algebra of H and let (H, z) ↪→ (Y , y) → (TH , 0) be thesemiuniversal deformation over it described in theorem 1.2.9. Since this deformationis semiuniversal, there exists a morphism ψ : (C, 0) → (TH , 0) such that ψ∗(Y , y) =(X , x).

For any point (W,H ′) ∈ Y , the factor morphism TH′Vd,H → TH′ ∼= TH , whereTH′Vd is the tangent space to Vd at H ′, defines an isomorphism W ∼= TH . Hence forany point (W,H ′) ∈ Y , ψ defines a morphism (C, 0) → (W,H ′) ⊂ (|H|, H ′). Thisdefines a morphism Ψ : (C, 0)× Y → (|H|, H) that satisfies

Ψ∗(Z , z) = pr∗1(X , x),

where (Z , z) → (|H|, H) denotes the deformation of H induced by its linear systemand pr1 : (C, 0)× Y → (C, 0) denotes the projection on the first coordinate.

Now, we define ϕ : (C, 0) \ 0 → (|H|, H) by ϕ := Ψ ◦ (Id × φ), and extend itto 0 by ϕ(0) := H. 2

Corollary 2.3.2 Let H ∈ Pn be a unisingular hypersurface of degree d ≥ 3 defined

by the equationn∑i=1

tαii t

d−αi0 +

n∑i=1

λitdi = 0. Suppose that H has one isolated singularity

and h1(JZea(H)/Pn(d)) = 1. Then, unless n = 2, d ≤ 6 or n = 4, d = 3, thedeformation of H induced by the linear system |H| is 1-complete.

Corollary 2.3.3 Let H ⊂ Pn be a hypersurface of degree d ≥ 3 with the uniquesingular point z = (1 : 0 : . . . : 0). Let F0 be the equation of H. For any m ≥ 1define Wm ⊂ Pn+m to be the hypersurface given by equation F0 +

∑mj=1 t

2n+jt

d−20 and

zm be the point (1 : 0 : . . . : 0).Suppose that h1(JZea(H)/Pn(2d−2)) = 0. Then for m ≥ h1 +1 the deformation

of Wm induced by the linear system |Wm| is 1-complete.

Let us now demonstrate one of known applications of 1-completeness. Sup-pose that we want to construct a hypersurface of degree d having m isolated singularpoints of prescribed analytic singularity types S1,...,Sm. Suppose that we can con-struct a hypersurface H with unique more complicated singularity that splits tosingularities of the given types Si after one-parameter deformation by hypersurfacesof higher degrees. If the deformation of H induced by the linear system |H| is 1-complete, there exists a deformation of H by hypersurfaces from |H| which containsdesired hypersurfaces.

Proposition 2.3.4 Let H ⊂ Pn be a hypersurface of degree d with one isolated

singular point z of the analytic singularity type S. Let (H, z)i↪→ (X , x)

φ→ (|H|, H)


be the deformation of (H, z) induced by the linear system |H|. Suppose that it is 1-complete. Let S1,..,Sm be analytic singularity types. Suppose also that there exists a

one-parameter deformation (j, ψ) : (H, z)j↪→ (Y , y)

ψ→ (C, 0) of (H, z) that includeshypersurfaces having m singularities of types S1,...,Sm. Then there exists a one-parameter deformation of (H, z) consisting of hypersurfaces of degree d that includeshypersurfaces having m singularities of types S1,...,Sm.

Proof. Since the deformation of H induced by |H| is 1-complete, there exists amorphism ϕ : (C, 0) → (|H|, H) such that (j, ψ) is isomorphic to the induced defor-mation (ϕ∗i, ϕ∗φ). Hence the induced deformation (ϕ∗i, ϕ∗φ) includes hypersurfaceshaving m singularities of types S1,...,Sm. On the other hand, the deformation (i, φ)consists of hypersurfaces of degree d, and hence the induced deformation (ϕ∗i, ϕ∗φ)also consists of hypersurfaces of degree d. 2

Chapter 3

The proof of the main results

3.1 Affine coordinates and the stratum V Ud

In this section we enter the notion of stratum V Ud that will be used in all the proofs,

in order to work in affine coordinates.

Definition 3.1.1 Let S be an analytic singularity type of projective hypersurfaces.Fix homogeneous coordinates on Pn and consider the open subset

U = {(t0 : . . . : tn)| t0 6= 0} ⊂ Pn.

We define V Ud (S) to be the space of all hypersurfaces of degree d that have a unique

singular point inside U of singularity type S.

Note that both V Ud (S) and Vd(S) are open subsets of the space of all hyper-

surfaces of degree d that have at least one isolated singular point of singularity typeS. Hence for any hypersurface H ∈ V U

d (S) ∩ Vd(S), the germs Vd,H := (Vd(S), H)and V U

d,H := (V Ud (S), H) coincide. Hence we will formulate statements on Vd,H and

prove them on V Ud,H .

Remark 3.1.2 There exists a natural embedding ν : V Ud (S) ↪→ V U

d+1(S) defined inthe following way. Let F (t0, . . . , tn) be an equation of the hypersurface H ∈ V U

d (S).Then we define ν(H) to be the hypersurface defined by the equation t0F (t0, . . . , tn) =0. Note that in the coordinate system xi = ti

t0on U , H and ν(H) will be given by

the same local equation.

Using the above embedding, the scheme theoretic structure on V Ud (S) can

be computed in the following way. By theorem 1.1.23 there exists N such thatV Ud+N(S) is a smooth variety. Then V U

d (S) is equal to the scheme-theoretic intersec-tion V U

d+N(S) ∩ |OPn(d)|.

39

40 CHAPTER 3. THE PROOF OF THE MAIN RESULTS

3.2 Proof of the theorems on quasihomogeneous

plane curve singularities

We could give a joint proof to theorems 2.1.2 and 2.1.3 but decided to write themseparately since the proof for the case of plane curves is more transparent. Theorem2.1.1 is clearly a special case of theorem 2.1.2.

First of all we remind the formulation of the theorem.Theorem 2.1.2 Let C be the projective plane curve given by the equation tk1t

d−k0 +

tl2td−l0 + λ1t

d1 + λ2t

d2 = 0, where d = k + l− 5, and λi are complex numbers such that

z = (1, 0, 0) is the unique singular point of C. Note that generic λi satisfies thiscondition. Suppose, for convenience, k ≥ l.

Let Vd,C be the germ at C of the equianalytic family of C. Then for any k, l ≥ 5,Vd,C is non-T-smooth and h1(JZea(C)/P2(d)) = 1.Furthermore:

(i) If d = 5 (i.e. l = k = 5), the germ V5,C is smooth of codimension less thanexpected by one.

(ii) If d = 6 (i.e. l = 5, k = 6), the germ V6,C is non-reduced. It is a double[V6,C ]red, and [V6,C ]red is smooth of expected codimension.

(iii) If d = 7 (i.e. l = 5, k = 7 or k = l = 6), the germ V7,C is reducibleand decomposes into two smooth components of expected codimension that intersectnon-transversally with multiplicity one. The intersection locus is smooth. Moreover,the germ V7,C has the sectional singularity type A1.

(iv) If d ≥ 8, the germ Vd,C is reduced irreducible non-smooth variety ofexpected codimension which has a smooth singular locus. Moreover, the germ Vd,Chas the sectional singularity type A1.

3.2.1 The structure of the proof

First of all we pass to affine coordinates x = t1t0

, y = t2t0

. In these coordinates, C is

given by the local equation f = xk + yl + λ1xd + λ2y

d = 0. It is a semiquasihomo-geneous polynomial with non-degenerate quasihomogeneous part g = xk + yl. It iseasy to see that f ∈ j(f) and j(f) = j(g). Hence f and g have the same Tjurinaideal and Tjurina algebra, are contact equivalent and Tf = Mf .

To verify T-smoothness we have to compute h1(JZea(C,z)/P2(d)).

h1(JZea(C,z)/P2(d)) = h0(JZea(C,z)/P2(d))− h0(OP 2(d)) + h0(OZea(C,z)).

The computations can be illustrated by means of Newton polygon (see figure 3.1).h0(OP 2(d)) equals the number of integer points in the triangle {(i, j)| i + j ≤ d}.h0(OZea(C,z)) equals the number of integer points in the rectangle {(i, j)| i ≤ k − 2,j ≤ l− 2}. h0(JZea(C,z)/P2(d)) equals the number of integer points that lie inside the

3.2. PROOF OF THEOREM 2.1.2 41

triangle but outside the rectangle. Hence h1(JZea(C,z)/P2(d)) equals the number ofinteger points that lie inside the rectangle but outside the triangle, i.e. equals one(see precise computation below in subsection 3.2.2 in the proof of the theorem ).

For the same reason, h1(JZea(C,z)/P2(d+ 1)) = 0.

Therefore Vd+1,C is smooth of expected codimension. Hence it is locally definedby τ analytic equations in coefficients with non-degenerate linearization.

We divide the proof into several cases. First, we consider the case k < 2l(subsections 3.2.4 and 3.2.5). In this case for any hypersurface H which lies inthe stratum germ there exists an affine coordinate change s.t. the equation ofH in the new coordinates does not include any terms that lie below the Newtonpolygon ∆(g) = {(i, j) ∈ Z2

≥0| ik + jl

= 1}. Hence we can consider substratum

Vd,C transversal to orbits of PGL3 consisting of hypersurfaces whose local equations

do not include any terms below ∆(g). In fact, instead Vd,C we will use several itssubstrata, described in details in the body of the proof.

By theorem 1.1.15, for any polynomial F which has no terms below ∆(g), theMilnor algebras MF and Mg are isomorphic. On the other hand, we have seen thatMg

∼= Tg ∼= Tf . By the Mather-Yau theorem 1.1.5, F is contact equivalent to f

if and only if TF ∼= Tf . Hence Fc∼ f iff MF

∼= TF , i.e., F ∈ 〈Fx, Fy〉. So thehypersurface HF defined by a polynomial F which has no terms below ∆(g) belongs

to Vd,C if and only if F ∈ 〈Fx, Fy〉.We check that condition using a computer algebra algorithm (3.2.1). In this

way we obtain a system of equations on Vd,C . In cases d = 5, 6, 7 we write downthe whole system and see that the substratum germ has desired properties (seesubsection 3.2.6). For d ≥ 8 we show that the system consists of a subsystemhaving diagonal linear part, and one more equation with quadratic principle part ofrank ≥ 3 (see subsection 3.2.7).

In case k > 2l, it is not enough to use linear coordinate changes in orderto bring any hypersurface of Vd,C to Vd,C . Nonlinear coordinate changes do notpreserve degree and hence do not preserve Vd,C . Hence we cannot pass to substratum

Vd,C . Instead, we take a polynomial F , pass to new coordinates in which F has noterms below ∆(g), write equations on the coefficients of the polynomial in the newcoordinates and express new coefficients through the old ones. Again we check thatthe obtained system consists of a subsystem having diagonal linear part, and onemore equation with quadratic principle part of rank ≥ 3. This is done in subsection3.2.8.

In both cases we show that for d ≥ 8 the last equation lies in the ideal gen-erated by elements that appear in its quadratic part. We deduce from this fact thesmoothness of the singular locus.


3.2.2 Proof of the relation h1(JZea(C,z)/P2(d)) = 1

First of all we pass to affine coordinates x = t1t0

, y = t2t0

. In these coordinates, C is

given by the local equation f = xk + yl + λ1xd + λ2y

d = 0. It is a semiquasihomo-geneous polynomial with non-degenerate quasihomogeneous part g = xk + yl. It iseasy to see that f ∈ j(f) and j(f) = j(g). Hence f and g have the same Tjurinaideal and Tjurina algebra, and Tf = Mf .

The Newton polygon of g is ∆(g) = {(i, j) ∈ Z2≥0| ik + j

l= 1}.

6

-

@@

@@

@@

@@

@@

@@@

d

ll − 2

k − 2 k

j

id

Td

P

6

-

@@

@@

@@@

@@

@@

@@

@@

@@

@@@

d

ll − 2

k − 2l k

j

id

D

D′∆

bij

eij

uij

vij◦

Figure 3.1: Newton polygon

We will now show that Vd,C is non-T-smooth and h1(JZea(C,z)/P2(d)) = 1.The Tjurina algebra of Tf = Txk+yl has a basis {xiyj, (i, j) ∈ P}, where P is

the rectangle P = {(i, j) ∈ Z2≥0| i ≤ k − 2, j ≤ l − 2}.

So τ(C, z) = |P| = (k−1)(l−1), where by |P| we denote the number of integerpoints in P . Hence h0(OZea(C,z)) = τ(C, z) = |P|,

H0(JZea(C,z)/P2(d)) = {∑

(i,j)∈Td\P

aijxiyj}

where Td is the triangle {(i, j) ∈ Z2≥0| i+ j ≤ d} (see figure 3.1). That means

h0(JZea(C,z)/P2(d)) = |Td| − |Td ∩ P|.

From the exact sequence

0 → H0(JZea(C,z)/P2(d)) → H0(OP 2(d)) → H0(OZea(C,z)) → H1(JZea(C,z)/P2(d)) → 0

we conclude that

h1(JZea(C,z)/P2(d)) = h0(JZea(C,z)/P2(d))− h0(OP 2(d)) + h0(OZea(C,z)) =


|Td| − |Td ∩ P| − |Td|+ |P| = |P \ Td| = 1.

The same argument shows that h1(JZea(C,z)/P2(d+ 1)) = |P \ Td+1| = 0.So by theorem 1.1.23 the germ Vd+1,C is T-smooth and the germ Vd,C is non-

T-smooth.Because of minimal obstructness (h1 = 1), Vd,C may be either non-smooth of

expected codimension or smooth of non-expected codimension.Now we would like to find out when it is non-smooth and when it has non-

expected codimension. For convenience, we divide the proof to three cases: l < k <2l, k = l and k ≥ 2l.

3.2.3 Notations that will be used during the whole proof

Let F (x, y) = xk + yl + λ1xd + λ2y

d +∑

i+j≤daijx

iyj. We introduce the following

notations:a) let eij be the coefficients of basis monomials which lie below ∆(g), i.e.

eij := aij for {(i, j)| 0 ≤ i ≤ k − 2, 0 ≤ j ≤ l − 2,i

k+j

l≤ 1};

and bij be the coefficients of basis monomials which lie above ∆(g), i.e.

bij := aij for {(i, j)| 0 ≤ i ≤ k − 2, 0 ≤ j ≤ l − 2,i

k+j

l> 1};

b) let gi := ai,l−1 and hj := ak−1,j, where 2 ≤ i ≤ d− (l−1) and 2 ≤ j ≤ d− (k−1);c) let uij := aij for (i, j) from the triangle {i ≥ 0, j ≥ l, i+ j ≤ d} and vij := aij for(i, j) from the triangle {i ≥ k, j ≥ 0, i+ j ≤ d}.The above notations are illustrated in figure 3.1.

Let A = C[aij] be the algebra of polynomials generated by aij. Letm = 〈aij〉 bethe maximal ideal in A generated by all aij, G := 〈g2, ..., gd−(l−1), h2, ..., hd−(k−1)〉 ⊂ Aand B := 〈bij〉 ⊂ A.

3.2.4 Case 1: l < k < 2l

First of all we will pass to a more convenient substratum, which has the samegeometric properties. Let S be the singularity type of (C, z).

The V Ud (S) is invariant under the action of the group of affine transformations

of C2. Consider the subgroup generated by x 7→ x, y 7→ y − ax and translations.We can choose section of V U

d (S) transversal to orbits of this group and given by theconditions that the singularity is in the origin and the tangent to the curve at it is{y = 0}. The section is the substratum V l

d(S) of all curves CF in V Ud (S) given by


polynomials F which have zero coefficients of xiyj for i + j ≤ l unless j = l. Notethat fixing of the singular point and the tangent has lowered the dimension by 3.

In the same way, by considering the subgroup generated by y 7→ y, x 7→ bx−cywe can reduce to substratum V l,0,1

d (S) of V ld(S) consisting of all curves CF given by

polynomials F which also don’t include the monomial xk−1y and have the samecoefficients of xk and yl. Its dimension is dim V l,0,1

d (S) =dim Vd(S)− 5.

So we will work with substratum germ V l,0,1d,C of Vd,C consisting of curves given

by polynomials of the form

F (x, y) = yl + xk + λ1xd + λ2y

d +∑

(i,j)∈D

aijxiyj,

where D = {l < i+ j ≤ d, (i, j) 6= (k − 1, 1), (i, j) 6= (k, 0)}.

Now we want to find out for which aij CF lies in V l,0,1d,C . Since F (x, y) does not

include the monomial xyl−1 , in order for F to be contact equivalent to f , all thecoefficients of monomials of F under the Newton polygon ∆(g) should vanish. Inother words, we have the following equations on aij:

eij = 0 fori

k+j

l≤ 1, where 0 ≤ i ≤ k − 2, 0 ≤ j ≤ l − 2 (1)

h0 := ak−1,0 = 0 if k > l + 1. (2)

So we continue to work with

F (x, y) = yl + xk + λ1xd + λ2y

d +∑

(i,j)∈D′aijx

iyj,

where D′ = {(i, j)| i, j ≥ 0,i

k+j

l> 1, i+ j ≤ d, (i, j) 6= (k − 1, 1)}.

By theorem 1.1.15 on Milnor algebras of semiquasihomogeneous polynomials,MF ' Mf . As we have shown, Mf = Tf . By the Mather-Yau theorem 1.1.5 F

c∼ f

iff TF ' Tf . But Tf = Mf 'MF hence Fc∼ f iff MF ' TF i.e. F (x, y) ∈ 〈Fx, Fy〉.

In order to check whether F (x, y) ∈ 〈Fx, Fy〉 we use the redNFBuchbergeralgorithm (see section 1.3.3). We refer to small neighborhood of the origin, hencewe consider F (x, y) and 〈Fx, Fy〉 in the local ring R = C[x, y]〈x,y〉. For computing inthis ring we define a local monomial ordering on C[x, y] such that the ring associatedto C[x, y] and this ordering will be C[x, y]〈x,y〉 . We choose the negative weighteddegree lexicographical ordering with w = (l, k) (see Example 1.3.3, (2)(iii)).

In general, the redNFBuchberger algorithm does not stop for local order-ings. However, in our case we can stop it manually when the leading monomialof the tail is less than xk−2yl−2. We are allowed to do that by lemma 1.3.10,for xk−2yl−2 is highest corner of 〈LM(Fx), LM(Fy)〉 = 〈xk−1, yl−1〉. Indeed, any


monomial smaller than xk−2yl−2 has either degree of x bigger or equal to k − 1,or degree of y bigger or equal to l − 1 and hence lies in 〈LM(Fx), LM(Fy)〉 andxk−2yl−2 /∈ 〈LM(Fx), LM(Fy)〉.

There is another explanation why we can stop the algorithm at this point.Consider Vd+1,C . As above, we compute h1(JZea(C,z)/P2(d + 1)) = 0. Hence Vd+1,C

is smooth of expected codimension. So V l,0,1d+1,C is also smooth and has expected

codimension which is equal to the number of basis elements which lie above theNewton polygon. Since we have exactly this number of independent equations onthis stage, there will be no more equations. Also since V l,0,1

d+1,C is smooth, all the

equations on it will have independent linear parts. When we return to V l,0,1d,C , linear

part of only one of them may vanish.So we rewrite the algorithm in the following way:

Algorithm 3.2.1 (Modified redNFBuchberger algorithm.)

1. p := 0, h := F ;

2. while (h 6= 0 and LM(h) ≥ xk−2yl−2)

(a) while (h 6= 0 and LM(h) ≥ xk−2yl−2 and(LM(Fx) divides LM(h) or LM(Fy) divides LM(h)))

i. if (LM(Fx) divides LM(h)){h := h− (LT (h)/LT (Fx)) · Fx}

ii. if (h 6= 0 and LM(Fy) divides LM(h)){h := h− (LT (h)/LT (Fy)) · Fy}

(b) if (h 6= 0 and LM(h) ≥ xk−2yl−2){p := p+ LT (h);h = tail(h)};

3. return p;

As a result, we obtain the normal form

NF (F |〈Fx, Fy〉) =∑

Rmn(aij)xmyn,

where xmyn, 0 ≤ m ≤ k − 2, 0 ≤ n ≤ l − 2 are elements of the basis of algebraC[x, y]/〈fx,fy〉 which lie above ∆(g) and Rmn(aij) are polynomials in aij. Hence Fbelongs to the ideal 〈Fx, Fy〉 if and only if all coefficients Rmn(aij) = 0.


Let G = 〈g2, ..., gd−(l−1), h2, ..., hd−(k−1)〉 and B = 〈bij〉. Transforming oursystem of equations to an equivalent one we obtain

bij = ψij(g2, ..., gd−(l−1), h2, ..., hd−(k−1), upq, vrs, bmn) for (i, j) 6= (k − 2, l − 2) and

0 = ψk−2,l−2(g2, ..., gd−(l−1), h2, ..., hd−(k−1), upq, vrs, bmn),

where ψij is a polynomial from G2 +Bm.Note, that this system is sorted by the quasihomogeneous degree of left hand

side and all bmn in the right hand sides are left hand sides of previous equations.Hence we can replace our system by an equivalent one:

bij = ψij(g2, ..., gd−(l−1), h2, ..., hd−(k−1), upq, vrs) for (i, j) 6= (k − 2, l − 2) (3)

where ψij ∈ G2 are polynomials in (gi, hj, upq, vrs) and

P :=

d−(l−1)∑i=2

l − 1

lgigk−2−i +

d−(k−1)∑j=2

k − 1

khjhl−2−j +

+ Θ(g2, ..., gd−(l−1), h2, ..., hd−(k−1), upq, vrs) = 0, (4)

where Θ(g2, ..., gd−l+1, h2, ..., hd−k+1, upq, vrs) is a polynomial from G2m.Merging this system with (1) and (2) we see that our substratum germ is

isomorphic to the germ at 0 of the affine variety given in the affine space withcoordinates {g2, ..., gd−l+1, h2, ..., hd−k+1, upq, vrs} by the last equation (4). We denote

this germ by V0.Let us now prove that the same is true in the case k = l, and then finish the

proof for both cases.

3.2.5 Case 2: k = l

As in the previous case, using PGL3 we reduce our consideration to the substratumV ld(S) consisting of curves having singular point at the origin with the same tangent

cone as C. By using the subgroup of PGL3 x 7→ x/a, y 7→ y we pass to thesubstratum V l,1

d (S) of curves given by equations F (x, y) = 0, where

F (x, y) = xl + yl + λ1xd + λ2y

d +∑

l<i+j≤d

aijxiyj.

Note that dim V l,1d (S) =dim Vd(S)− 5. Using modified redNFBuchberger algo-

rithm (algorithm 3.2.1), we obtain system of equations on V l,1d,C . As in the previous

case, we transform this system of equations to an equivalent system of equationswith diagonal linear part. The first equations express all the basis elements abovei+ j = l through other coefficients and the last one is (4).


So, as in the previous case, our substratum germ is isomorphic to thegerm V0 at 0 of the affine variety given in the affine space W with coordinates{g2, ..., gd−l+1, h2, ..., hd−k+1, upq, vrs} by the last equation (4).

3.2.6 Proof of statements (i), (ii) and (iii)

(i) For k = l = 5, d = 5 = l and hence V 5,15,C consists of one point. Hence V5,C is

also a smooth algebraic variety of dimension 0 + 5 = 5. The expected dimension isd(d+3)

2− (k − 1)(l − 1) = 4. So, V5,C has non-expected codimension.

(ii) For l = 5, k = 6, d = k + l − 5 = 6 the points of V 5,0,16,C are curves

given by equations of the form

F (x, y) = y5 + (1 + λ1)x6 + λ2y

6 + b42x4y2 + b33x

3y3 + g24x2y4 + u15xy

5 + u06y6.

Using modified redNFBuchberger algorithm (algorithm 3.2.1) we obtainb42 = 0, b33 = 0, and (g24)

2 = 0, which implies the statement (i).

(iii) For l = 5, k = 7, d = k + l − 5 = 7 the variety V 5,0,17,C is given in the

affine space W by the equation

g24(g34 −1

2g24u15) = 0.

So it is reducible and decomposes into two smooth components of expected codimen-sion that intersect transversally in W . Hence, the two components of V7,C intersectin V|C| non-transversally with multiplicity one and the sectional singularity is of typeA1.

For k = l = 6, d = k+ l− 5 = 7 the variety V 6,17,C is given in the affine space W

by the equation h252 + g2

25 = 0. So it is reducible and decomposes into two smoothcomponents of expected codimension that intersect transversally in W . Hence, thetwo components of V7,C intersect in V|C| non-transversally with multiplicity one andthe sectional singularity is of type A1.

3.2.7 Finish of the proof of statement (iv) for cases 1 and 2

In this case k, l ≥ 5, d = k + l − 5 ≥ 8. The quadratic part of equation (4) is

Q :=

d−(l−1)∑i=2

Aigigd−l+3−i +

d−(k−1)∑j=2

Bjhjhd−k+3−j.


Since Ai and Bi are positive, it is a quadratic non-degenerate form of rank r =d − l + d − k = 2d − (l + k) = d − 5. Since for d ≥ 8 the rank r = d − 5 ≥ 3, oursubstratum germ is reduced irreducible non-smooth variety of expected codimension.

Now we are going to prove that the singular locus Y of our substratum germcoincides with the germ X0 at the origin of the affine subspace X = Z(G). Sinceequation (4) lies in G2, all its first order partial derivatives lie in G, and hence thesingular locus includes X0. Let Z be the variety given by the equations

∂P

∂g2

= 0, ...,∂P

∂gd−l+1

= 0,∂P

∂h2

= 0, ...,∂P

∂hd−k+1

= 0.

Since the linear part of this system of equations is non-degenerate, the germ Z0 of Zat the origin is smooth and hence irreducible. Clearly Y ⊂ Z0 and hence X0 ⊂ Z0.They have the same dimension and Z0 is irreducible hence X0 = Z0 which impliesX0 = Y .

So our substratum germ is reduced irreducible non-smooth variety of expectedcodimension, which has smooth singular locus.

It is left to show that the sectional singularity type is A1.

The sectional singularity type of a variety germ with smooth singular locusis the singularity type of transversal intersection of the singular locus with a linearspace.

Consider the linear subspace L spanned by gi and hj. We have seen that it is

transversal to the singular locus of V0 in W . Hence the sectional singularity type ofV0 is the singularity type of the scheme theoretic intersection L∩ V0. L∩ V0 is givenin L by one equation with quadratic principle part which is not degenerate in thevariables gi and hj. Hence its singularity type is A1.

3.2.8 Case 3: k ≥ 2l

In the same way as in case 1, we will pass to a more convenient substratum, whichhas the same geometric properties. First, we pass to the substratum V l

d(S) of allcurves CF in V U

d (S) given by polynomials F which have zero coefficients of xiyj fori + j ≤ l unless j = l. Then by considering the subgroup generated by y 7→ y,x 7→ x − by we can reduce to substratum V l,0

d (S) of V ld(S) consisting of all curves

CF in V ld(S) given by polynomials F which also don’t include the monomial xk−1y.

Note that dimV l,0d (S) = dimVd(S)− 4.

Thus, we will work with substratum germ V l,0d,C of Vd,C consisting of curves

given by polynomials of the form

F (x, y) = yl+xk+λ1xd+λ2y

d+∑

(i,j)∈D

aijxiyj, where D = {l < i+j ≤ d, (i, j) 6= (k−1, 1)}.


Now we make the following series of [k/l] − 1 coordinate changes. The firstone will be x 7→ x, y 7→ y − g2

lx2. The coefficients of the polynomial F in the

new coordinates are expressed through the coefficients in the old coordinates by theformula

a(2)ij = aij +

[i/2]∑n=1

(−1)nCnj+nai−2n,j+n(

g2

l)n,

We continue in the same way. The coordinate change number s− 1 will be x 7→ x,y 7→ y − gs

lxs and

a(s)ij = a

(s−1)ij +

[i/s]∑n=1

(−1)nCnj+na

(s−1)i−sn,j+n(

g(s−1)s

l)n. (5)

We stop when s = [k/l]. Clearly the degrees of polynomials of degree ≤ d remainuniversally bounded after these coordinate changes. Denote this bound by N .

Let a′ij be the coefficients of our polynomials after all these coordinate changes,

i.e. a′ij := a([k/l])ij . Clearly g′i = 0, i = 2, ..., [k/l]. Since the coordinate changes (5)

were local analytic diffeomorphisms, the curve CF given by the polynomial equationF (x, y) = 0 lies in V l,0

d,C if and only if the curve CF given by F (x, y) = 0 lies in V l,0N,C

where

F (x, y) = yl + xk + λ1xd + λ2y

d +∑

a′ijxiyj.

Now we want to find out for which a′ij CF lies in V l,0N,C . Since F (x, y) does

not include the monomials xiyl−1 for i = 1, ..., [k/l], in order for F to have quasi-homogeneous singularity of type (1/k, 1/l, 1) all the coefficients of monomials underthe Newton polygon ∆(g) should vanish. In other words, we have the followingequations on a′ij:

e′ij = 0 fori

k+j

l≤ 1, where 0 ≤ i ≤ k − 2, 0 ≤ j ≤ l − 2 (6)

h′0 := a′k−1,0 = 0 (7)

So we continue to work with

F (x, y) = yl + (1 + a′k0)xk + λ1x

d + λ2yd +

∑(i,j)∈D′′

a′ijxiyj,

where D′′ = {(i, j)| i, j ≥ 0,i

k+j

l> 1, i+ j ≤ N}.

According to theorem 1.1.15 on Milnor algebras of semiquasihomogeneouspolynomials, MF ' Mg. As we have seen, Mg

∼= Tf . The Mather-Yau theorem


1.1.5 states that Fc∼ f iff TF ' Tf . But Tf ' MF hence F

c∼ f iff MF ' TF i.e.F (x, y) ∈ 〈Fx, Fy〉.

As in the previous cases, in order to check whether F (x, y) ∈ 〈Fx, Fy〉 we usethe modified redNFBuchberger algorithm (algorithm 3.2.1).


NF (F |〈Fx, Fy〉) =∑

Rmn(a′ij)x

myn,

where xmyn, 0 ≤ m ≤ k − 2, 0 ≤ n ≤ l − 2 are elements of the basis of algebraC[x, y]/〈fx,fy〉 which lie above ∆(g) and Rmn(a

′ij) are rational functions in a′ij whose

denominators are polynomials in a′k0 non vanishing at zero. Hence F belongs to the

ideal 〈Fx, Fy〉 if and only if all coefficients Rmn(a′ij) = 0.

As in previous cases, we transform this system of equations to an equivalentone and obtain

b′ij =ψ′ij(g

′[k/l]+1, ..., g

′l−3−[k/l], h

′[l/k]+1, ..., h

′k−3−[l/k], u

′pq, v

′rs)

(1 + v′k0)γi,j

(8)

where ψ′ij ∈ G′2 for G′ = 〈g′[k/l]+1, ..., g′l−3−[k/l], h

′[l/k]+1, ..., h

′k−3−[l/k]〉.

The number of equations in system (8) is equal to the number of basis coeffi-cients above the Newton polygon ∆(g).

Now we express new coefficients through the old ones. As we can see from5, a′ij = aij + ϕij(apq) for (i, j) ∈ D and a′ij = ϕij(apq) for (i, j) /∈ D, whereϕij(apq) ∈ mI.

Consider system (6). It is sorted by the quasihomogeneous degree of left handside. After substituting old coefficients it will be eij = −ϕij(g2, ..., g[k/l], emn), whereall emn are left hand sides of previous equations. Hence we can replace our systemby an equivalent one:

eij = −ϕij(g2, ..., g[k/l]) for all (i, j).

Now we substitute −ϕij instead of eij in all the other equations. The additionalequation (7) acquires the form h0 = ϕk−1,0(g2, ..., g[k/l]). Consider system (8) exceptthe last equation i.e. the equation on b′k−2,l−2. After substituting the old coefficientsand multiplying by denominators it will become

bij = ψij(g2, ..., gd−l+1, h2, ..., hd−k+1, upq, vrs, bmn) (9)

where ψ ∈ m2.

The last equation is of particular interest. In the old coordinates its right hand


side will be

d−(l−1)+2−([k/l]+1)∑i=[k/l]+1

Aigigd−(l−1)+2−i +

d−(k−1)∑j=2

Bjhjhd−(k−1)+2−j +

+ Ψ(g2, ..., gd−l+1, h2, ..., hd−k+1, upq, vrs, bmn), (10)

where Ψ is rational function regular at the origin whose numerator lies in G2.The left hand side will be 0 +

∑[k/l]i=2 Aigigd−l+3−i + Φ(g2, ..., gd−l+1, upq), where

Φ(g2, ..., gd−l+1, upq) is a polynomial from G2m and Ai are positive real numbers.

Multiplying by the denominator of Ψ we obtain the equation

P :=

d−(l−1)∑i=2

Aigigd−l+3−i +

d−(k−1)∑j=2

Bjhjhd−k+3−j +

+ Θ(g2, ..., gd−l+1, h2, ..., hd−k+1, upq, vrs) = 0, (11)

where Θ(g2, ..., gd−l+1, h2, ..., hd−k+1, upq, vrs) lies in G2m and Ai, Bj are positive realnumbers.

So our substratum germ V l,0d,C is isomorphic to the germ at 0 of the affine variety

given in the affine space with coordinates {g2, ..., gd−l+1, h2, ..., hd−k+1, upq, vrs, bij} bythe system of equations (9) with diagonal linear part and the last equation (11).

Since in this case d = k + l − 5 ≥ 3l − 5 ≥ 10, we have to prove that V l,0d,C is

reduced irreducible non-smooth variety of expected codimension which has a smoothsingular locus with the sectional singularity of type A1.

The system (9) has a diagonal linear part. Hence all the equations in it areindependent and the variety W , defined by it, is smooth at 0. The last equation(11) does not depend on the preceding ones and does not have a linear part. HenceV l,0d,C is non-smooth of expected codimension.

The quadratic part of equation (11) is

Q :=

d−(l−1)∑i=2

Aigigd−l+3−i +

d−(k−1)∑j=2

Bjhjhd−k+3−j.

Since Ai, Bj are positive, it is quadratic non-degenerate form of rank

r = d− l + d− k = 2d− (l + k) = d− 5 ≥ 5. (12)

Hence the variety {P = 0} is reduced and irreducible and hence our germ is alsoreduced and irreducible.

Now we are going to prove that the singular locus Y of our substratum germcoincides with the germ X0 at the origin of the affine subvariety X given in W bythe ideal G.


Since equation (11) lies in G2, all its first order partial derivatives lie in G, andhence X0 lies in the singular locus. Consider the jacobian of the system obtainedby merging system (9) with the last equation (11). Let M(i) be its minor givenby columns that include partial derivatives by bmn and gi and N(j) be its minorgiven by columns that include partial derivatives by bmn and hj. The linear part ofM(i) is Aigd−l−3−i and of N(j) is Bjgd−k−3−j. Let Z0 be the germ at the origin ofsubvariety of W given by the system of equations M(i) = 0 and N(j) = 0. Since thelinear part of this system is diagonal, Z0 is irreducible and has the same dimensionas X0. As X0 ⊂ Y ⊂ Z0, it implies X0 = Y = Z0.

So the substratum germ V l,0d,C is reduced irreducible non-smooth variety of

expected codimension which has a smooth singular locus.The sectional singularity type is computed in a way similar to cases 1 and 2.Consider the linear subspace L spanned by bij, gi and hj. We have seen that

it is transversal to the singular locus of V l,0d,C in the affine space with coordinates

{g2, ..., gd−l+1, h2, ..., hd−k+1, upq, vrs, bij}. Hence the sectional singularity type of V l,0d,C

is the singularity type of the scheme theoretic intersection L∩V l,0d,C . L∩V l,0

d,C is givenin L by system of equations (9) with the linear part non-degenerate in bij andone more equation with a quadratic principle part which is not degenerate in thevariables gi and hj. Hence the singularity type of L ∩ V l,0

d,C is A1. 2

3.3 Proof of the theorem on quasihomogeneous

hypersurface singularities

This section is dedicated to the proof of theorem 2.1.3. For convenience, we remindits formulation.Theorem 2.1.3 Let H ⊂ Pn, n ≥ 3 be the projective hypersurface given by the

equationn∑i=1

tαii t

d−αi0 +

n∑i=1

λitdi = 0 where d =

n∑i=1

αi − (2n + 1) and λi are complex

numbers such that z = (1, 0, ..., 0) is the unique singular point of H. Note thatgeneric λi satisfy this condition.

Let Vd,H be the germ at H of the equianalytic family of H. Then for any{αi}ni=1 ≥ 2 such that d ≥ αi for all i, Vd,H is non-T-smooth and h1(JZea(H)/Pn(d)) =1.Furthermore:

(i) If n = 4 and d = 3 (i.e. α1 = α2 = α3 = α4 = 3), the germ V3,H(S) issmooth variety of non-expected codimension (one less than expected).

(ii) Otherwise, the germ Vd,H(S) is reduced irreducible non-smooth variety ofexpected codimension which has a smooth singular locus. Moreover, the germ Vd,H(S)has the sectional singularity type A1.



As in the case of curves, we pass to affine coordinates xi = tit0

. In these coordinates,

H is given by local equation f =∑n

i=1 xαii +

∑ni=1 λix

di = 0. Again, we show that f

has the same Tjurina ideal and Tjurina algebra as g =∑n

i=1 xαii (and hence f

c∼ g)and Tf = Mf .

Then we prove the equality h1(JZea(H,z)/Pn(d)) = 1 (see subsection 3.3.2).

Next (in subsection 3.3.3) we switch to substratum germ V 0,0d,H of Vd,H consisting of

hypersurfaces given by polynomials of the form

F (x1, ..., xn) =n∑i=1

xαii +

n∑i=1

λixdi +

∑I∈D

aIxI , where D = {I ∈ Zn

≥0|αn < |I| ≤ d}\

\⋃

1≤i6=j≤n

{(0, ..., 0, αi − 1, 0, ..., 0, 1, 0, ..., 0)} \⋃

1≤i≤n

{(0, ..., 0, αi, 0, ..., 0)}.

We prove that this substratum is transversal to the orbits of the group of affinetransformations of Cn.

Again we divide the proof into two cases. In case α1 < 2αn (see section 3.3.4),a hypersurface HF lies in V 0,0

d,H if and only if F has no terms below and on theNewton polytope ∆(g) and F (x1, . . . , xn) ∈ 〈Fx1 , . . . , Fxn〉, for the same reason asin the case of curves. We again use a computer algebra algorithm in order to checkwhether F (x1, . . . , xn) ∈ 〈Fx1 , . . . , Fxn〉. In this way we obtain a system of equationson V 0,0

d,H . In case d = 3, n = 4 the substratum germ consists of one point. We showthat otherwise the system consists of a subsystem having a diagonal linear part, andone more equation with quadratic principle part of rank ≥ 3.

In the case of α1 ≥ 2αn, there are hypersurfaces HF in V 0,0d,H whose equations

include some terms below or on the Newton polytope ∆(g). For every such polyno-mial F , we pass to new coordinates in which F has no terms below and on ∆(g),write equations on the coefficients of F in the new coordinates and express newcoefficients through the old ones. In this theorem we will make more complicatedcoordinate changes than in the case of curves.

Again we check that the obtained system consists of a subsystem having diag-onal linear part, and one more equation with quadratic principle part of rank ≥ 3.This is done in subsection 3.3.5.

As in the previous theorem, we show that in both cases the last equation liesin the ideal generated by elements that appear in its quadratic part. We deducefrom this fact the smoothness of the singular locus.

3.3.2 Proof that h1(JZea(H,z)/Pn(d)) = 1

Suppose, for convenience, α1 ≥ α2 ≥ ... ≥ αn ≥ 2.


First, we pass to affine coordinates xi = tit0

. In these coordinates, H is given by

local equation f =∑xαii +

∑λix

di . It is a semiquasihomogeneous polynomial with

non-degenerate quasihomogeneous part g =∑xαii . It is easy to see that f ∈ j(f)

and j(f) = j(g). Hence f and g have the same Tjurina ideal and Tjurina algebra,which also coincides with their Milnor algebras.

The polynomial g is quasihomogeneous of type (1/α1, . . . , 1/αn; 1) hence (H, z)is quasihomogeneous hypersurface singularity. The Newton polytope of g is ∆(g) =

{I ∈ Zn≥0|

∑nj=1

Ijαj

= 1}.

6

-�

��

��

��

@@

@@

@@

@@

@@

@@@

��

d

α1 − 2

P

Td

dα2 − 2

d

α3 − 2

6

-�

��

��

��

@@

@@

@@

@@

@@

@@@

��

d

α1

dα2

d

α3

Figure 3.2: Newton polytope

We will now show that Vd,H is non-T-smooth at H and h1(JZea(H,z)/Pn(d)) = 1.The Tjurina algebra of f has a basis {xI , I ∈ P}, where P is the parallelepiped

P = {I ∈ Zn≥0| Ij ≤ αj − 2 for all 0 ≤ j ≤ n}. So τ(H, z) = |P| =

∏ni=1(αi − 1),

where by |P| we denote the number of integer points in P .Hence h0(OZea(H,z)) = τ(H, z) = |P|,

H0(JZea(H,z)/Pn(d)) = {∑

I∈Td\P

aIxI}

where Td is the simplex {I ∈ Zn≥0| |I| ≤ d} (see figure 3.2). That means that

h0(JZea(H,z)/Pn(d)) = |Td| − |Td ∩ P|


From the exact sequence

0 → H0(JZea(H,z)/Pn(d)) → H0(OPn(d)) → H0(OZea(H,z)) → H1(JZea(H,z)/Pn(d)) → 0

we conclude that

h1(JZea(H,z)/Pn(d)) = h0(JZea(H,z)/Pn(d))− h0(OPn(d)) + h0(OZea(H,z)) =

|Td| − |Td ∩ P| − |Td|+ |P| = |P \ Td| = 1.

The same argument shows that h1(JZea(H,z)/Pn(d + 1)) = |P \ Td+1| = 0. Thus bytheorem 1.1.23 the germ Vd+1,H is T-smooth and the germ Vd,H is non-T-smooth.

Because of minimal obstructness (h1 = 1), Vd,H may be either non-smooth ofexpected codimension or smooth of non-expected codimension.

Now we would like to find out when it is non-smooth and when it has non-expected codimension. First we will pass to a more convenient substratum, whichhas the same geometric properties.

3.3.3 Switch to substratum and notations

First of all let us shift the singularity to the origin. Let S be the singularity typeof (H, z). The family V U

d (S) is invariant under the action of affine transformationsof Cn. Consider the subgroup generated by translations. We switch to the sectionV ′d(S) of V U

d (S) transversal to orbits of this group and given by the conditions thatthe singularity is in the origin.

In the same way, using the subgroup GLn consisting of the linear coordinatechanges, we want to reduce to substratum V 0,0

d (S) of V ′d(S) consisting of all hy-

persurfaces HF given by polynomials F which also do not include the monomialsxαi−1i xj for i 6= j, and include the monomials xαi

i with coefficient 1 if αi 6= d or withcoefficient 1 + λi if αi = d . For this purpose we will prove the following lemma.

Lemma 3.3.1(i) THV

′d(S) = THV

0,0d (S)⊕ THGLnH

(ii) GLnV0,0d (S) = V ′

d(S) in a neighborhood of H and GLnH ∩ V 0,0d (S) = {H} in a

neighborhood of H.The same is true for V ′

d+1(S):

(iii) THV′d+1(S) = THV

0,0d+1(S)⊕ THGLnH

(iv) GLnV0,0d+1(S) = V ′

d+1(S) in a neighborhood of H and GLnH ∩ V 0,0d+1(S) = {H}

in a neighborhood of H.


Proof.(i): For any point HF ∈ V ′

d(S) denote by cij the coefficient of the monomial xαi−1i xj

and by dij the coefficient of the monomial xd−1i xj for 0 ≤ i, j ≤ n in the polynomial

F − f . The tangent space to V 0,0d (S) at H is given inside THV

′d(S) by equations

(v, ∂cij) = 0. On the other hand THGLnH = Span{αi∂cij + λid∂dij}. HenceTHV

′d(S) = THV

0,0d (S)⊕ THGLnH.

(iii) is proven in the same way.(iv) follows from (iii) since V 0,0

d+1(S) is smooth (see section 3.3.2).(ii) follows from (iv) since GLn preserves degree:

GLnV0,0d (S) = GLn(V

0,0d+1(S) ∩ |OPn(d)|) = (GLnV

0,0d+1(S)) ∩ |OPn(d)| =

= V ′d+1(S) ∩ |OPn(d)| = V ′

d(S).

Here, |OPn(d)| = |H| is the linear system of hypersurfaces of degree d. Also GLnH∩V 0,0d (S) ⊂ GLnH ∩ V 0,0

d+1(S) = {H} in a neighborhood of H. 2

From this lemma we see that it is enough to prove our statement for the germ V 0,0d,H

of V 0,0d (S). Consider now arbitrary hypersurface HF ∈ V 0,0

d,H given by polynomialequation F = 0. Since HF is obtained from H by local analytic diffeomorphism andboth have their only singularity at the origin, F has no terms of degree less thanαn. Also F has the same terms of degree αn as f . Indeed, HF is obtained from Hby local analytic diffeomorphism Φ without free terms. Let us decompose Φ = QLwhere L is linear transformation on variables xi s.t. αi = αn, and Q is the rest.Since Q cannot effect terms of lowest degree and QL(H) = HF ∈ V 0,0

d,H , LHF also lies

in V 0,0d,H . However, GLnH ∩ V 0,0

d,H = {H} hence LH = H so HF = QH and thereforeF has the same terms of the lowest degree αn as f .

So we will work with substratum germ V 0,0d,H of Vd,H consisting of hypersurfaces

given by polynomials of the form

F (x1, ..., xn) =n∑i=1

xαii +

n∑i=1

λixdi +

∑I∈D

aIxI , where D = {I ∈ Zn

≥0|αn < |I| ≤ d}\

\⋃

1≤i6=j≤n

{(0, ..., 0, αi − 1, 0, ..., 0, 1, 0, ..., 0)} \⋃

1≤i≤n

{(0, ..., 0, αi, 0, ..., 0)}.

For convenience, we introduce the following notations:a) let bI be the coefficients of basis monomials above the Newton polytope ∆(g) ,i.e. bI := aI for I ∈ D such that Ij ≤ αj − 2 for all j and w(I) > 1;b) let eI be the coefficients of basis monomials below the Newton polytope ∆(g) butof degree more than αn , i.e. eI := aI for I ∈ D such that Ij ≤ αj − 2 for all j andw(I) ≤ 1;c) let gI := aI for I ∈ E where E = {I ∈ D| Ij = αj − 1 for some j, Ik ≤ αk − 2 for


all k 6= j and Ik > 0 for some k 6= j};d) let uI := aI for I ∈ D such that Ij ≥ αj for some j or Ij = αj−1 and Ik = αk−1for some k 6= j;e) let qI := aI for I = (0, ..., 0, αj − 1, 0, ..., 0) for some j;f) for I = (i1, ..., , ik−1, αk − 1, ik+1, ...in) ∈ E denote

dual(I) := (α1 − 2− i1, ..., , αk−1 − 2− ik−1, αk − 1, αk+1 − 2− ik+1, ..., αn − 2− in).

Note that dual(I) also lies in E and dual(dual(I)) = I.Let A = C[aI ] be the algebra of polynomials generated by aI , I ∈ D. Let

m = 〈aI〉 be the maximal ideal in A generated by all aI , G = 〈gI〉 be the ideal in Agenerated by all gI and B = 〈bI〉 be the ideal in A generated by all bI .

3.3.4 Proof of the theorem for the case α1 < 2αn

In this case the Newton polytope ∆(g) lies below the hyperplane |I| = 2αn.We want to find out for which {aI} HF lies in V 0,0

d,H . Our F doesn’t include

monomials xI for I below and on the Newton polytope and satisfying Ij = αj − 1for some j. Hence in order to belong to our substratum, F should include no termsbelow and on the Newton polytope except of xαi

i .By theorem 1.1.15 on Milnor algebras of semiquasihomogeneous polynomials,

MF ' Mg ' Mf . Since g is quasihomogeneous, Mg = Tg. By the Mather-Yau

theorem 1.1.5 Fc∼ f iff TF ' Tf . But Tf ' Tg = Mg ' Mf ' MF hence F

c∼ f iffMF ' TF i.e. F (x1, ..., xn) ∈ 〈Fx1 , ..., Fxn〉.

In order to check whether F (x1, ..., xn) ∈ 〈Fx1 , ..., Fxn〉 we use the redNF-Buchberger algorithm. We refer to a small neighborhood of the origin, hence weconsider F (x1, ..., xn) and 〈Fx1 , ..., Fxn〉 in the local ring R = C[x1, ..., xn]〈x1,...,xn〉. Tocompute in this ring, we define a local monomial ordering on C[x1, ..., xn] such thatthe ring associated to C[x1, ..., xn] and this ordering will be C[x1, ..., xn]〈x1,...,xn〉.We choose the negative weighted degree lexicographical ordering with w =(1/α1, ..., 1/αn) (see Example 1.3.3, ordering (2)(iii)).

In general, the redNFBuchberger algorithm does not stop for local order-ings. However, in our case we can stop it manually when the leading monomial ofthe tail is less than xα1−2

1 · ... ·xαn−2n . We are allowed to do that by lemma 1.3.10, for

xα1−21 · ... · xαn−2

n is highest corner of 〈LM(Fx1), ..., LM(Fxn)〉 = 〈xα1−11 , ..., xαn−1

n 〉.Indeed, any monomial smaller than xα1−2

1 · ... · xαn−2n has degree of xj bigger

or equal than αj − 1 for some j and hence lies in 〈LM(Fx1), ..., LM(Fxn)〉 andxα1−2

1 · ... · xαn−2n /∈ 〈LM(Fx1), ..., LM(Fxn)〉.

There is another explanation why we can stop the algorithm at this point.Consider Vd+1,H . It is smooth of expected codimension (see section 3.3.2). ThereforeV 0,0d+1,H is also smooth and has expected codimension which is equal to the number

of basis elements which lie above the Newton polytope. Since we have exactly this


number of independent equations on this stage, there will be no more equations.Also since V 0,0

d+1,H is smooth, all the equations on it will have independent linear

parts. When we return to V 0,0d,H , linear part of only one of them may vanish.

So we rewrite the algorithm in the following way:

Algorithm 3.3.2 (Modified redNFBuchberger algorithm.)

1. p := 0, h := F ;

2. while (h 6= 0 and LM(h) ≥ xα1−21 · . . . · xαn−2

n )

(a) while (h 6= 0 and LM(h) ≥ xα1−21 · . . . · xαn−2

n

and exists i such that LM(Fxi) divides LM(h))

{h := h− (LT (h)/LT (Fxi)) · Fxi

}

(b) if (h 6= 0 and LM(h) ≥ xα1−21 · . . . · xαn−2

n ){p := p+ LT (h);h = tail(h)};

3. return p;


NF (F |〈Fx1 , ..., Fxn〉) =∑

RJ(aI)xJ ,

where xJ , Jk ≤ αk − 2, for all k are elements of the basis of algebraC[x1, ..., xn]/〈fx1 ,...,fxn 〉 which lie above ∆(g) and RJ(aI) are polynomials in aI . HenceF belongs to the ideal 〈Fx1 , ..., Fxn〉 if and only if all coefficients RJ(aI) = 0.

Transforming our system of equations to an equivalent one we obtain

bI = ψI(gJ , uK , bL) for (I) 6= (α1 − 2, ..., αn − 2) and

0 = ψα1−2,...,αn−2(gJ , uK , bL),

where ψI are polynomials from G2 +Bm. Recall that G = 〈gI〉 and B = 〈bI〉.Note, that this system is sorted by the quasihomogenious degree of left hand

side and all bL in the right hand sides are left hand sides of previous equations.Hence we can replace our system by an equivalent one:

bI = ψI(gJ , uK) for I 6= (α1 − 2, ..., αn − 2) (13)

where ψI ∈ G2 are polynomials in gJ and uK and the last equation

R :=∑I∈E

AkgIgdual(I) + Θ(gJ , uK) = 0, (14)


where Ak are positive rational numbers and Θ(gJ , uK) is a polynomial from G2m.

We see that our substratum germ is isomorphic to the germ at 0 of the affinevariety given in the affine space with coordinates {gJ , uK} by the last equation(14). The quadratic part Q of this equation is a non-degenerate quadratic form in{gI |I ∈ E}. We will now show that unless n = 4 and α1 = α2 = α3 = α4 = 3,the quadratic form Q has rank at least 3. For this it is enough to find 3 pointsI ∈ E. Indeed for any such point I = (j1, ..., jk−1, αk − 1, jk+1, ..., jn) the dual pointI ′ = (α1 − 2 − j1, ..., αk−1 − 2 − jk−1, αk − 1, αk+1 − 2 − jk+1, ..., αn − 2 − jn) alsosatisfies these conditions. Consider several cases separately:

0. For n = 4, α1 = α2 = α3 = α4 = 3, d = 12 − 9 = 3, there are nogI . Moreover, D is empty. So V 0,0

d,H consists of one point. Hence Vd,H is also asmooth algebraic variety of dimension n2 − 1 + n = 19. The expected dimension isCnn+d−1−

∏ni=1(αi−1) = C4

7−1−24 = 18. So, Vd,H has non-expected codimension.Till the end of the proof we assume that this is not the case.

1. α1 ≥ 4. In this case we have points (1, 1, 0, ..., 0, αn− 1) and (2, 0, 0, ..., 0, αn− 1).Since d ≥ α1, α2 ≥ 3.1.1 α2 = 3. In this case n ≥ 4 and α3 ≥ 3 since d ≥ α1. Hence we have one morepoint (1, 0, 1, 0, ..., 0, αn − 1).1.2 α2 ≥ 4. In this case we have point (0, 2, 0, ..., 0, αn − 1).

2. α1 = 3. In this case n ≥ 4 and α1 = α2 = α3 = α4 = 3 for the samereason. So unless n = 4 we have points (1, 1, 0, ..., 0, αn − 1), (0, 1, 1, 0, ..., 0, αn − 1)and (1, 0, 1, 0, ..., 0, αn − 1).

Note that for all mentioned points in cases 1, 2, |I| = αn+1 which is less or equal tod. Note also that the weight of all these points is at least 2

α1+ αn−1

αn> 2

2αn+ αn−1

αn= 1.

Since the quadratic form is non-degenerate of rank ≥ 3, our substratum germis reduced irreducible non-smooth variety of expected codimension and of order two.

Now we are going to prove that the singular locus Y of our substratum germcoincides with the germ X0 at H of the affine subspace X = Z(G). Since equation(14) lies in G2, all its first order partial derivatives lie in G, and hence the singularlocus includes X0. Let Z be the variety given by the equations

∂R

∂gI= 0

for I ∈ E. Since the linear part of this system of equations is non-degenerate, thegerm Z0 of Z at f is smooth and hence irreducible. Clearly Y ⊂ Z0 and henceX0 ⊂ Z0. They have the same dimension and Z0 is irreducible hence X0 = Z0 whichimplies X0 = Y .


So our substratum germ is reduced irreducible non-smooth variety of expectedcodimension which has a smooth singular locus.

3.3.5 Proof of the theorem for the case α1 ≥ 2αn.

We will make now a series of coordinate changes so that in new coordinates F willnot have monomials below and on the Newton polytope except xαi

i . The first onewill be xi 7→ xi for i < n and xn 7→ xn − 1

αna〈2,0,...,0,αn−1〉x

21. The coefficients of the

polynomial F in the new coordinates are expressed through the coefficients in theold coordinates by the formula

anewI = aI +

[i1/2]∑s=1

(−1)sCsk+sai1−2s,i2,...,in−1,in+s(

a〈2,0,...,0,αn−1〉

αn)s. (15)

After this coordinate change the coefficient anew〈2,0,...,0,αn−1〉 will vanish.In the same way, we get rid of all the coefficients of the form a〈j1,...,jn−1,αn−1〉

in ascending order of the corresponding monomials. We remind that the mono-mial ordering we use is the negative weighted degree lexicographical orderingwith w = (1/α1, ..., 1/αn) (see Example 1.3.3). The coordinate change indexedJ = (j1, ..., jn−1, αn−1) will be xi 7→ xi for i < n and xn 7→ xn− 1

αnaj1,...,jn−1,αn−1x

j11 ·

... ·xjn−1

n−1 . The coefficients of the polynomial F in the new coordinates are expressedthrough the coefficients in the old coordinates by the formula

anewI = aprevI +

min{[il/jl]|jl 6=0, 0≤l≤n−1}∑s=1

(−1)sCsk+sa

prev〈i1−sj1,...,in−1−sjn−1,in+s〉(

aprevJ

αn)s. (16)

As can be seen from formula 16, gI can be effected only during a coordinate changewhose index does not exceed I by any coordinate, and hence has lower weight. Thusafter all these coordinate changes, all coefficients gj1,...,jn−1,αn−1 will be zero. Denoteby a′I the coefficient of the monomial xI after the coordinate changes. It is easy tosee that a′I = aI + φ′I(aJ) where φ′I(aJ) ∈ mG. Note that

a′α1−2,...,αn−2 =∑

AJgj1,...,jn−1,αn−1gα1−2−j1,...,αn−1−2−jn−1,αn−1 + Φ′(aI)

where AJ ∈ R \ {0} for all J , and Φ′(aI) ∈ mG2.We continue with the coordinate changes and, in the same way as before, we get

rid of the coefficients of the form g′〈j1,...,jn−2,αn−1−1,0〉 starting from g′〈1,0,...,0,αn−1−1,0〉.

This time it might be non-zero since φ′1,0,...,0,αn−1−1,0(aJ) may be non-zero. However,it lies in the ideal G.

We do the same for g′〈j1,...,jk−1,αk−1,0,...,0〉 for all k ≥ 2 in the descending orderof k.


Denote by aI the coefficient of the monomial xI after all these coordinatechanges. Again, aI = aI + φI(aJ) where φI(aJ) ∈ mG and

aα1−2,...,αn−2 =n∑k=1

∑AJgj1,...,jk−1,αk−1,0,..,0gα1−2−j1,...,αk−1−2−jk−1,αk−1,αk+1−2,...,αn−2+Φ(aI)

where AJ 6= 0 for all J and Φ(aJ) ∈ mG2.

Now we want to find out for which {aI} HF lies in V 0,0d,H . Let F (x1, ..., xn) be

the polynomial F in new coordinates. F doesn’t include monomials xI for I belowand on the Newton polytope and satisfying Ij = αj − 1 for some j. Hence in order

to belong to our substratum F should include no terms below and on the Newtonpolytope except of xαi

i .In other words, we have the following equations on aI :

eI = 0 (17)

vI = 0 (18)

By theorem 1.1.15 on Milnor algebras of semiquasihomogeneous polynomials,MF ' Mf . As we have seen, Mf = Tf . By the Mather-Yau theorem 1.1.5 F

c∼ f

iff TF ' Tf . But Tf = Mf ' MF hence Fc∼ f iff MF ' TF i.e. F (x1, ..., xn) ∈

〈Fx1 , ..., Fxn〉.As in case one, in order to check that we use the redNFBuchberger

algorithm with the negative weighted degree lexicographical ordering with w =(1/α1, ..., 1/αn) (see Example 1.3.3).

Again, we can stop the algorithm manually when the leading monomial of thetail is less than xα1−2

1 · ... · xαn−2n (see algorithm 3.3.2).


NF (F |〈Fx1 , ..., Fxn〉) =∑

RJ(aI)xJ ,

where xJ are elements of the basis of algebra C[x1, ..., xn]/〈fx1 ,...,fxn 〉 which lie above

∆(g) and RJ(aI) are polynomials in aI . Hence F belongs to the ideal 〈Fx1 , ..., Fxn〉if and only if all coefficients RJ(aI) = 0. Transforming our system of equations toan equivalent one we obtain

bI =ψI(gJ , uK)∏n

i=1 uγI,i

i

(19)

where ψI ∈ G2 for G = 〈gJ〉 and ui = a(0,...,0,αi,0,...,0).The number of equations in system (19) is equal to the number of basis coef-

ficients above the Newton polytope ∆(g).


Now we express new coefficients through the old ones. Recall that ui = 1 +φ(0,...,0,αi,0,...,0)(aJ), aI = aI + φI(aJ) for other I ∈ D, and aI = φI(aJ) for I /∈ D,where φI(aJ) ∈ mG.

Consider system (17). After substituting old coefficients it will be eI =−φI(aJ), where φI ∈ m2. The same with the system (18).

Consider system (19) except the last equation i.e. the equation on bα1−2,...,αn−2.After substituting the old coefficients and multiplying by denominators it will be-come

bI = ψI(aJ), (20)

where ψI ∈ m2.The last equation is of particular interest. After passing to the old coordinates

and multiplying by the denominator its right hand side will become

ψ(α1−2,...,αn−2)(gJ , uK) =∑J∈S

AJgJgdual(J) + Θ′(aK),

where S ⊂ E is the set of indexes of terms that did not vanish during the coordinatechanges, Θ′(gJ , uK) lies in mG2 and AJ are positive rational numbers. The left handside will be

bα1−2,...,αn−2

n∏i=1

uγI,i

i = (0+φ(α1−2,...,αn−2))(n∏i=1

(1+φi)γI,i) =

∑J∈T

AJgJgdual(J)+Ψ(aK),

where T = {J ∈ E| J /∈ S, and dual(J) /∈ S}, Ψ(aK) ∈ mG2 and AJ are positiverational numbers. Moving right hand side to the left we obtain the equation

P :=∑J∈E

BJgJgdual(J) + Θ(aK) = 0, (21)

where Θ(aK) ∈ mG2, BJ are non-zero rational numbers and sign(BJ) =sign(Bdual(J)).

So our substratum germ V 0,0d,H is isomorphic to the germ at the origin of the

affine variety given in the affine space with coordinates {aI} by the system of equa-tions

eI + φI(aJ) = 0 (22)

qI + φI(aJ) = 0

bI − ψI(aJ) = 0

and the last equation (21).We have to prove that V 0,0

d,H is reduced irreducible non-smooth variety of ex-pected codimension which has a smooth singular locus.


The system (22) has a diagonal linear part. Hence all the equations in it areindependent and the variety W , defined by it, is smooth at 0. The last equation(21) does not depend on the preceding ones and does not have linear part. HenceV 0,0d,H is non-smooth of expected codimension.

The quadratic part of equation (21) is

Q :=∑J∈E

BJgJgdual(J)

Since BJ are non-zero, it is quadratic non-degenerate form of rank r equal tothe number of integer points in E.

Now we will show that r ≥ 3. We do that by exhibiting 3 integer points inE: (2, 0, ..., 0, αn − 1), (1, 1, 0, ..., 0, αn − 1) and (2, 1, 0, ..., 0, αn − 1). Since α1 ≤ dwe have α2 ≥ 3, and hence Ij ≤ αj − 2 for j < n for all the 3 points. Hence, it isenough to show that their degrees don’t exceed d. Indeed, their maximal degree isαn + 2 ≤ 2αn ≤ α1 ≤ d. So r ≥ 3. Hence the variety defined by the principle partof our system of equations on V 0,0

d,H is reduced and irreducible and hence our germ is

reduced and irreducible. Since the quadratic form Q is non-zero V 0,0d,H has order two.

Now we are going to prove that the singular locus Y of our substratum germcoincides with the germ X0 at the origin of the affine subvariety X given in W bythe ideal G.

Since equation (21) lies in G2, all its first order partial derivatives lie in G, andhence X0 lies in the singular locus. Consider the jacobian of the system obtainedby merging system (22) with the last equation (21). Fix I ∈ E. Let M(I) be theminor of the jacobian given by columns that include partial derivatives by gI andall eJ , bJ and qJ . The linear part of M(I) is AIgdual(I). Let Z0 be the germ at theorigin of subvariety of W given by the system of equations M(I) = 0 for all I ∈ E.Since the linear part of this system is diagonal, Z0 is irreducible and has the samedimension as X0. As X0 ⊂ Y ⊂ Z0, it implies X0 = Y = Z0.

So the substratum germ V 0,0d,H is reduced irreducible non-smooth variety of

expected codimension which has a smooth singular locus.

3.3.6 Proof that the sectional singularity type is A1 for allcases

The sectional singularity type of a variety germ with smooth singular locus is thesingularity type of transversal intersection of the singular locus with a linear space.

Consider the linear subspace L spanned by eI , qI , bI and gI . We have seenthat it is transversal to the singular locus of V 0,0

d,H in the affine space with coordinates

{aI}. Hence the sectional singularity type of V 0,0d,H is the singularity type of the

scheme theoretic intersection L∩V 0,0d,H . L∩V 0,0

d,H is given in L by system of equations


(22) with linear part non-degenerate in eI , qI and bI and one more equation withquadratic principle part which is not degenerate in the variables gI . Hence thesingularity type of L ∩ V 0,0

d,H is A1. 2

3.4 Proof of the theorem on stability properties

of obstructed equianalytic families

This section is dedicated to the proof of theorem 2.2.1. For convenience, we remindits formulation.

Theorem 2.2.1 Let H ⊂ Pn be hypersurface of degree d ≥ 3 with the uniquesingular point z = (1 : 0 : . . . : 0). Let h1 := h1(JZea(H)/Pn(d)) and τ := degZea(H).Let F0 be the equation of H. Suppose h1 > 0.

For any m ≥ 1 define Wm ⊂ Pn+m to be the hypersurface given by equationF0 +

∑mj=1 t

2n+jt

d−20 and zm be the point (1 : 0 : . . . : 0). Denote by V U

d,Wm the germat Wm of the family of all hypersurfaces of degree d that have a unique singularpoint inside U = {(t0 : ... : tn+m)|t0 6= 0} ⊂ Pn+m, and are analytically equivalent to(Wm, z) near the singular point. Then

(a) h1(JZea(Wm)/Pn+m(d)) = h1 and degZea(Wm) = τ .

(b) If h1(JZea(H)/Pn(2d − 2)) = 0 then the germs V Ud,Wm for m ≥ h1 + 1 have

a reduced component of expected dimension.

(c) If H is a plane curve then already h1(JZea(H)/P2(2d − 4)) = 0 and hencefor m ≥ h1 + 1 the germs V U


(d) If h1 < d − 1 then h1(JZea(H)/P2(2d − 2)) = 0 and hence for m ≥ h1 + 1the germs V U


(e) If h1 = 1 then the germs V Ud,Wm are non-smooth of expected dimension

for all m ≥ 1, reduced for m ≥ max{1, 5−d} and irreducible for m ≥ max{1, 6−d}.

(f) Suppose that F0 =∑n

i=1 tαii t

d−αi0 and h1(JZea(H)/Pn(t)) = 0 for some

d < t ≤ 2d − 2. Then for m ≥ dh1 (t−d+1)2(d−2)

e + 1 the germs V Ud,Wm have a re-

duced component of expected dimension. In particular, if h1(JZea(H)/Pn(2d− 3)) = 0

then for m ≥ dh1

2e + 1 the germs V U

d,Wm have a reduced component of expecteddimension.



First of all, we pass to local coordinates xi = tit0

, and denote f0(x1, ..., xn) :=F0(1, x1, ..., xn).

It is known (lemma 1.1.12) that polynomials g1 = f1 + x2n+1(1 + h1) and

g2 = f2 + x2n+1(1 + h2), where fi ∈ C{x1, ..., xn} and hi ∈ m ⊂ C{x1, ..., xn+1}, are

contact equivalent if and only if fi are contact equivalent.

For any polynomial F = f0 + x2n+1 +

∑aI,jx

Ixjn+1 ∈ m2 ⊂ C{x1, ..., xn+1}there exists an analytic diffeomorphism of (Cn+1, 0) that brings F to the form f0 +x2n+1(1 + h). One can write explicit formulas for this diffeomorphism that depend

polynomially on the coefficients of F (see section 3.4.2). Now we build a map ofgerms φ : (|OPn+1(d)|,W 1) → (|OPn(τ + 1)|, H) in the following way: take equationof the hypersurface which includes x2

n+1 with coefficient 1, bring it to the formf0 +x2

n+1(1+h) (using the above diffeomorphism) and take the hypersurface definedby (τ+1)-jet of f0. By lemma 1.1.12 and finite determinacy theorem 1.1.8, preimageof Vτ+1,H will be V U

d,W 1 .

In section 3.4.2 we obtain an explicit formula for φ (formula (24)). The linearpart of φ is identity and the quadratic part depends only on coefficients of monomialswhich include xn+1 with degree 1.

Since Vτ+1,H is T-smooth, it is locally defined by a system (*) of τ equationshaving non-degenerate linear part. Using φ we obtain a system (**) of equations onV Ud,W 1 . This is done in section 3.4.3. Since the linear part of φ is identity, the linear

part of (**) is obtained from linear part (*) by substituting zeros for coefficients ofmonomials of degree more than d. Since the tangent space to Vd,H has codimensionτ − h1, the rank of the linear part of (**) will be τ − h1. This proves statement (a).

The quadratic part of (**) is the sum of two systems. First is obtained fromquadratic part of (*) by substituting zeros for coefficients of monomials of degreemore than d. The second summand is obtained from linear part of (*) by composingit with quadratic part of φ and then substituting zeros for coefficients of monomialsof degree more than d. We show that if h1(JZea(H)/Pn(2d− 2)) = 0 then the secondsummand is non-zero.

In case h1 = 1, (**) has τ −1 equations with independent linear parts and onemore equation with quadratic principal part which is independent of linear partsof previous equations. In section 3.4.5 we show that it has rank at least 2 form ≥ max{1, 5− d} and at least 3 for m ≥ max{1, 6− d}. This finishes the proof ofstatement (e).

In section 3.4.4 we analyze the system (**) in the general case and provestatement (b).

In sections 3.4.6 and 3.4.7 we prove statements (c) and (d) using lemmas 1.1.27and 1.1.28 on Castelnuovo function.

In sections 3.4.8 we prove statement (f).


3.4.2 Coordinate changes

First of all, we pass to local coordinates xi = tit0

, and denote f0(x1, ..., xn) :=F0(1, x1, ..., xn).

Let F (x1, ..., xn+1) = f0 + x2n+1 +

∑aI,jx

Ixjn+1 be a polynomial of degree ≤ d.We want to find equations on the coefficients aI,j that F should satisfy in order todefine a hypersurface that belongs to V U

d,W 1 . First of all we may suppose that the

coefficient of monomial x2n+1 is one.

Now we want to get rid of the coefficients of monomials xIxn+1. In order todo that we make the following series of coordinate changes: xi 7→ xi for 1 ≤ i ≤ n,xn+1 7→ xn+1 − 1/2aIx

I . After this coordinate change the new coefficients will beexpressed from the previous ones by the following formula:

a(I)J,k =

maxl{[Jl/Il]}∑s=0

(s+ k

k

)(−1

2)saprevJ−sI,s+k(a

prevI,1 )s (23)

We start from I having smallest degree, and continue in increasing order of degrees.All the coefficients of the form aJ,1 influenced during coordinate change indexed Ihave degree more than degree of aI,1 (except aI,1 which vanishes). Hence we cancontinue making such coordinate changes until F has no coefficients aI,1 of degreeless than τ + 2. Denote the final coefficients by a′I,j. From the formula (23) we seethat they can be expressed through the original coefficients by

a′I,0 = aI,0 − 1/2∑

aJ,1aI−J,1 − 1/4a2I/2,1 + φI(aK,k) (24)

where φI ∈ 〈aJ〉3.

3.4.3 Equations defining V Ud,W 1 and proof of statement (a)

We want to find equations on the coefficients aI,j that F should satisfy in order tobe analytically equivalent to f0. By finite determinacy theorem we can suppose thatall coefficients of F of degree more than τ + 1 are zero. Then we can present F inthe form

F = f0 +∑

a′I,0xI + x2

n+1(1 +∑

a′I,jxIxj−2

n+1).

Then, by lemma 1.1.12 Fc∼ g0 if and only if f0

c∼ f0 +∑a′I,0x

I . Therefore,in order for F to lie in V U

τ+1,W 1 , f0 +∑a′I,0x

I should lie in Vτ+1,H . Hence in orderto find the needed equations on aI,j we have to take equations that define Vτ+1,H

inside the linear system |OPn(τ + 1)|, substitute there a′I,0 and then using formulas(24) express a′I,0 through old coefficients aI,j.

Let us now investigate equations on Vτ+1,H . Since h1(JZea(H)/Pn(2d− 2)) = 0,both Vτ+1,H and V2d−2,H are smooth and have expected codimension τ . That means


that there is a system of τ local equations on Vτ+1,H with a non-degenerate linear partwhich remains non-degenerate after substituting zeroes in place of all coefficients ofmonomials of degree bigger than 2d− 2.

Replacing the system of equations by an equivalent one, we can suppose thatthe linear part of this system has echelon form such that every row has a specialelement aLi

which appears only in linear part of this row and satisfies |Li| ≤ 2d− 2.

When we substitute in these equations zeros instead coefficients of degree > dthe rank of the linear part of the system drops by h1. That means that h1 rowsof the linear part include only coefficients of degrees > d. Renumber the rows sothat those will be the last h1 equations. Denote the index of the special element ofequation number i by Li.

As can be seen from formula (24), the linear part of a′I,0 is aI,0, and thequadratic part is −1/2

∑aJ,1aI−J,1− 1/4a2

I/2,1. Hence the linear parts of the last h1

equations remain 0 when we express new coefficients through the old ones.

Therefore the stratum germ V Ud,W 1 is locally defined by a system of τ equa-

tions, of which only τ − h1 have linear part, which is non-degenerate. Henceh1(JZea(W 1)/Pn+1(d)) = h1. By induction the same is true for all V U

d,Wm for anym ≥ 1. By lemma 1.1.12 degZea(Wm) = τ . Statement (a) is now proven.

3.4.4 Proof of statement (b)

We have to show that V Ud,Wm has an irreducible component which is reduced of

expected dimension for m ≥ h1 + 1. Consider W 2. It is obtained from W 1 by thesame procedure of adding a square. Hence we know the form of the equations onV Ud,W 2 . The system consisting of the first τ−h1 equations has a non-degenerate linear

part, and it is the same linear part as in equations on Vd,H . The last h1 equationsstart from quadratic forms.

Let us analyze the quadratic form of equation number τ − h1 + j. Let wj0 bethe quadratic form appearing in the equation number τ − h1 + j on Vd,H (it may bezero). The quadratic part of the corresponding equation on V U

d,W 1 is wj0 + wj1 where

wj1 depends only on coefficients aI,1, and its form was described above. Hence thequadratic part of the corresponding equation on V U

d,W 2 is wj0 + wj1 + wj2 where wj1depends only on coefficients aI,1,0, and wj2 is the same as wj1 but with variables aI,0,1.

Continuing in the same way we see that for general m, the quadratic part ofequation number τ − h1 + j will be wj0(aI,0,...,0) + wj1(aI,1,0,...,0) + wj2(aI,0,1,0,...,0) +· · · + wjm(aI,0,...,0,1). For convenience, we denote 0m := (0, ..., 0) ∈ Zm and ei :=(0, ..., 0, 1, 0, ..., 0) ∈ Zm.

Let Xm be the variety defined by principle part of the system of equations onV Ud,Wm . In order to show that V U

d,Wm has a reduced component of expected dimension,it is enough to show that Xm has a reduced component of expected dimension. To


do that we will prove that there is a point in Xm in which the jacobian of theprinciple part of the system of equations on V U

d,Wm has maximal rank.For every 1 ≤ j ≤ h1 we choose Jj and Kj (not necessary different) such that

Jj + Kj = Lτ−h1+j. Here, Li is the index of the special element of linear part of

equation number i on Vτ,H . Let A be the linear subspace spanned by all {aKj ,ej}h1

j=1

and by all {aI,em}1≤|I|≤d−1.Consider the minor defined by derivatives with respect to aLi,0m for 1 ≤ i ≤

τ − h1 and to aJj ,ejfor 1 ≤ j ≤ h1. We claim that the restriction of this minor on

the linear subspace A is C · aK1,e1 · · · · · aKh1 ,eh1 where C is non-zero real number.This minor is a determinant of a block matrix in which the first block is identitymatrix Id(τ−h1)×(τ−h1). It is left to show that the second block is diagonal matrixwith Cj · aKj ,ej

on the diagonal.Indeed, consider, for example, equation number τ − h1 + 1. The derivative of

w10 w.r. to aJ1,e1 is 0 since w1

0 doesn’t depend on it at all. The same is true aboutw1≥2. The derivative of w1

1 w.r. to aJ1,e1 contains aK1,e1 with non-zero coefficient C1.It may also contain other aI,e1 , but they vanish on our subspace A. Consider nowthe derivative of equation number τ − h1 + j (for j > 1) w.r. to aJ1,e1 . Again, thederivatives of w0 and wj≥2 is zero. The derivative of wj1 does not contain aK1,e1 sincethe coefficient aJ1+K1 = aL1 does not appear in the linear part of equation numberτ − h1 + j on Vτ,H . Hence this derivative also vanishes on the subspace A spannedby all {aKj ,ej

}h1

j=1 and by all {aI,em}1≤|I|≤d−1.By the same reason, the restriction to A of the derivative of equation number

τ − h1 + j with respect to aJi,eiis equal to 0 if j 6= i and Ci · aKi,ei

if j = i.Hence on A the minor is a determinant of diagonal matrix with entries

(CjaKj ,ej) and hence the minor is CaK1,e1 · · · · · aKh1 ,eh1 . Clearly, every neighbor-

hood of 0 in Xm ∩ A contains a point in which CaK1,e1 · · · · · aKh1 ,eh1 6= 0. At thispoint, a maximal minor of the jacobian is non-zero, hence the jacobian has maximalrank. Therefore the tangent space to Xm at this point has codimension τ and henceXm has a reduced component of expected codimension τ . Hence V U

d,Wm also has areduced component of expected codimension for m ≥ h1 + 1.

3.4.5 Proof of statement (e)

Now we have h1(JZea(H)/Pn(d)) = 1. By lemma 1.1.27, (d) on the Castelnuovofunction h1(JZea(H)/Pn(d + 1)) = 0. Hence from section 3.4.3 V U

d,W 1 is defined by asystem of τ − 1 equations with non-degenerate linear part and one more equationwithout linear part (since h1 = 1). Hence V U

d,W 1 is non-smooth.

In order to show that V Ud,Wm has expected codimension (respectively is reduced,

respectively irreducible) it is enough to show that the scheme defined by the principleparts of the above system of equations has expected codimension (respectively isreduced, respectively irreducible).


For 1 ≤ i ≤ τ − 1, we can just express aLifrom equation number i. Therefore,

this scheme is isomorphic to the scheme Xm defined by the quadratic part w of thelast equation in the affine space of coefficients {aI,J |2 ≤ |I + J | ≤ d} \ {aLi,0}τ−1

i=1 .Since w is non-zero, Xm has expected codimension. Xm is reduced if rank(w) ≥ 2and irreducible if rank(w) ≥ 3.

As in section 3.4.4, w = w0(aI,0,...,0) + w1(aI,1,0,...,0) + w2(aI,0,1,0,...,0) + · · · +wm(aI,0,...,0,1) where rank(w1) = · · · = rank(wm) ≥ 1. Hence Xm is reduced form ≥ 2 and irreducible for m ≥ 3. It is left to deal with m = 1 and m = 2. Itis enough to show that for d = 4 we have rank(w1) ≥ 2 and for d ≥ 5 we haverank(w1) ≥ 3.

We start with the case d ≥ 5. The quadratic form w1 can be expressed by

w1 =∑

(I,J) s.t. |I+J |=d+1

CI,JaI,e1aJ,e1 .

Note that if I + J = Lτ then CI,J 6= 0. Since |Lτ | = d + 1, it can be presented asLτ = I1 + I2 where |I1| = 2 and |I2| = d− 1 and also as Lτ = I3 + I4, where |I3| = 3and |I4| = d − 2. Note that I1, I2 and I3 are different since they have differentdegrees since d ≥ 5. If d = 5, it is possible that I3 = I4. Consider the reduction w′1of w1 on the (3 or 4 dimensional) linear subspace spanned by aI1,e1 , aI2,e1 , aI3,e1 , aI4,e1 .The only sums of pairs of those multiindexes whose degrees are d+1 are I1 + I2 andI3 + I4. Hence w′1 = CLτ (aI1,e1aI2,e1 + aI3,e1aI4,e1) which has rank 3 or 4. Hence therank of w1 is at least 3. Hence V U

d,Wm is reduced and irreducible for m ≥ 1.

Consider now d = 4. As in the previous case, we can find I1 and I2 such that|I1| = 2, |I2| = d−1 = 3 and I1 + I2 = Lτ . By reducing w1 on the subspace spannedby aI1,e1 and aI2,e1 we see that the rank of w1 is at least 2. Hence V U

d,Wm is reducedfor m ≥ 1 and irreducible for m ≥ 2.

3.4.6 Proof of statement (c)

By lemma 1.1.27 (d), it is enough to show that CZea(H)(2d−3)) = 0. Let Z(j(f0)) bezero-dimensional scheme defined by the jacobian j(f0) where f0 is the local equationof H. Then Zea(H) is a subscheme of Z(j(f0)) and hence by lemma 1.1.27 (e) it isenough to show that CZ(j(f0))(2d − 3)) = 0. Let C1 = Z( ∂f0

∂x1) and C2 = Z( ∂f0

∂x2) and

let Z ′ be the complete intersection C1∩C2. Then Z(j(f0)) is a subscheme of Z ′ andhence by lemma 1.1.27 (e) it is enough to show that CZ′(2d− 3)) = 0. Let k be thedegree of ∂f0

∂x1and l be the degree of ∂f0

∂x2. By lemma 1.1.28, CZ′(l+k−1)) = 1−1 = 0.

Since k, l ≤ d − 1 we obtain CZ′(2d − 3) = 0. Therefore h1(JZea(H)/P2(2d − 4)) = 0and hence by (b) the germs V U

d,Wm have a reduced component of expected dimensionfor m ≥ h1 + 1.


3.4.7 Proof of statement (d)

Suppose h1(JZea(H)/Pn(2d − 2)) > 0. Then h1(JZea(H)/Pn(l)) > 0 for all d ≤ l ≤2d− 2. Hence by lemma 1.1.27,(d) the Castelnuovo function CZea(H)(l) > 0. Hence

h1(JZea(H)/Pn(d)) − h1(JZea(H)/Pn(2d − 2)) =∑2d−2

l=d+1CZea(H)(l) ≥ d − 2. Henceh1(JZea(H)/Pn(2d−2)) ≤ h1(JZea(H)/Pn(d))−d−2 ≤ 0. So h1(JZea(H)/Pn(2d−2)) = 0and by (b) the germs V U

d,Wm have a reduced component of expected dimension form ≥ h1 + 1.

3.4.8 Proof of statement (f)

Since f0 =∑xαii , the Tjurina ideal Tj(f0) =< f0,

∂f0∂x1, ..., ∂f0

∂xn> is generated by

{xαi−1i }. Hence the monomial basis of the Tjurina algebra is given by monomials

xI where I lies in the rectangular parallelepiped P := {I ∈ Zn≥0| for all 1 ≤ j ≤

n, Ij ≤ αj−2}. The {xI , I /∈ P} form a monomial basis of the Tjurina ideal Tj(f0).Therefore we can prove the following lemma

Lemma 3.4.1 There exists a system of equations (Sys) that defines Vτ+1,H insidethe linear system |OPn(τ + 1)| such that the linear part of (Sys) is diagonal matrixconsisting of coefficients aI , I ∈ P that correspond to basis monomials.

Proof of the lemma. The tangent space to Vτ+1,H in H is the intersection of Tj(f0)with the space of polynomials of degree ≤ τ + 1. Hence it is spanned by {aI | I /∈ Pand |I| ≤ τ + 1}.

Let (Sys′′) be any system of equations that defines Vτ+1,H inside the linearsystem |OPn(τ + 1)|. Since {aI | I /∈ P and |I| ≤ τ + 1} span the tangent spaceto Vτ+1,H , (Sys′′) can be replaced by an equivalent system (Sys′) whose linear partconsists only of aI , I ∈ P . Since Vτ+1,H is smooth of codimension τ , (Sys′) consists ofτ equations with independent linear parts. Hence it can be replaced by an equivalentsystem (Sys) such that the linear part of (Sys) is diagonal matrix consisting ofcoefficients aI , I ∈ P . 2

Denote the coefficient which stands in linear part of row number i in (Sys)by Li. We sort the system by decreasing order of degree |Li|. In other words,|Li| ≤ |Lj| for i ≥ j. Note that after this sorting, |Li| ≥ d+ 1 for 1 ≤ i ≤ h1. Sinceh1(JZea(H)/Pn(t)) = 0, |Li| ≤ t ≤ 2d− 2 for any i. If h1(JZea(H)/Pn(d+ 1)) = 0 thenh1 = 1 and the statement follows from (e). So we suppose h1(JZea(H)/Pn(d+ 1)) > 0and hence |L1| ≥ d+ 2. The rest of the proof is similar to section 3.4.4.

Let Xm be the variety defined by principle part of the system of equations onV Ud,Wm . In order to show that V U

d,Wm has a reduced component of expected dimensionit is enough to show that Xm has a reduced component of expected dimension. For


that we will prove that there is a point in Xm in which the jacobian of the principlepart of the system of equations on V U

d,Wm has maximal rank.We will choose h1 pairs (Jj, Kj) and natural numbers s(j) such that the minor

that contains derivatives with respect to all aJj ,es(j)restricted on the linear subspace

A spanned by all {aKj ,es(j)}h1

j=1 and by all {aI,em}1≤|I|≤d−1 will become a minor ofnon-degenerate upper triangular matrix. Sufficient condition for that is that(i) Jj +Kj = Lj(ii) If (Jj +Kk = Li and s(j) = s(k)) then i ≤ j.

For any j we find natural numbers s(j) and f(j) such that 2 ≤ f(j) ≤ d − 1and

f(j) + (d− 2)(s(j)− 1) =

j∑i=1

(|Li| − d).

Now for any j we choose Kj such that Lj − Kj ∈ Zn≥0 and |Kj| = f(j). We

choose Jj to be Lj −Kj.Now we have to show two things:

1) s(j) ≤ dh1 (t−d+1)2(d−2)

e for all j

2) Jj, Kj and s(j) satisfy the conditions (i) and (ii).In the proof of 1) we will use the following elementary lemma

Lemma 3.4.2 Let {ai}ni=1 be an increasing sequence of positive real numbers, and{bj}nj=1 be a non-increasing sequence of real numbers. Then

n∑i=1

aibi ≤n∑i=1

ai

∑nj=1 bj

n=

1

n(n∑i=1

ai)(n∑i=1

bi).

Proof of 1.It is enough to show that∑h1−1

i=1 (|Li| − d)

(d− 2)≤ h1 (t− d+ 1)

2(d− 2)− 1 +

d− 1

d− 2

and hence enough to showh1−1∑i=1

(|Li| − d) ≤ h1 (t−d+1)2

.

Let b(q) be the number of indexes Li such that |Li| = q. This number is equalto the Castelnuovo function CZea(H)(q). Note that |Li| ≤ t for all i.

h1∑i=1

(|Li| − d) =t∑

q=d+1

(q − d)b(q)


By lemma 1.1.27 on Castelnuovo function, b(q) does not increase. Hence by lemma3.4.2

t∑q=d+1

(q−d)b(q) ≤ 1

t− d(

t∑q=d+1

(q−d))(t∑

q=d+1

b(q)) ≤ (t− d+ 1)(t− d)h1

2(t− d)= h1 (t− d+ 1)

2.

Proof of 2. Condition (i) is satisfied by definition of Jj. Let us now prove (ii).Suppose that Jj +Kk = Li and s(j) = s(k). Then

|Li| = |Lj| − f(j) + f(k) = |Lj| −j∑l=1

(|Ll| − d) +k∑l=1

(|Ll| − d).

If k < j then

|Li| = |Lj|−j∑l=1

(|Ll|−d)+k∑l=1

(|Ll|−d) = |Lj|−j∑

l=k+1

(|Ll|−d) ≤ |Lj|−(|Lj|−d) ≤ d

which is impossible.If k > j then

|Li| = |Lj| −j∑l=1

(|Ll| − d) +k∑l=1

(|Ll| − d) = |Lj|+k∑

l=j+1

(|Ll| − d) > |Lj|

and hence i < j.If k = j then Li = Jj +Kj = Lj and hence i = j.So i ≤ j in any case and hence (2) is now proven.

Therefore on A the minor is a determinant of upper triangular matrix withentries (CjaKj ,es(j)

) on the diagonal and hence the minor is CaK1,es(1)· . . . · aKh1 ,es(h1)

where C 6= 0. Clearly, every neighborhood of 0 in Xm ∩ A contains a point inwhich CaK1,es(1)

· . . . · aKh1 ,es(h1)6= 0. In this point, a maximal minor of the jacobian

is non-zero, hence the jacobian has maximal rank. Therefore Xm has a reducedcomponent of expected dimension and hence V U

d,Wm has a reduced component of

expected dimension for m ≥ dh1 (t−d+1)2(d−2)

e+ 1. 2

Remark 3.4.3 If one wishes Wm to be unisingular, they can be defined by F0 +∑mj=1 t

2n+jt

d−20 +

∑mj=1 λjt

dn+j, where λj are generic. The same proof shows that an

analogous theorem will hold about the equisingular family germ Vd,Wm .

Remark 3.4.4 Statement (c) can be strengthened as follows: let H be a projectiveplane curve of degree d which is not a union of d lines through the same point. Thenh1(JZea(H)/P2(2d− 5)) = 0 .


Proof. Consider the linear system∑αi

∂f0∂xi

. A generic curve of this system is irre-ducible (see [GrL], lemma 3.8). Let C ′ be an irreducible curve from this system.Zea can be viewed as subscheme of C ′. By Bezout theorem, degJZea(H)/C′(d

′) =d′(d − 1) − τ . By Riemann - Roch theorem (see also [GrK] for its proof for sin-gular curves), in order to show that h1(JZea(H)/C′(d

′)) = 0 enough to show thatdegJZea(H)/C′(d

′) > 2ga(C′) − 2 = (d − 2)(d − 3) − 2. Substituting d′ = 2d − 5

we see that it is enough to show (2d − 5)(d − 1) − τ > (d − 2)(d − 3) − 2, i.e.d2 − 2d + 1 > τ which is true since H is a plane curve of degree d which is not aunion of d lines through the same point. Hence h1(JZea(H)/C′(2d − 5)) = 0. Henceby 1.1.24 h1(JZea(H)/P2(2d− 5)) = 0. 2

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