zariski geometries, lecture 2non-standard analysis notion of tangency getting a group.. zariski...

Non-standard analysisNotion of tangency

Getting a group

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Zariski Geometries, Lecture 2

Masanori Itai

Dept of Math Sci, Tokai University, Japan

August 30, 2011 at Kobe university

Masanori Itai Zariski Geometries, Lecture 2


Getting a group

Table of Contents

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1 Non-standard analysis

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2 Notion of tangency

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3 Getting a group



Getting a group

1. Non-standard analysis



Getting a group

Elementary extensions of Zariski structures

Recall that if M is a Zariski structure and M ′ is an elementaryextenstion of M ′, we can make M ′ to be a Zariski structure aswell.

This makes it possible to introduce the notion of infinitesimalneiborhood of a point.



Getting a group

Specializations

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Definition (Def 2.2.2 )

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M � M∗, M ⊂ A ⊆ M∗

A mapπ : A → M is called a(partial) specialization if

for everya ∈ An and ann-ary M-closedSwith a ∈ S∗ = S(M∗)we haveπ(a) ∈ S.

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Remark

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Specialization is just a model-theoretic homomorphism.



Getting a group

Infinitesinal neighborhood

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Definition (Defn 2.2.18)

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M � M∗

π : M∗ → M , a universal specialization

For a ∈ M n,Va = π−1(a)

is called aninfinitesimal neighborhood of a.



Getting a group

2. Notion of tangency



Getting a group

Ampleness (AMP)

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Definition (AMP)

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Zariski structure C is ample if there is a two-dimensional,irreducible, faithful family L of curves on C2. L is locally isomorphicto an open subset of C2.

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Remark

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AMP ≡ non-locally modular≡ non-linear

We assume (AMP) for the rest of today’s talk.



Getting a group

Lemma 3.8.5

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Lemma (Lemma 3.8.5)

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There exists an irreducible, faithful, one-dimensional smooth familyN of curves through (a, b).



Getting a group

Example 3.8.8. (p. 71)

Work with C ≺ C∗

I ={(u, v, x, y, z) :

ux + v(y− 1) + z(z− 1) = 0 andux2 + v(y− 1)z + z(z− 1) = 0

}

For each u, v ∈ C (u , 0∨ v , 0), put

gu,v = I (u, v,C3) = {(x, y, z) : (u, v, x, y, z) ∈ I }

gu,v passes through (x, y, z) = (0, 1, 0) and (x, y, z) = (0, 1, 1)



Getting a group

Example 3.8.8. (p. 71) Cont’d


Eliminating z, we haveuv2x(y− 1)2 + v3(y− 1)3 + u2x2(x− 1)2− uvx(x− 1)(y− 1) = 0

Projection of gu,v on the (x, y)-plane are curves through (0,1)with a nodal singularity at (0, 1)

gu,v is non-singular at both (0, 1 , 0) and (0, 1, 1)

gu,v defines two families of local functions:

g0u,v : V0 → V1,0 and g1

u,v : V0 → V1,1



Getting a group

Example 3.8.8. (p. 71) Cont’d


The first coordinates of the functions define the branches ofthe planar curves through (0,1) with the correspondingtrajectory z = 0 and z = 1.



Getting a group

Branches of a curve at a point

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Definition (Def 3.8.7)

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Let (a, b) ∈ C2. γ ⊂ V(a,b) is said to be abranch of a curve at(a, b) if

there are somec ∈ Cm−2, family of curvesG through(a, b) ^ c,and a curveg ∈ G such that

the cover(u, (x, y) ^ z) 7→ (u, x) is regular and unramifiedγ = {(x, y) ∈ Va,b) : ∃z ∈ Vc (g′, (x, y) ^ z) ∈ I }



Getting a group

We define the notion of two curves g1, g2 are tangent at a point,written

g1T g2

such that

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Proposition (Prop 3. 8. 14)

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TFAE

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1 g1Tg2

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2 ∀x ∈ Va ∀g′1∈ Vg1 ∃g′

2∈ Vg2 (g′

1(x) = g′

2(x))

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3 ∀x ∈ Va ∀g′2∈ Vg2 ∃g′

1∈ Vg1 (g′

1(x) = g′

2(x))



Getting a group

Intuitive idea

Two curves g1 and g2 are tangent at (a, b)

iff they have a common tangent line at (a, b)

iff they have the same derivative.

In Zariski structures we don’t have the notion of derivative. Hencewe use the notion of branches of a curve in order to say that twocurves have the same derivative.

Two curves are tangent at (a, b) iff they have the same branch at(a, b).



Getting a group

Tangency is an equivalence relation

G(a,b) =⋃

Gc,I

where c is a trajectory and I incidence relation.

G(a,b) is a collections of infinitesimal pieces of smooth(regular) curves passing trough the point (a, b).

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Corollary (Cor. 3.8.18)

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T is an equivalence relation onG(a,b)



Getting a group

3. Getting a group



Getting a group

Notion of tangency gives rise to a group

Consider the composition of local functionsVa → Va modulothe tangency

Compositon defines an associative operation, pre group ofjets.

Any Zariski pre-group can be extended to a group with apre-smooth structure on it. (Weil’s group chanks theorem)



Getting a group

Composition of curves

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Definition (Def 4.1.3)

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g1 ∈ G1, g2 ∈ G2 with G1,G2 ⊂ G(a,b) define thecompositioncurveg−1

2◦ g1 as

{(x1, x2) ∈ C2 : ∃y, z1, z2 ((x1, y, z1) ∈ g1 ∧ (z2, y, x2) ∈ g−12

)}

Branch at(a, a)

(g−12◦ g1)(a,a) = {(x1, x2) ∈ Va × Va : x−1

2= g−1

2(g1(x1))}



Getting a group

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Lemma (Lemma 4.1.2)

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For every g1, g2 ∈ G(a,b);

g1 T g2 iff g−11

T g−12

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Lemma (Lemma 4.1.4)

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Tangency relation T is preserved by composition of branches ofsufficiently generic pairs of curves;

g1 T h1 ∧ g2 T h2 =⇒ g−12◦ g1 T h−1

2◦ h1



Getting a group

Recall (AMP) and the irreducible, faithful, one-dimensional smoothfamily N of curves through (a, b) from Lemma 3.8.5.

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Lemma (Lemma 4.1.7)

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Given a generic pair (`1, `2, n1) ∈ N3, there is n2 ∈ N such that

n−11◦ `1 T n−1

2◦ `2



Getting a group

Composition gives rise to a group operation

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Lemma (Lemma 4.1.8)

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Given a generic( f1, f2) ∈ Haa × Haa, there is a genericg ∈ Haa

such thatg is tangent to the compositionf1 ◦ f2.

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Remark

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This gives the group operation!



Getting a group

Proof of Lemma 4.1.8 (1)

Fix an ` ∈ N generic over ( f1, f2). By Lemma 4.1.7, there aren1, n2 ∈ N such that

f1 T n−11◦ ` and f2T n−1

2◦ `.

Claim: f1 ◦ f2 T n−11◦ n2

Notice that (n1, n2) is a generic pair as well.



Getting a group

Proof of Lemma 4.1.8 (2)

Consider x ∈ Va and (n′1, n′

2) ∈ V(n1,n2). We need to find

( f ′1, f ′

2) ∈ V( f1, f2) such that

f ′1 ◦ f ′2(x) = n′−11 ◦ n′2(x).

For this we choose first f ′2∈ V f2 so that f ′

2(x) = y = `−1 ◦ ◦n′

2(x).

Next we choose f ′1∈ V f1 so that f ′

1(y) = n′−1

1◦ `(y).

Recall the criterion of tangency given by Prop 3.8.14.



Getting a group

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Proposition (Prop 4.1.9 (pre-group of jets))

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There is a one-dimensional irreducible manifold U and aconstructible irreducible ternary relation P ⊆ U3 which is the graphof a partial map U2 → U and determines a partial Z-groupstructure on U.



Getting a group

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Theorem (Thm 4.1.13 (Z-version of Weil’s group chanks thm))

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For any partial irreducible Z-group U, there is a connectedZ-group G and a Z-isomorphism between some dense openU′ ⊆ U and dense open G′ ⊆ G.



Getting a group

Getting a group from partial group U

Define G to be a semi-group of partial functions U → Ugenerated by shifts by elements in U.

For h, g ∈ G, h = g if h(u) = g(u) on an open subset of U.

The semi-group operation is defined by the composition offunctions.

With some arguments, we can identify G with the constructiblesort (U × U)/E where E is an equivalence relation;

(a1, a2) E (b1, b2) iff

dim{u ∈ U : ∃v,w, x ∈ U P(a2, u, v) ∧ P(a1, v, x)∧P(b2, u,w) ∧ P(b1,w, x)} = dim U



Getting a group

Semi-group G is in fact a group.

This is a general fact about semi-groups without 0 being definablein stable structure.

If g ∈ G does not have an inverse, then

gn+1G ( g nG, for all n ∈ Z>0.

This defines the strict order property on the structure, contradictingstability.

We have now a group!



Getting a group

We show that G is a Z-group. (1)

Let d = dim U2. Choosing generic pairwise independenta1, · · · , ad ∈ U we can show that

G = {a1, · · · , ad} · U−1 =

d⋃

i=1

{ai · v : v ∈ U}.



Getting a group


Each ai · U−1 can be identified with {ai} × U thus consideredas a manifold.

Partial bijections ai · U−1 → a j · U−1 defined by

(ai , v) 7→ (a j ,w) if ai · w = a j · v

The relation (ai , v) ↔ (a j ,w) defines a constructibleequivalence relation on

S =

d⋃

i=1

ai · U−1



Getting a group


The relation (ai , v) ↔ (a j ,w) defines a constructibleequivalence relation on

S =

d⋃

i=1

ai · U−1

The relation E and the ternary operation P corresponding themultiplications are closed.

View G as a manifold defined setwise as S/E. And G iscovered by smooth subsets ai ·U−1. The multiplication P on Gis closed.

Hence G is a Z-group!



Getting a group

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Corollary (Cop 4.1.14)

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The groupJ of jets ata on the curveC generated byU = Haa/T is aconnected Z-group of dimension 1.

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Proposition (Prop 4.1.15, Reineck’s theorem)

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A one-dimensional connected group G is Abelian. In particular J isAbelian.

Tomorrow we start with this J i.e., an irreducibleZariski curve C with an Aberian group structure onit.


zariski geometries, lecture 2non-standard analysis notion of tangency getting a group.. zariski...

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