Zariski Geometries, Lecture 3 - Kobe UniversityTitle Zariski Geometries, Lecture 3 Author Masanori Itai Created Date 8/28/2011 4:02:22 PM

Getting a fieldMain Theorem

Applications

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Zariski Geometries, Lecture 3

Masanori Itai

Dept of Math Sci, Tokai University, Japan

August 31, 2011 at Kobe university

Masanori Itai Zariski Geometries, Lecture 3


Applications

Table of Contents

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1 Getting a field

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2 Main Theorem

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3 Applications



Applications

Assumption

There is an irreducible Zariski curve J with an abeliangroup strucure on it.



Applications

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Lemma (Lemma 4.2.1)

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There is a one-dimensional, smooth, faithful family N ofcurves on J2 through (0, 0).



Applications

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Corollary (Cor 4.2.5)

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There are a one-dimensinal irreducible manifol U andconstructible irreducible ternary relations P,S ⊂ U2 whichdetermine a partial Z-field structure on U



Applications

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Lemma (Lem 4.2.6)

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There is a non-commutative meta abelian Z-group structuteT(U) on U × U.For some dense open U′ ⊆ U, there is a Z-embedding of thepre-group T(U′) into a connected Z-group G such that

dim G = 2

G is a solvable group with finite center.



Applications

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Theorem

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There is a Z-field in M .



Applications

Outline of proof (1)

Start with the Z-group G with the finite center C(G).With some arguments, we may assume that G is centerless.

Consider the commutater subgroup [G,G] of G, and1 , a ∈ G.

Claim: dim aG = 1.

This is because aG is a normal subgroup of [G,G] and0 < dim aG < dim G.



Applications


If b is another non-unit element of [G,G], we have

aG = bG or aG ∩ bG = ∅.

We have a partition of [G,G]/{1} into one-dimensionalorbits.

By dimension argumet, we see that there are only finitelymany one-dimensional orbits.



Applications


[G,G] is a constructible group of dimension 1.

[G,G] contains a connected normal subgroup [G,G]◦ ofdimension 1.

Thus [G,G]◦ = aG ∪ {1} for some non-unit a ∈ G.Fix a for the rest of the proof.

Put K+ := [G,G]◦, and write the group operation of K+

additively.



Applications






additively.



Applications






additively.



Applications






additively.



Applications


The group G acts on K+ by conjugation; for g ∈ G andx ∈ K+

gx := g−1xg

x 7→ gx is an automorphism of the Z-group K+.

Recall we have fixed non-unit a ∈ G. C(a) is a normalsubgroup of G.

Notice that g1 and g2 induces the same action if

g−11

g2 ∈ C(a)

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Denote K× the quotient group G/C(a).



Applications



gx := g−1xg




g−11

g2 ∈ C(a)

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Applications



gx := g−1xg




g−11

g2 ∈ C(a)

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Applications



gx := g−1xg




g−11

g2 ∈ C(a)

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Applications



K× acts transitively on K+ − {0}.K× is a connected one-dimensional group, hencecommutative.



Applications



K× acts transitively on K+ − {0}.

K× is a connected one-dimensional group, hencecommutative.



Applications



K× acts transitively on K+ − {0}.K× is a connected one-dimensional group, hencecommutative.



Applications


For g, g1, g2 ∈ K×, we have;

If ga = g1a + g2a then gx = g1x + g2x for allx ∈ K+ − {0}.If g1a = −g2a then g1x = −g2x for all x ∈ K+ − {0}.

We can identify g ∈ K× with ga ∈ K+ − {0}.

Transfer the multiplication of K× to K+.

The manifold K+ gets the addition and the multiplication,both operations are Z-closed, hence K+ is a Z-field!



Applications

Example 4.2.9 (1)

WorkV1 ⊆ (K+)∗. Consider a family of curves of the form

g(v) = a · v + b.

We need g(1) = 1. Hence, put b = 1− a. Thus,

ga(v) = a · v + 1− a



Applications

Example 4.2.9 (2)

For ga(v) = a · v + 1− a, g(v) = b · v + 1− b, we have

ga(gb(v)) = a(b · v + 1− b) + 1− a = ab · v + 1− ab

On the other hand,

gab(v) = ab · v + 1− ab

Thus, we have

ga ◦ gb(v) = ga(gb(v)) = gab(v)



Applications

Example 4.2.9 (3)

We use the multiplication to introduce

ga(v) ⊕ gb(v) = (a · v + 1− a) · (b · v + 1− b)= ab · v2 + (a + b− 2ab) · v + ab + 1− a− b= f (v)

This curve has a derivative at 1; f ′(1) = a + b.Thus f (v) is tangent to ga+b = (a + b) · v + 1− a− b at (1, 1).Therefore,

ga ⊕ gbTga+b



Applications

Main Theorem

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Theorem (Thm 4.4.1)

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Let M be a Zariski structure satisfying (EU) andC be apre-smooth Zariski curve inM . Assume thatC is non-linear(i.e., ample). Then there is a non-constant continous map

f : C→ P1(K)

for some algebraically closed fieldK.



Applications

Chow Theorem and Purity Theorem

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Theorem (Thm 4.3.9 generalized Chow Theorem)

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Any closed subset of Pn is an algebraic subvariety of Pn.

Let K be the acf interpreted in an ample Zariski geometry M .

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Theorem (Thm 4.3.10, purity Theorem)

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Any relation R ⊆ Km definable in the natural language of Mis constructible i.e., a boolean conbination of algebraic sets.



Applications

Simple Zariski groups are algebraic

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Theorem (Thm 4.4.6)

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G is a simple Z-group satisfying (EU)

Some one-dim irred Z-subsetC ⊆ G is pre-smooth.

ThenG is Z-isomorphic to an algbraic groupG(K) for somealgebraically closed fieldK.



Applications

Z-meromorphic functions

Given a Zariski set N and an algebraically closed field K.

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Definition (Def 4.4.2)

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A continous function g : N → K with the domain containingan open subset of N is called Z-meromorphic.

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Proposition

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The set of all Z-meromorphic functions from N to K forms afield KZ(N).

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Definition (p. 101)

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KZ(N) is called the set of all Z-meromorphic functions fromN to K.



Applications

Outline of proof of Theorem 4.4.6. (1)

Let m = rk( G).

(General property) For any G, simple group of finiteMorley rank, Th(G) is almost strongly-minimal, henceℵ1-categorical.



Applications


Let C be a strongly minimal definable subetset of G.There is a definable relation F ⊆ G× Cm giving afinite-to-finte correspondence between a subset R ⊆ Gand a subset D ⊆ Cm.

For G in the theorem, there exists a non-constantZ-meromorphic function G→ K.



Applications


Let C be a strongly minimal definable subetset of G.There is a definable relation F ⊆ G× Cm giving afinite-to-finte correspondence between a subset R ⊆ Gand a subset D ⊆ Cm.

For G in the theorem, there exists a non-constantZ-meromorphic function G→ K.



Applications


Consider the field KZ(G) of Z-meromorphic functionsfrom G to K.

Each g ∈ G acts on KZ(G) by f (x) 7→ f (g · c) giving arepresentation of G as a group of automorphisms ofKZ(G).

By Rosenlicht’s Theorem and the purity theorem, wesee that there is a G-invariant finite-dimensionalK-subspace V of KZ(G).



Applications







Applications







Applications


G can be represented as a definable subgroup G(K) ofGL(V).

By the purity theorem again, the definable subgroupG(K) is algebraic.

The representation is isomorphism since G is simple.



Applications







Applications







Applications

Geometric Mordel-Lang conjecture

Hrushovski proved Geometric Mordel-Lang conjecture usingZariski geometry argument.



Applications

A recent application by Zilber, 2010

Zilber reproved a theorem of Bogomorph, Korotaev andTschinkel with the help of Zariski geometry.

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Theorem

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For any algebraically closed fields K1 and K2 there is a fieldisomorphism β : K1 → K2 inducing an isomorphism of pairsand a bijective isogeny ψ such that

α = ψ ◦ β : (J1; C1,+) → (J2 : C2,+)


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Zariski Geometries, Lecture 3 - Kobe UniversityTitle Zariski Geometries, Lecture 3 Author Masanori Itai Created Date 8/28/2011 4:02:22 PM