Zariski Geometries, Lecture 1 - Kobe UniversityHZ96 Ehud Hrushovski, Boris Zilber, Zariski geometries, J of the AMS, 1996 Z10 B. Zilber, Zariski Geometries, London Math. Soc. Lect

A little bit of HistoryTopological Structures with good dimension

Quantifier EliminationElementary extensions

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Zariski Geometries, Lecture 1

Masanori Itai

Dept of Math Sci, Tokai University, Japan

August 30, 2011 at Kobe university

Masanori Itai Zariski Geometries, Lecture 1



References

HZ96 Ehud Hrushovski, Boris Zilber, Zariski geometries, J of theAMS, 1996

Z10 B. Zilber, Zariski Geometries , London Math. Soc. Lect NoteSer. 360, Cambridge, 2010




Table of Contents

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1 A little bit of History

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2 Topological Structures with good dimension

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3 Quantifier Elimination

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4 Elementary extensions




Zilber Conjecture

Strongly minimal

trivial (no structure)linear (locally modular) : vector spacesnon-linear (non-locally modular) : alg. closed fields

Zilber conjectured that any non-locally modular stronglyminimal set interprets an acf!

Counter example was constructed by Hrushovski usinggeneric model construction.




Zilber Conjecture

Strongly minimal







Zilber Conjecture

Strongly minimal







ZC is true for Zariski geometries

In algebraic geometry; algebraically closed field⇒ Zariskitopology, dimension notion

Model theory of Zariski structures; Noetherian topology,dmension notion⇒ algebraically closed field

Ample (non-linear, non-locally modular) Zariski geometryinterprets an algebraically closed field.


















Topological structures, p. 12

Consider a collection of topological spaces {M n : n ∈ N}.each Mn is Noetherian

proj is continuous

graph of equality is closed

fibers of closed sets are closed

Cartesian products of closed sets is closed

etc




Good dimension

In subsection 3.1.1, we have a list of postulates for the dimesionnotion;

(DP) dimesion of a point

(DU) dimension of unions

(SI) strong irreducibility

(AF) addition formula

(FC) fiber condition




Zariski structures

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Definition (Def 3.1.3)

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{M n : n ∈ N} is Zariski structures if

Noetherian topology

dimension notion

semi-proper :Let S ⊆cl M n be irreducible, then there exists a proper closed

subsetF ⊂ prSsuch that

prS− F ⊆ prS

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Zariski geometry

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Zariski geometry= Zariski structure+ (sPS)+ (EU)

(sPS) : strongly Pre-Smooth

(EU) : Essentially Uncountable




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Theorem (Thm 3.6.21)

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A one-dimensional, uncountable, pre-smooth, irreducible ZariskistructureM is a Zariski geometry.




Basic Examples (1)

Algebraic varaity M over an acf. (Thm 3.4.1)Zariski structure is complete (ie. projection of closed is closed)if M is.(PS), pre-smooth if M is.(EU), essentially uncountable, if M .

Compact complex manifold M with the notion of analyticdimension. It satisfies (PS) and (EU). (Thm 3.4.3)




Basic Examples (1)

Algebraic varaity M over an acf. (Thm 3.4.1)Zariski structure is complete (ie. projection of closed is closed)if M is.(PS), pre-smooth if M is.(EU), essentially uncountable, if M .

Compact complex manifold M with the notion of analyticdimension. It satisfies (PS) and (EU). (Thm 3.4.3)




Basic Examples (2)

Proper varieties of rigid anlaytic geometry (Thm 3.4.7)

Definable sets of finite Morley rank and Morley degree 1 indifferentially closed fields. (Thm 3.4.9, Pillay)




Basic Examples (2)

Proper varieties of rigid anlaytic geometry (Thm 3.4.7)

Definable sets of finite Morley rank and Morley degree 1 indifferentially closed fields. (Thm 3.4.9, Pillay)




Semi-properness

M is a Zariski structure.

We want the theory of the structure M to have the elimination ofquantifiers.For this, semi-properness axiom is needed:

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Definition (SP)

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Let S ⊆cl M n be irreducible, then there exists a proper closed

subset F ⊂ prS such that

prS− F ⊆ prS

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QE for Zariski structures

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Theorem (Thm 3.2.1)

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A Zariski structureM admits elimination of quantifiers; that is anydefinable ssubsetQ ⊆ M n is constructible, i.e., boolean conbinationof closed sets.




Proof of QE (1)

We must show that the projection of a constructible subset isconstructible.

For this let Q = S− P where both S, P closed.

We show the theorem by induction on dim S.

LetdS = min{dim S(a, M) : a ∈ prS},F = {b ∈ prS : dim P(b, M) ≥ dS}.




Proof of QE (2)

Let F be the closure of F.

F is a proper closed subset of prS, by (FC).

Since prS is irreducible, we have

dim F < dim prS.

Let S′ = S∩ pr−1(F).

Since F ∩ pr(S) , pr(S), we have S′ ( S.

Thus, dim S′ < dim S.




Proof of QE (2)




dim F < dim prS.







Proof of QE (2)




dim F < dim prS.







Proof of QE (2)




dim F < dim prS.







Proof of QE (2)




dim F < dim prS.







Proof of QE (2)




dim F < dim prS.







Proof of QE (3)

Notice that pr(Q) = pr(S− P) ⊆ pr(S′ − P) ∪ pr(S− F).

If b ∈ pr(S− P), then P(b, M) ( S(b, M). Thus

(prS− F) ⊆ prQ.

Therefore, pr(Q) = pr(S− P) = pr(S− P) ∪ (pr(S) − F).

Notice that pr(S) − F is already in the desired form byinduction hypothesis.

Apply induction to S− P to finish the proof.




Proof of QE (3)



(prS− F) ⊆ prQ.







Proof of QE (3)



(prS− F) ⊆ prQ.







Proof of QE (3)



(prS− F) ⊆ prQ.







Proof of QE (3)



(prS− F) ⊆ prQ.







QE for Zariski geometry

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Proposition (Prop 3.3.7)

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M : one-dimensional Zariski Geometry. Then the theory ofM admitsquantifier elimination.




Axioms for one-dimensional Zariski geometry, p. 31

(Z1) (QE) prS ⊇ prS− F, for some proper closed F ⊂cl prS

(Z2) (SM) For S ⊆cl M n+1, there is m such that for all a ∈ M n

S(a) = M or |S(a)| ≤ m

(Z3) dim M n ≤ n.Given a closed irreducible S ⊆ M n. Every component of thediagonal F ∩ {xi = x j} is of dimension ≥ dim S− 1.

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Remark

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These are the axioms given in the paper [HZ].




strong minimality

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Corollary (Cor 3.3.8)

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M is strongly minimal.




Proof of S.M. with (Z2)

Let E ⊆ M n × M be definable.

Show E(a) is finite or co-finite, with a uniform bound for alla ∈ M n.

WLOG, E = S− F, with S, F closed.

If S(a) is finite, then E(a) is finite with the same bound.

Apply (Z2) to both S, F to finish the proof.








































Language

M : Zariski structure

C ⊆ M n is closed, introduce a predicate symbol for eachclosed C ⊆ M n




elementary extensions (1)

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a-closed : LetS(x, y) be l + m-ary closed set.S ⊆ M ′l .Postulate that eachS(a, M ′m) is closed inM ′m.This gives the topology on eachM ′.Dimension

dim S(a, M ′) = max{k ∈ N : a ∈ P(S, k)} + 1

whereP(S, k) = {a ∈ prS : dim S∩ pr−1(a) > k}.




elementary extensions (2)

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Theorem (Thm 3.5.25)

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M : Zariski structure satisfying (EU)

M � M ′

M ′ : Zariski structure




Summary

Start with M : Noerthrian topology + dimension notion

Introduce language for M , hence model theory of M ispossible

Theory of M admits quantifier elimination

M is strongly minimal

Elementary extension of M is Zariski structure (geometry)




Summary









Summary









Summary









Summary







Documents

Zariski Geometries, Lecture 1 - Kobe UniversityHZ96 Ehud Hrushovski, Boris Zilber, Zariski geometries, J of the AMS, 1996 Z10 B. Zilber, Zariski Geometries, London Math. Soc. Lect