Zilber's theory of Zariski structures

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Università di Pisa

Dipartimento di Matematica

Corso di Laurea Magistrale in Matematica

Tesi di Laurea

Zilber's theory of

Zariski structures

22 settembre 2017

Candidata Relatore

Angela Veronese Prof. Alessandro Berarducci

Controrelatrice Prof.ssa Rita Pardini

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Introduction

In this work we are going to analyse an important class of structures, introduced by Hrushovski and Zilber in 1993, in the general setting of Geometric Stability Theory: Zariski geometries.

Geometric Stability Theory is a branch of Model Theory that studies the properties of categorical theories and their generalisations. We say that a theory T is κ-categorical, where κ is an innite cardinal, if T admits a model M of cardinality κ, which is unique up to isomorphisms. The theory T is said uncountably categorical if it is κ-categorical for every uncountable cardinal κ. Familiar examples of uncountably categorical theories are the theory of algebraically closed elds and the theory of vector spaces over a division ring.

The rst remarkable result, that historically marked the beginning of Stability Theory, is the following famous theorem by Morley.

Theorem (Morley's Categoricity Theorem, 1960s). Let T be a κ-categorical theory for some uncountable cardinal κ. Then T is categorical in all uncountable cardinals.

Uncountably categorical theories have thus become a main subject of study; a great work was made in particular by Boris Zilber in the successive decades, with the aim to classify such theories and study their properties.

Many of the tools and ideas in Stability Theory are inspired from classical topology and algebraic geometry, which are major sources of examples. A crucial role is played by pregeometries: a pregeometry on a set X is an analogue of the operations of the (relative) algebraic closure of a subset of a eld A ⊆ k or of the linear span of a set of vectors in a vector space. To every pregeometry we can associate a geometry, built such that the empty set and each singleton are closed, that is, coincident with their closure.

There are three classical examples of geometries:

• The trivial geometry on a set X, where every subset of X is closed;

• The geometry of vector spaces, where the closure of a subset is its linear span; • Algebraically closed elds, where the closure of a subset is its algebraic closure. In this setting, we can isolate an important class of sets, the strongly minimal sets, that can be endowed in a natural way with a pregeometry, given by the model-theoretic algebraic closure.

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Let M be a structure and A ⊆ M. We say that b ∈ M is algebraic over A if there exists a formula φ(x, ¯a) with parameters ¯a ∈ A, such that φ(b, ¯a) holds and the set of the realisations of φ, φ(M, ¯a), is nite. The model-theoretic algebraic closure of the set A is then the set of elements b ∈ M algebraic over A. For instance, if K is an algebraically closed eld and A ⊆ K, the model-theoretic algebraic closure of A in K coincides with the usual algebraic closure.

We say that a subset D ⊆ M of a structure M is minimal if every denable subset of D is nite or has nite complement; D is strongly minimal if it is minimal in every elementary extension of M.

In particular, we can prove that a strongly minimal theory, namely a theory whose models are strongly minimal, is uncountably categorical. In strongly minimal structures, the model-theoretic algebraic closure satises all the properties of a pregeometry. It follows from quantiers elimination that algebraically closed elds and vector spaces over a division ring are strongly minimal sets: in these two cases the model-theoretic algebraic closure coincides with the classical algebraic closure and the closure given by the linear span of a subset, respectively.

Of utmost importance in Stability Theory is dimension theory: starting from a pregeometry, we can give the notions of independence and basis. The dimension of a set will be the cardinality of any of its bases. In the three examples above, the notion of dimension is given by:

• The size of the set in the case of the trivial geometry; • The usual linear dimension in the case of vector spaces;

• The transcendence degree in the case of algebraically closed elds.

At rst, it appeared that all the geometries in an uncountably categorical theory could be based on one of these three types of dependence. This was a consequence of a more general conjecture formulated by Zilber in [25].

Conjecture (Trichotomy Conjecture). The geometry on any strongly minimal structure M is either trivial, locally projective or isomorphic to a geometry of an algebraically closed eld. In the latter case, M interprets an algebraically closed eld K and all the relations induced by M on K are denable in the eld structure itself.

However, in 1993, Hrushovski introduced a new construction that produced a counter-example to the conjecture. This gave rise to new studies aimed at nding the best possible class of structures for which the Trichotomy Conjecture holds: it is in this setting that Hrushovski and Zilber introduced the notion of Zariski structures.

We explain in the following the main ideas that come with the introduction of Zariski structures and what has been collected in this work. We start by recalling the model-theoretic setting in Chapter 1: here we describe in more details some properties of strongly minimal structures and the Trichotomy Conjecture.

Chapter 2 gives the rst denitions about Zariski structures and provides some ex-amples, together with the study of the tools of specialisations.

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vii The starting point is given by the very basics of algebraic geometry: consider an algebraically closed eld K and its powers Kn_{. A Zariski-closed set (or algebraic set) of}

Kn is a subset of Knof the form

V(I) = {(a1, . . . , an) ∈ Kn| f (a1, . . . , an) = 0 ∀ f ∈ I}

where I is an ideal in the ring of polynomials K[x1, x2, . . . , xn]. The Zariski-closed sets

form a Noetherian topology on Kn_{, called the Zariski topology. The Krull dimension of}

a Zariski closed set C ⊆ Kn _{is given by the maximal length n of a chain of irreducible,}

closed sets C0 ( C1 ( · · · ( Cn⊆ C. The behaviour of these sets is studied by classical

algebraic geometry.

The main idea in developing the theory of Zariski structures is a sort of converse: we start from a set M and a collection C of subsets of its powers Mn_{, which will play the}

part of the closed sets. We require that the elements of C satisfy some topological axioms (denition 2.1.1), reecting some properties of algebraic sets in an algebraically closed eld; the most important ones are the following:

• For each power Mn_{, the subsets of M}n _{in C are the closed sets of a Noetherian}

topology on C.

• The product topology on Mn_{× M}l _{is coarser than the given topology on M}n+l_;

• The projection maps Mn+l _{→ M}n _{are continuous.}

Moreover, we assume that a natural number, called dimension is associated to every closed subset of Mn_{: the dimension notion veries some axioms as well and has a good}

behaviour with respect to the projection map pr : Mn+l _{→ M}n _(denition _2.1.5_{). In}

particular, we require that the dimension satises the following: (AF) Addition formula: for any irreducible, closed S ⊆ Mn+l_,

dim S = dim pr(S) + min

a∈pr(S)dim(pr

−1_{(a) ∩ S)}

holds;

(FC) Fibre condition: for any irreducible, closed S ⊆ Mn+l_{, there exists V ⊆ pr(S)}

relatively open in pr(S) such that min

a∈pr(S)dim(pr −1

(a) ∩ S) = dim(pr−1(v) ∩ S)

for any v ∈ V .

The axioms (AF ) and (F C) remind us the well-known bre dimension theorem of algeb-raic geometry.

A crucial additional property of Zariski structures is presmoothness: we say that a Zariski structure M is presmooth if for any two closed irreducible subsets S1, S2 ⊆ Mn

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and any irreducible component S0 of the intersection S1∩ S2, the following inequality

holds:

dim S0 ≥ dim S1+ dim S2− dim Mn.

In other words, presmoothness gives a uniform bound on the dimension of the irreducible components of an intersection. This property turns out to be what is really needed in order to develop an interesting theory of Zariski structures.

Notice that we do not need to assume the presence of any eld operation and this can all be developed in the abstract setting of rst order structures. However, one of the main results is that, starting from these axioms of geometric nature and under certain hypotheses of richness of the structures, we can dene an algebraically closed eld in M and thus recover the classical algebraic geometry and Zariski topology. In particular, this implies that the family of one-dimensional, irreducible, uncountable Zariski structures satisfying the presmoothness property veries the Trichotomy Conjecture.

Not surprisingly, the easiest examples of Zariski structures arise from algebraic geo-metry.

Theorem (Theorem 2.5.1). Let V be a quasi-projective variety over an algebraically closed eld K. We consider the usual Zariski topology on V (an its powers) together with the Krull dimension.

Then V is a Zariski structure; if V is a smooth algebraic variety, then it is a presmooth Zariski structure.

Moreover, in the case of an algebraic curve C, we can tell whether or not C is presmooth as Zariski structure: this happens if and only if the normalisation map ˜C → C is injective, or, equivalently, that the singular points of C are all cusps (theorem3.4.1). The fact that the eld is algebraically closed is of maximal importance.

Example. The eld of real numbers R with the Zariski topology and Krull dimension is not a Zariski structure.

Indeed, consider the projection pr : R3 _{→ R given by (x, y, z) 7→ z and the}

Zariski-closed set S = V(x2_{+ y}2_{− z) ⊂ R}3_{. Both axioms (AF ) and (F C) fail in this case, as}

there is only a point in pr(S) (the origin of R) whose inverse image has dimension 0. However, algebraic varieties are not the only remarkable example of Zariski structures: they can be found also in complex geometry (compact complex manifold are Zariski structures), in rigid analytic geometry (proper analytic varieties) and in the setting of dierentially closed elds of characteristic zero. The latter ones have played a crucial role in Hrushovski's proofs of the Mordell-Lang conjecture for function elds and of the Manin-Mumford conjecture; the model-theoretic proof of the Mordell-Lang conjecture has been the rst ever found.

The rst step needed in order to study the theory of Zariski structures is developing a sort of innitesimal analysis: this is done with the help of specialisations.

Let M be a Zariski structure and consider an elementary extension ∗_{M M}_{; since}

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ix S ⊆ Mn we can consider the corresponding set S(∗M ) ⊆ (∗M )n. Let A ⊆ ∗M be such that M ⊆ A. A map π : A → M is a specialisation if S is closed in Mn_{, then}

π(A ∩ S(∗M )) ⊆ S. Our standard example of specialisations is the following (see also example 2.4.3): consider M = k an algebraically closed eld of characteristic zero and let ∗_{M = K} _{be the algebraically closure of the eld of Laurent series k((t)). We know}

that K is the eld of Puiseux series given by K = [

n∈N

k((tn1)).

Then K is a valued eld; if R ⊂ K is the associated valuation ring, the map π : R → k sending f ∈ R to its constant term is a specialisation of Zariski structures. The map π extends to a total specialisation of the projective lines P1_{(K) → P}1_(k)_.

We focus upon a special class of specialisations, called universal specialisations (den-ition 2.4.5), that satisfy a sort of universal property: the main idea is that a universal specialisation incorporates all the specialisations π : A ∪ M → M where A is a nite subset of an elementary extension of M. In the example above, a universal specialisation is obtained by repeating the construction with Puiseux series countably many times.

Given a specialisation π :∗_{M → M}_{, we can dene the innitesimal neighbourhood of}

an element a ∈ Mn_:

Va= π−1(a).

This is a well dened notion (lemma 3.3.4). In model-theoretic terms, the innitesimal neighbourhood of a ∈ Mn _{coincides with the set of realisations of a certain n-type built}

from open subsets containing a (denition 2.4.11and lemma2.4.12); more intuitively, it is a formalisation of the small moves that were used in algebraic geometry to dene, for instance, the tangency and the multiplicity of intersection between two curves.

Through innitesimal neighbourhoods we can somehow dene, in the setting of Zariski structures where we lack the familiar eld operations, a notion of multiplicity. This is done as follows: let M be a presmooth, irreducible Zariski structure. A covering F/D is a surjective projection map pr : F ⊆ D × Ml _{→ D}_{, where we suppose that F is closed,}

irreducible in D ×Ml _{and D ⊆ M}n_{is presmooth. We consider coverings F/D with nite}

bres; then if (a, b) ∈ F , we dene the multiplicity of the covering F/D at the point (a, b) as

multb(a, F/D) = #(F (a0,∗M ) ∩ Vb),

for a generic a0 _{∈ D(}∗_{M ) ∩ V}

a, where

F (a0,∗M ) = {c ∈∗M | (a0, c) ∈ F (∗M )}.

The idea beyond this denition is the following: the multiplicity of F/D at (a, b) is the number of points in the inverse image of a generic element of D which is very close to a ∈ D. We give an example to clarify the meaning of this multiplicity.

Example. Consider the curve C = {x3 _{= y}2_{} ⊆ C}2_{: the projection (x, y) 7→ x gives}

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(u, v) 6= (0, 0) is 1, while the multiplicity at (0, 0) is 2. In fact, if we move a little the point 0 ∈ C and consider an α very close to 0 (but α 6= 0), there are always two distinct solutions to the equation α3_{= y}2_{, both very close to 0.}

This notion of multiplicity has its counterpart in classical algebraic geometry; in the case of a nite morphism between two smooth projective curves (which can be seen as a covering in the sense of Zariski structures), the multiplicity dened above coincides with the ramication index. More precisely, following the work by T. De Piro (see [4]), the following theorem can be proven.

Theorem. Let ϕ: C → D be a nite morphism of smooth projective curves over an algebraically closed eld K of characteristic zero, and let F ⊆ D × C be the graph of ϕ. Then for any point (q, p) ∈ F we have that

multp(q, F/D) = eϕ(p),

where eϕ(p) is the ramication index of ϕ at the point p ∈ C.

We study coverings and the Zariski multiplicity in Chapter3.

The multiplicity of a covering thus dened permits us to develop some intersection theory in our Zariski structure M, without the need of an algebraic structure. Here we restrict our study to a special class of Zariski structures, that of one-dimensional, irreducible and presmooth Zariski geometries. Let C be such a Zariski structure: we want to describe the notion of families of curves in Cn _{and exhibit a way to dene the}

intersection index of two families of curves.

A family of curves in Cm_{, m ≥ 2, is a triple (P, L, I) such that:}

• P ⊆ Cm _{is the set of points of the curves.}

• Lis a presmooth Zariski structure which is locally isomorphic to Ck_{for some k ≥ 1.}

The set L is the parametrisation of a set of curves • I ⊆ P × Lis irreducible and closed in P × L.

The set I is called incidence relation and if (p, l) ∈ I, we say that the point p ∈ Cm

belongs to the curve l ∈ L. We also ask that all l ∈ L are of dimension 1 and that they are irreducible for generic l.

The index of intersection of two families of curves (P1, L1, I1) and (P2, L2, I2) is

dened as follows, under the assumption that for almost all l1 ∈ L1 and l2 ∈ L2, the

intersection l1∩ l2 is nite and non-empty. Consider l1, l2 that satisfy this property; if

p ∈ l1∩ l2, the index of intersection of l1, l2 at the point p is

indp(l1, l2/L1, L2) = #l01∩ l 0 2∩ Vp, where (l0 1, l02) ∈ V(l1,l2) ∩ L1( ∗_{C) × L}

2(∗C) is a generic element. Then the index of

intersection of the families L1, L2 will be

ind(L1, L2) =

X

p∈l1∩l2

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xi where l1∈ L1and l2 ∈ L2are any two curves with nite and non-empty intersection. The

index of intersection is well dened, as it can be seen as a special case of the multiplicity for the coverings dened before (proposition 4.2.5).

Indeed, if we let D ⊆ L1× L2 be the set of pairs of curves with nite and non-empty

intersection, and consider

F = {(p, l1, l2) ∈ P × L1× L2| (l1, l2) ∈ D, p ∈ l1∩ l2},

then the presmoothness of P, L1and L2implies that each irreducible component F1, . . . , Fr

of F is a covering of D with nite bres, and that indp(l1, l2/L1, L2) =

r

X

i=1

multp((l1, l2), Fi/D).

This is still related to the intuitive idea of small moves: the index of intersection at the point p ∈ l1∩ l2 is the number of points very close to p in the intersection of two curves

obtained by slightly modifying l1 and l2 near p.

Example. Consider the two curves L = {y = 0} and C = {x2 _{= yz}} _{in the projective}

space P2_{(C). The point P = [0 : 0 : 1] is the only common point of C and L, but the}

index of intersection at P of C and L is 2. In fact, by moving a little the line L and the curve C, the number of points in the intersection near P is always 2.

In general, we can see L and C as members of the families of curves of degree 1 and of degree 2 on P2_{(C), respectively. The index of intersection of these two families is 2,}

since a generic line intersects a generic conic in two distinct points.

In this way, without using any eld operation, we nd ourselves in a universe with no numbers, but with a lot of geometric objects that interact in a nice way. One of the biggest achievements of Hrushovski and Zilber in their rst paper [9] was the nding that, if our structure C is rich enough, then C interprets an algebraically closed eld. Precisely, the condition we need C to satisfy is the following:

(AMP) Ampleness: there exists a 2-dimensional, irreducible, faithful family of curves (P, L, I)in C2 (namely, L is 2-dimensional and irreducible).

The recovering of a eld structure from a family of geometric objects is not a new idea. For instance, consider a eld k; then, if we are given all the subsets of kn_{dened by}

linear equations, it is quite easy to recover the eld operations by studying the behaviour of these subsets (see [1]).

Interpreting a eld in our ample Zariski structure C is more complex and requires a very technical intermediate stage, that consists in dening a notion of tangency between branches of curves, which we do not discuss in this work. In the end, we obtain the following result.

Theorem (Theorem 4.3.3 and remark 4.3.5). Let C be a one-dimensional, irreducible and presmooth Zariski structure; assume C is ample. Then C interprets an algebraically closed eld K, which is a one-dimensional and irreducible Zariski structure.

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Let K be the algebraically closed eld interpreted in the Zariski structure C: in particular, K inherits from C a topology that makes it a Zariski structure. This topology on K does not coincide, a priori, with the usual Zariski topology given by algebraic sets; anyway, the construction implies that algebraic sets are closed (so the topology on K renes the usual one). Studying the geometry on the eld K, we can prove the following result.

Theorem (Theorem 4.4.12). Every relation induced on the eld K by the Zariski geo-metry C is denable within the eld structure.

To prove this theorem, we analyse the geometry on the projective space P2_(K)_{, in}

particular, we are interested in how the closed subsets of K interact with the classical fam-ilies of algebraic curves of xed degree d. The purity theorem is then implied by suitable adaptations of Bezout's theorem (theorem4.4.8) and Chow's theorem (theorem 4.4.11). The intersection theory in Zariski structures, an overview on how the Zariski geometry C interprets a eld and the proof of the purity theorem can be found in Chapter4.

The nal result of this thesis, to which we devote Chapter5, is a classication theorem for ample Zariski geometries by Hrushovski and Zilber, showing that our Zariski structure C is always almost an algebraic curve, that is, it a nite cover of a quasi-projective algebraic curve.

Theorem (Theorem5.1.1 and theorem 5.2.5). Let C be a one-dimensional, irreducible, ample Zariski geometry. Then there exists an algebraically closed eld K and a Zariski continuous map f : C → P1_(K) _{such that}

• f has nite bres;

• for any denable set T ⊆ Cn_{, the image f(T ) ⊆ (P}1_(K))n _{is constructible in the}

sense of classical algebraic geometry.

Moreover, there exists a smooth quasi-projective curve X over K and a surjective Zar-iski morphism ψ : C → X with the following universal property: for any smooth quasi-projective curve Y over K and any surjective Zariski function τ : C → Y , there exists a surjective Zariski function σ : X → Y such that σ ◦ τ = ψ.

The map ψ : C → X is not necessarily injective: an example built starting from an elliptic curve was given by Zilber and Hrushovski in their paper [9]. Thus we have the following theorem.

Theorem. There exists an ample, irreducible, one-dimensional Zariski geometry C that cannot be interpreted in an algebraically closed eld.

However, it is possible to specify a geometric condition stronger than ampleness (called very ampleness), that makes C an exact algebraic curve.

Zilber's theory of Zariski structures has been developed in the last twenty years, mostly in the direction of studying analytic Zariski geometries and the consequences of these results in other branches of mathematics. This constitutes a growing eld of study, with promising expectations.

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Chapter 1

Model theoretic preliminaries

In this initial chapter, we will briey resume some notions in Model Theory that are necessary to the further understanding of the thesis. We assume that the reader is familiar with the basic model theoretic concepts and fact, such as those regarding structures, theories, completeness and elementary extensions; we refer to [14] and to [20] for a complete treatise on these subjects.

We will introduce saturated and stable structures, together with the most important tools and theorems that will be of use in the following.

We will also give an introduction to geometries and strongly minimal structures, in order to analyse the weak trichotomy theorem (theorem 1.4.7). The chapter ends with a Conjecture by Zilber and an outline of the main reasons why Zariski geometries have been introduced.

Almost all theorems we expose in this chapter come without proof: we refer to the books cited above for these and more facts about the basics of Model Theory.

1.1 Basic model theoretic notions

We recall some model theoretic notions that will be used in the study of Zariski structures in the following chapters.

Remark 1.1.1. In the following, we will denote in the same way a structure and its domain. When we say that M is an L-structure, we mean that we are dealing with a one-sorted structure M = (M; L) whose domain is M and whose language is L.

Quantier elimination

Denable subsets of a structure M can be very complicated, since they can be dened by formulas with many alternations of quantiers. Subsets dened by quantier-free formulas are easier to deal with and we are interested in those theories in which all denable subsets can be dened by such formulas.

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Denition 1.1.2. We say that a theory T has quantier elimination if for every formula φthere exists a quantier-free formula ψ such that

T ` φ ←→ ψ.

Quantier elimination is a very important tool to show completeness or decidability of a theory, as well as many other properties.

We give some examples of theories that have quantier elimination.

Example 1.1.3. The theory DLO of dense linear orders without endpoints has quantier elimination.

Example 1.1.4. Consider the theory of Presburger arithmetic T = Th(Z; +, −, <, 0, 1). This theory does not admit quantier elimination in the language L = {+, −, <, 0, 1}; for instance, the formula

ψn(v) := ∃y(v = y + y + · · · + y

| {z }

n-times

)

meaning that v is a multiple of n, is not equivalent to any quantier-free formula. Nevertheless, if we consider the language L∗ _{= L ∪ {R}

p}p∈P, where P is the set of

prime numbers and Rp is a unary relation satised by the multiples of p, then T is a

complete and decidable theory in L∗ _{with quantier elimination.}

Example 1.1.5. The theory of algebraically closed elds ACF has quantier elimination in the language of rings Lring = {+, −, ·, 0, 1}.

In this case, quantier elimination implies a bunch of very useful facts about algeb-raically closed elds, like:

• Model-completeness of the theory ACF ;

• Completeness of the theory ACFpof algebraically closed elds of xed characteristic

p, where p = 0 or p is prime;

• A geometric interpretation: if K is an algebraically closed eld, then a subset X ⊆ Kn is denable if and only if it is a boolean combination of Zariski-closed subsets;

• Chevalley's Theorem: The image of a constructible set under a polynomial map is constructible;

• Hilbert's Nullstellensatz;

• The theory ACF is strongly minimal (see below).

We will see in section 2.3that Zariski structure have quantier elimination in their natural language.

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1.1. Basic model theoretic notions 3

Categorical theories

Denition 1.1.6. Let κ be a cardinal. We say that a theory T is κ-categorical if it has models of cardinality κ and any two models of T of cardinality κ are isomorphic. Example 1.1.7. Let K be a eld and consider the theory of K-vector spaces: a model V of this theory is a structure in the language L = {+, 0, λq, q ∈ k}, where λqis a symbol

of function representing the multiplication by the scalar q ∈ k.

Consider a cardinal κ > max{ℵ0, |K|}: then any K-vector space of cardinality κ is

of dimension κ over K. This implies that the theory of K-vector spaces is κ-categorical. In particular, the theory of Q-vector space is uncountably categorical, that is, κ-categorical for any uncountable κ.

Example 1.1.8. Consider p = 0 or p a prime number. Then the theory of algebraic-ally closed elds of characteristic p is κ-categorical for any uncountable κ. Indeed, two algebraically closed elds of the same characteristic are isomorphic if and only if they have the same transcendence degree and any algebraically closed eld of cardinality κ, for uncoutable κ, is of transcendence degree κ.

Example 1.1.9. The theory of dense linear orders without endpoints DLO is ℵ0

-categorical, but it is not κ-categorical for any κ > ℵ0.

Categoricity is a test for completeness.

Lemma 1.1.10. Let T be a satisable L-theory with no nite models and suppose that T is κ categorical for some κ ≥ |L|. Then T is complete.

Proof. Suppose that T is not complete; then there is a sentence φ such that T 6 φ e T 6 ¬φ, therefore the theories T1 = T ∪ {φ}and T2 = T ∪ {¬φ} are satisable. Since T

has no nite models, and by the Löwenheim-Skolem theorem, we can nd M1 T1 and

M2 T2 both of cardinality κ.

The models M1 and M2 are not elementary equivalent, since they disagree on φ, and

this contradicts the κ-categoricity of T .

A key role in Zilber's studies leading to the weak trichotomy theorem is played by uncountably categorical theories, that is, theories which are κ-categorical for any uncount-able κ.

An important result of Morley is the following fundamental theorem.

Theorem 1.1.11 (Morley's Categoricity Theorem). Let T be a κ-categorical theory for some uncountable cardinal κ.

Then T is categorical in all uncountable cardinals. Proof. See [14, Theorem 6.1.1].

For our purpose of studying Zariski geometries, we will be more interested in strongly minimal structures (see section 1.2), which are, as theorem 1.2.19shows, a special case of uncountably categorical theories.

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Types

Let M be an L-structure.

Denition 1.1.12. Given an n-tuple of distinct variables ¯x, a n-type in M is a set of L-formulas

p(¯x) = {φi(¯x) | i ∈ I}

which is nitely satisable: for any φ1(¯x), . . . , φk(¯x) ∈ p(¯x) there exists ¯a ∈ Mn such

that

M ^

j≤k

φj(¯a).

A n-type p(¯x) is realised in M if there exists ¯a ∈ Mn _{such that M φ(¯a) for any}

φ(¯x) ∈ p(¯x). If a type p(¯x) is not realised in M, we say that p(¯x) is omitted or that M omits p(¯x).

A type p(¯x) is complete if for any l formula ψ(¯x), either ψ(¯x) ∈ p(¯x) or ¬ψ(¯x) ∈ p(¯x). Example 1.1.13. Consider the structure (Q; +, ·, <): there are 2ℵ0 complete 1-types in

this structure and among these, ℵ0 are realised.

Given an L-structure M and ¯a ∈ Mn_{, we can consider the type of the element ¯a in}

M:

tp_M(¯a) = {φ(¯x)L-formula | M φ(¯a)} . The type tpM(¯a)is always a complete type which is realised in M.

Given a type p(¯x), we denote by p(M) the set of realisations of the type p(¯x) in M. All the denitions can be generalised to types over A, where A ⊆ M, using LA

-formulas instead of L--formulas.

Denition 1.1.14. Consider an L-theory T . A set of L-formulas p(¯x) is an n-type if there exists M T and ¯a ∈ Mn _{such that M p(¯a), that is, p(¯x) is a realised n-type in}

M.

The rst important result we need to recall is the following.

Proposition 1.1.15. Let M be an L-structure, A ⊆ M and p(¯x) an n-type over A. Then there is an elementary extension N M such that p(¯x) is realised in N.

Proof. The proof is an easy consequence of the Compactness theorem: see for instance [14, Proposition 4.1.3].

We will denote by SM

n (A)the set of complete n-types over A in the structure M. The

set SM

n (A) can be endowed in a natural way with a topology that makes it a compact

topological space.

Denition 1.1.16. The Stone topology on SM

n (A) is the topology generated by taking

as basic open sets, the sets of the form

[φ] = {p ∈ S_nM(A) | φ ∈ p} for any LA-formula φ.

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1.1. Basic model theoretic notions 5 It is an easy exercise to verify that the sets [φ] are the basis of a topology on SM

n (A):

each [φ] is both open and close and the resulting topological space is compact and totally disconnected.

It is interesting to give a characterisation of the space Sn of complete n-types in the

case of algebraically closed elds.

Example 1.1.17. Let K be an algebraically closed eld and k ⊆ K a subeld of K. Consider the space of n-types over k, SK

n (k); we can associate each type p ∈ SKn(k)with

a prime ideal of the ring k[x1, x2, . . . , xn](that is, an element Ip ∈ Spec k[x1, x2, . . . , xn])

in the following way:

Ip = {f (¯x) ∈ k[x1, . . . , xn] | f (¯x) = 0 ∈ p}.

Then the map p 7→ Ip is a continuous bijection from SnK(k)to Spec k[x1, . . . , xn](see, for

instance, [14, Example 4.1.14 and Proposition 4.1.16]).

Saturated models

Denition 1.1.18. Let M be an L-structure and κ an innite cardinal. We say that M is κ-saturated if for any subset A ⊆ M such that |A| < κ, all the 1-types over A are realised in M.

We say that M is saturated if it is κ-saturated for κ = |M|.

Example 1.1.19. An algebraically closed eld K of innite transcendence degree is a saturated model of ACF .

We briey analyse the existence of κ-saturated models of complete theories.

Theorem 1.1.20. Let M be an L-structure and κ an innite cardinal. Then there exists an elementary extension N M which is κ+_{-saturated and such that |N| ≤ |M|}κ_.

Proof. The proof of this theorem involves a standard process in model theory: we give it here as an example, since other theorems in the next chapters will follow in an analogous way.

We begin by showing that M has an elementary extension M0 _{such that |M}0_{| ≤ |M |}κ

and every type over a subset A ⊆ M, |A| ≤ κ, is realised in M0_.

Notice that

|{A ⊆ M | |A| ≤ κ}| ≤ |M |κ and that |SM

1 (A)| ≤ 2κ for every A ⊆ M, |A| ≤ κ. Hence we can list all the types over

a set of parameters of cardinality ≤ κ with

(pα : α < |M |κ).

We build a chain of elementary extensions of M, (Mα)α<|M |κ, in the following way:

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ii) for limit ordinal α,

Mα=

[

β<α

Mβ;

iii) Mα+1 Mα is such that |Mα+1| = |Mα|and pα is realised in Mα+1.

Now we dene

M0 = [

α<|M |κ

Mα

and it is easy to see that this M0 _{is the desired elementary extension of M.}

Now we prove the theorem by building another elementary chain (Nα)α<κ+:

i) N0= M;

ii) for limit ordinal α, dene

Nα=

[

β<α

Nβ;

iii) Nα+1 Nα is obtain as we did before: Nα+1 = Nα. That is, Nα+1 realises all the

types p ∈ SNα

1 (A)with A ⊆ Nα and |A| ≤ κ. Moreover, by induction, since

|Nα| ≤ |M |κ =⇒ |Nα|κ ≤ (|M |κ)κ = |M |κ,

we can ask that |Nα+1| ≤ |M |κ.

Consider

N = [

α<κ+

Nα.

Then N M and since κ+_{≤ |M |}κ_{, we have that |N| ≤ |M|}κ_.

We are left to show that N is κ+_{-saturated. Consider A ⊆ N, |A| < κ}+ _{and p ∈}

S_aN(A); then |A| ≤ κ and, because κ+ is a regular cardinal, there exists α < κ+ such that A ⊆ Nα. Thus p is realised in Nα+1 and consequently in N.

1.2 Stability theory and Morley rank

ω-stable theories

Let T be a complete theory in a countable language with innite models.

Denition 1.2.1. Let κ be an innite cardinal. We say that T is κ-stable if for every model M T and A ⊆ M such that |A| ≤ κ, than the set of 1-types over A is of cardinality |S1(A)| ≤ κ.

A structure M is κ-stable if its complete theory Th(M) is.

Remark 1.2.2. It is equivalent, in denition 1.2.1, to ask that |S1(A)| ≤ κ or that

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1.2. Stability theory and Morley rank 7 Example 1.2.3. The theory ACF of algebraically closed elds is ω-stable.

The theory of dense linear orders without endpoints DLO is not ω-stable, since Q DLO and |S1(Q)| = 2ℵ0 (each type can be identied with a real number).

We will be interested most in ℵ0-stable theories in the following, which are often

addressed to as ω-stable theories. A rst important result is the theorem below.

Theorem 1.2.4. Let T be a complete theory in a countable language. If T is ω-stable, then it is κ-stable for all innite cardinals κ.

Proof. See [14, Theorem 4.2.18].

We know that κ-stable theories play a signicant role in the construction of saturated model.

Theorem 1.2.5. Let κ be a regular cardinal. If the theory T is κ-stable, than there exists a saturated model M T of cardinality κ.

In particular, if T is ω-stable, then for any regular cardinal κ there is a saturated model M T such that |M| = κ.

Proof. We can prove that, given any M0 T of cardinality κ, there exists an elementary

extension M M0 such that |M| = κ and M is saturated. The proof follows a

con-struction similar to that in the proof of theorem1.1.20. See [14, Theorem 4.3.15] for the details.

Remark 1.2.6. Theorem1.2.5 is true also in the case that the cardinal κ is singular, but the proof is more subtle and makes use of Morley sequences and indiscernibles. See [14, Theorem 6.5.4].

Morley rank

We will introduce the Morley rank, which gives a notion of dimension for denable subsets and is one of the most important tools for analysing ω-stable theories.

Let M be an L-structure. For any formula φ(¯v) (with parameter from M) in the free variables ¯v, we denote by φ(M) the set of realisations of φ.

Denition 1.2.7. Let φ(¯v) be an LM-formula. The Morley rank of φ in M, RMM(φ),

is dened inductively as follows:

• RMM_{(φ) ≥ 0} _{if and only if φ(M) 6= ∅;}

• If α is a limit ordinal, then RMM(φ) ≥ αif and only if RMM(φ) ≥ β for all β < α; • For any ordinal α, RMM(φ) ≥ α + 1 if and only if there exist countably many LM-formulas ψ1(¯v), ψ2(¯v), . . . such that the sets ψ1(M ), ψ2(M ), . . . are pairwise

disjoint subsets of φ(M) and RMM_(ψ

i) ≥ α for all i ≥ 1.

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• RMM(φ) = −1 if and only if φ(M) = ∅;

• RMM(φ) = αif and only if RMM(φ) ≥ αand RMM(φ) 6≥ α + 1; • RMM_{(φ) = ∞}_{if RM}M_{(φ) ≥ α}_{for all ordinals α.}

Thus dened, the Morley rank of a formula depends on the model in which it is calculated: if N M, than an LM-formula φ is also an LN-formula, but in general

RMM(φ) 6= RMN(φ). However, in ω-saturated structures the Morley rank has a good behaviour: in this case, we can prove that RM(φ) depends only on the type of the parameters in φ and the following result follows.

Theorem 1.2.8. Suppose that M ≺ N are ω-saturated models of a theory T and that φ is an LM-formula. Then RMM(φ) = RMM(φ).

Consequently, if M is an L-structure, φ is an LM-formula and N M is an

ω-saturated elementary extension of M, RMN_(φ) _{does not depend on the chosen N.}

Denition 1.2.9. Let M be an L-structure and φ an LM-formula. The Morley rank of

φis

RM(φ) = RMN(φ) where N is any ω-saturated elementary extension of M.

The Morley rank of a denable subset X ⊆ Mn _{is RM(X) = RM(φ) where φ is the}

LM-formula dening X.

The Morley rank has the properties of a good dimension notion.

Lemma 1.2.10. Let M be an L-structure. For any denable subsets X, Y ⊆ MN_{, the}

following hold:

• X ⊆ Y implies RM(X) ≤ RM(Y ); • RM(X ∪ Y ) = max{RM(X), RM(Y )}; • RM(X) = 0 if and only if X is nite.

Proof. The proof is a quite simple exercise; it can be found, for instance, in [14, Lemma 6.2.7].

Theories where all formulas have a rank (that is, the rank is an ordinal α), are a special class of theories.

Denition 1.2.11. An L-theory T is called totally transcendental if for every M T and every LM-formula φ, RM(φ) < ∞.

Together with Morley rank, we can use another way to describe more deeply the denable subsets of Mn _{of nite Morley rank, that is an analogue of the irreducible}

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1.2. Stability theory and Morley rank 9 Lemma 1.2.12. Let M be an L-structure; we assume for simplicity that M is ω-saturated. Let φ be an LM-formula of Morley rank α. Then there is a maximum natural

number d such that there exist ψ1, ψ2, . . . , ψd LM-formulas with the following properties:

• RM(ψ₁) = RM(ψ2) = · · · = RM(ψd) = α;

• ψ1(M ), ψ2(M ), . . . , ψd(M ) are disjoint subsets of φ(M).

Proof. See [14, Proposition 6.2.9].

Denition 1.2.13. The natural number d of lemma 1.2.12 is called the Morley degree of φ, Mdeg(φ) = d.

Strongly minimal structures

In the following, M will be an L-structure and T will denote a complete theory in a countable language with innite models.

Denition 1.2.14. Let φ be an LM-formula and let D ⊆ Mn be the set dened by φ.

We say that φ and D are minimal in M if, for any denable Y ⊆ D, then either Y is nire or D \ Y is nite.

We say that φ and D are strongly minimal if φ is minimal in every elementary extension N M.

The theory T is strongly minimal if every N T , N is strongly minimal.

We can characterize strongly minimal structures using Morley rank and Morley de-gree, as the following theorem shows.

Theorem 1.2.15. A formula φ is strongly minimal if and only if RM(φ) = Mdeg(φ) = 1. Proof. See [14, Corollary 6.2.10].

Example 1.2.16. Quantier elimination in ACFpimplies that the theory of algebraically

closed elds of xed characteristic is strongly minimal ([14, Corollary 3.2.9]).

The theorem we have stated above allows us to prove an important property of Morely rank in denable sets, namely that it is a denable notion.

Lemma 1.2.17. Let D be a strongly minimal set and suppose that C ⊆ Dm+n _is

den-able. Consider the projection C → Dm _{and, for each a ∈ D}m_{, the bre over a}

C(a, D) = {x ∈ Dn| (a, x) ∈ C}. Then the set

Jk= {a ∈ Dm| RM(C(a, D)) ≥ k}

is denable for each k ≤ n. Proof. See [13, Lemma 3].

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Finally, we state a crucial result relating strongly minimal and uncountably categor-ical theories.

Theorem 1.2.18. Let T be a complete theory in a countable language with innite models. Suppose that T is κ-categorical for some uncountable cardinal κ. Then T is ω-stable.

Proof. See [14, Corollary 5.2.10].

Theorem 1.2.19. Let T be a strongly minimal theory. Then T is κ-categorical for any κ ≥ ℵ1.

Proof. See [14, Corollary 6.1.12].

Model-theoretic algebraic closure

Denition 1.2.20. Consider the L-structure M and a subset A ⊆ M. We say that an element b ∈ M is algebraic over A if there exists a formula φ(x, ¯a) with parameters ¯

a ∈ A, such that φ(b, ¯a) holds and the set of the realisations of φ, φ(M, ¯a), is nite. The algebraic closure of the set A is

acl(A) = {b ∈ M | bis algebraic over A} .

Example 1.2.21. Consider an algebraically closed eld K and a subset A ⊆ K. Then the (model theoretic) algebraic closure coincides with the usual algebraic closure: acl(A) is the algebraic closure in K of the subeld generated by A.

Example 1.2.22. Let K be a eld and V a K-vector space. If A ⊆ V , then acl(A) is the linear subspace of V generated by A.

The algebraic closure operator has some remarkable properties.

Lemma 1.2.23. The algebraic closure on a structure M satises, for any A ⊆ M: i) acl(acl(A)) = acl(A) ⊇ A;

ii) If A ⊆ B, then acl(A) ⊆ acl(B);

iii) If a ∈ acl(A), then a ∈ acl(A0) for some nite A0 ⊆ A.

Moreover, if M is strongly minimal, then acl satises the exchange property: for any subset A ⊆ M and elements a, b ∈ M,

a ∈ acl(A ∪ {b}) \ acl(A) =⇒ b ∈ acl(A ∪ {a}). Proof. See [14, Lemma 6.1.3 and Lemma 6.1.4].

Lemma 1.2.23 states that in a strongly minimal structure M, the algebraic closure aclgives a pre-geometry on M.

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1.3. Geometry of strongly minimal sets 11

1.3 Geometry of strongly minimal sets

We will now introduce the concept of pre-geometry on a strongly minimal set X and give some basic example of geometric structures.

Denition 1.3.1. Let X be a set and let cl : P(X) → P(X) be an operator on the power set of X. Then (X, cl) is a pre-geometry if:

• For any A ⊆ X, A ⊆ cl(A); • For any A ⊆ X, cl(cl(A)) = cl(A);

• It satises Steinitz exchange property: if A ⊆ X and a, b ∈ X such that a /∈ cl(A), then

a ∈ cl(A ∪ {b}) =⇒ b ∈ cl(A ∪ {a});

• It is of nite character: for any A ⊆ X and a ∈ cl(A), then there is a nite subset A0 ⊆ Asuch that a ∈ cl(A0).

Subsets A ⊆ X such that cl(A) = A are called closed.

A pre-geometry (X, cl) is a geometry if cl(∅) = ∅ and cl({x}) = {x} for every x ∈ X. Remark 1.3.2. Given a pre-geometry (X, cl), there is a natural way to build a geometry ( ˆX, ˆcl). Consider X0 = X \ cl(∅)and dene

ˆ

X = X0/ ∼

where, if a, b ∈ X0, we have that

a ∼ b ⇐⇒ cl({a}) = cl({b}). Consider ˆA ⊆ ˆX and A ⊆ X such that A/ ∼= ˆA. Then

ˆ

cl( ˆA) = cl(A)/ ∼ denes a geometry ( ˆX, ˆcl).

Denition 1.3.3. Given a pre-geometry (X, cl) and a subset D ⊆ X, the localisation of the pre-geometry with respect to D is another pre-geometry (X, clD)obtained by setting

clD(A) = cl(D ∪ A) ∀ A ⊆ X.

It can be veried that (X, clD) is a pre-geometry.

The classical example of pre-geometry is given by algebraic closure in strongly minimal structures (see lemma1.2.23).

In pre-geometries, we can generalise the notions of independence and of dimension, in the following way.

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Denition 1.3.4. Let (X, cl) be a pre-geometry and A ⊆ X. We say that A is inde-pendent if for all a ∈ A, a /∈ cl(A \ {a}).

We say that B ⊆ A is a basis for A if B is independent and A ⊆ cl(B).

As in the more familiar examples of algebraic closure and linear closure in vector spaces, the following lemma can be proven.

Lemma 1.3.5. Let (X, cl) be a pre-geometry. Consider Y ⊆ X and B1, B2 ⊆ Y such

that each Bi is a basis for Y . Then B1 and B2 have the same cardinality.

Proof. See [14, Theorem 6.2.19].

Denition 1.3.6. Let (X, cl) be a pre-geometry and Y ⊆ X. The dimension of Y dim Y is the cardinality of any B ⊆ Y which is a basis for Y .

Denition 1.3.7. Let (X, cl) be a pre-geometry. We say that Y ⊆ X is independent over A ⊆ X if Y is independent in the localisation (X, clA). We then call dimension of

Y over A, dim(Y/A), the dimension of Y in (X, clA).

We now give some examples of pre-geometries, other than the ones given by algebraic closure in vector spaces and in algebraically closed elds that we have already examined. Denition 1.3.8. A pre-geometry (X, cl) is trivial if for any A ⊆ X,

cl(A) = [

a∈A

cl({a}). Example 1.3.9. Consider a set D with no structure. Then

acl(∅) = ∅, acl({a}) = {a} ∀ a ∈ X. In this case, (D, acl) is a trivial geometry.

Example 1.3.10. Let M be a model of Th(Z, S), where S(x) = x + 1 is the successor of x. Then acl(∅) = ∅ and acl(A) = {Sn_{(a) | a ∈ A, n ∈ Z} for any A ⊆ M . This is an}

example of trivial pre-geometry which is not a geometry.

Example 1.3.11. Consider a vector space V over a division ring F . By dening cl(A) to be the smallest ane space of V containing A ⊆ V , we obtain a geometry on V .

We conclude this section with an important property of algebraic closure in strongly minimal structures, namely that the induced notion of dimension coincides with Morley rank.

Denition 1.3.12. Let (M, acl) be an ω-saturated pre-geometry and consider an A-denable X ⊆ Mn _{(we can assume A is nite). Then we can dene the dimension of X}

as

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1.4. The Weak Trichotomy Theorem and Zilber's conjecture 13 If X is A-denable and B ⊃ A, then X is also B-denable. It can be veried that this notion of dimension does not depend on the set A over which X is dened.

Theorem 1.3.13. Let M be a strongly minimal structure and let X ⊆ Mn_{be a denable}

set. Then dim(X) = RM(X). Proof. See [17, Lemma 2.6].

Example 1.3.14. Let K be an algebraically closed eld. Then the notions of (model-theoretical) dimension, Morley rank and transcendental degree coincide.

1.4 The Weak Trichotomy Theorem and Zilber's conjecture

The weak trichotomy theorem gives a classication of geometries on a strongly minimal structure M; it states that such a geometry is either trivial or belongs to one of the following special classes: locally projective geometries and geometries in which a pseudo-plane is denable.

We are going to briey discuss this types of geometries; in the following, (M, cl) will always be a strongly minimal structure with the pregeometry given by algebraic closure. Denition 1.4.1. An abstract projective geometry is a set of objects, points and lines, such that:

• There is a unique line through any two points; • Each line has at least three points;

• For any distinct points a, b, c, d, if the lines though a, b and c, d intersect, then also the lines through a, c and b, d intersect.

Example 1.4.2. Let F be a division ring and V an innite vector space over F . We have that V is strongly minimal and the algebraic closure of any A ⊆ V is its linear span. The algebraic closure in this case is not a geometry, since acl(∅) = {0} and for any a ∈ V, acl(a) is equal to the line through a and 0.

The associated geometry is the projective space associated to V , P(V ): the points are the lines through 0 in V , and the closure of a set of lines is made by all the lines in their linear span.

Thus the geometry associated to (V, acl) is a projective geometry in the sense of den-ition 1.4.1, whose points and lines are respectively one-dimensional and 2-dimensional linear subspaces of V .

Denition 1.4.3. The geometry on M is called locally projective if for any a ∈ M, the geometry associated to the localisation clais isomorphic to a projective geometry over a

division ring.

The last class of geometries we will introduce is that of geometries in which a pseudo-planes is denable. This is also the case that we will consider when we will talk about ample Zariski geometries in section 4.3.

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Denition 1.4.4. An incidence system is a 2-sorted structure (P, L, I) where P and L are sorts and I ⊆ P × L is a binary relation, called incidence relation.

Denition 1.4.5. We say that a pseudo-plane is denable in M if there is an incidence system (P, L, I) denable in MA, for some A ⊆ M, such that:

• RM(P ) = 2 and Mdeg(P ) = 1; • RM(L) ≥ 2;

• If we denote by I(P, l) the set of points p ∈ P such that pIl, then RM(I(P, l)) = 1 for all l ∈ L;

• If l1, l2 ∈ Lare distinct, then I(P, l1) ∩ I(P, l2) is nite or empty.

Example 1.4.6. There is an easy way to dene a pseudo-plane in any algebraically closed eld K: consider P = K2_{, L the lines of K}2 _{and I is the incidence relation}

pIl ⇐⇒ p ∈ L. We are ready to state Zilber's trichotomy theorem.

Theorem 1.4.7 (Weak Trichotomy Theorem). The geometry on a strongly minimal structure M satises one and only one of the following:

• The geometry is trivial;

• The geometry of M is locally projective; • A pseudo-plane is denable in M.

Proof. The whole subject and the proof of the trichotomy theorem can be found, for the general setting of uncountably categorical structures, in Chapter II of the book by Boris Zilber [24].

We will often refer to the pseudo-plane case of the trichotomy theorem as the non-linear case, while we will say that the geometry is locally modular if is it either trivial or locally projective.

Zilber's Conjecture

As we have seen, there are three basic examples of strongly minimal structures that illustrate the three cases in the trichotomy theorem. These are:

Trivial structures: the associated geometry is trivial and the dimension of a nite X ⊂ M is the size of X;

Vector spaces over a countable division ring: the associated geometry is locally projective and the dimension of a nite X ⊂ M is the linear dimension, the dimension of the linear vector space spanned by X;

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1.4. The Weak Trichotomy Theorem and Zilber's conjecture 15 Algebraically closed elds: here a pseudo-plane can be dened and the dimension of

a nite X ⊆ M is its transcendental degree.

These considerations led Zilber to formulate the following conjecture (see [25]). Conjecture 1.4.8 (Zilber, 1983). The geometry on a strongly minimal structure M is either trivial, locally projective or isomorphic to a geometry of an algebraically closed eld.

In the latter case, M interprets an algebraically closed eld and the only structure induced by M on K is denable in the eld structure itself (purity of the eld).

Zilber's conjecture was proven false by Ehud Hrushovski in his paper [6], in which he introduced a new construction, that would become a great source of counterexamples afterwards.

However, the conjecture turned out to be true in many special classes of strongly minimal structures; for instance, in 1996 Peterzil and Starchenko considered the context of o-minimal structures, where the conjecture can still be formulated with appropriate adjustments, and proved it true (see [16]).

The main class for which conjecture1.4.8holds is that of Zariski geometries. Zariski geometries were introduced as an attempt to give light to this subject, by adding a topological component.

As we will see in more details in section4.3, any one-dimensional irreducible Zariski structure satisfying the property (AMP) of ampleness (which corresponds to the non-linear case of 1.4.7) interprets an algebraically closed eld. A deeper inspection of the geometry inherited by the eld will bring us to theorem 4.4.12, which states the purity of the eld, thus proving Zilber's Conjecture in this setting.

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Chapter 2

Zariski structures

In this chapter Zariski geometries are introduced: we will give the axioms of a Zariski structure, the main denitions and the theorems that made up the basis for this theory. We will give the main examples of Zariski structures: the most important one is that of algebraic varieties over an algebraically closed eld, but we will briey analyse also other remarkable examples such as compact complex manifolds and Zariski structures in dierentially closed elds.

2.1 Topological structures and dimension notion

We are going to introduce topological structures, the basic structures from which Zariski geometries are built.

Consider a structure M (whose domain will be called M as well), equipped with a collection C of denable subsets of the powers M, M2_{, M}3_{, . . .}_{. We will refer to the}

elements of the family C as closed sets. Complements of the closed sets in C are called open sets.

Denition 2.1.1. The structure (M, C) is a topological structure if it satises the fol-lowing axioms:

(L) Language: The n-ary relations of the language of M are the ones given by the closed subsets in C. Moreover, the family C is closed with respect to positive boolean combinations. This means that:

1. Arbitrary intersection of closed sets is closed; 2. Finite union of closed sets is closed;

3. The domain of the structure M and the empty set are closed; 4. The diagonals (i.e. the graph of equality) in Mn _{are closed;}

5. Any singleton in the domain of M is closed; 6. Cartesian products of closed sets are closed;

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7. If S ⊆ Mn _{is closed and σ : M}n _{→ M}n _{is a permutation of the coordinates,}

then σ(S) is closed;

8. If S ⊆ Mn+l _{is closed and pr : M}n+l _{→ M}n _{is the projection onto the rst n}

coordinates, then for all a ∈ pr(S) the subset pr−1_{(a) ∩ S ⊆ M}n+l _{is closed.}

Consider the projection map pr : Mn+l _{→ M}n _{and a closed relation S = S(x, y) ⊆}

Mn+l, the following notation will be often used, for a ∈ Mn: S(a, Ml) = {y ∈ Ml| M S(a, y)} .

Notice that the sets S ∩ pr−1_(a) _{and S(a, M}l₎ _{are homeomorphic as topological spaces;}

this is why sometimes they will be identied with each other. Moreover, we will use both notation a ∈ S and S(a) to mean that a ∈ Mn _{satises the closed relation S.}

Topological spaces are obvious examples of topological structures.

Remark 2.1.2. Projection maps pr : Mn+l_{→ M}n_{are continuous maps, that is, the inverse}

image of a closed subset S ⊆ Mn _{is a closed subset (equal to S × M}l_{) of M}n+l_.

A subset S ⊆ Mn_{is called locally closed if it can be written as intersection of a closed}

subset of Mn _{and an open one. We will also write S}

1 ⊆cl S ⊆ Mn, meaning that S1 is

relatively closed in the denable subset S, i.e. S1 = S ∩ C, C being a closed subset of

Mn.

A subset S ⊆ Mn_{is a constructible set if it is a boolean combination of closed subsets}

of Mn _{or, equivalently, if it is a nite union of locally closed subsets of M}n_{. A subset}

S ⊆ Mn is called projective if it is a nite union of projections of locally closed subsets in some products Mn+ki.

A denable set S ⊆ Mn_{is irreducible if whenever S}

1, S2 are relatively closed subsets

of S and S = S1∪ S2 holds, then either S = S1 or S = S2.

Denition 2.1.3. Let M be a topological structure; M is called complete if the following property of projection maps holds:

(P) Properness: if pr : Mn+l _{→ M}n _{is a projection and S ⊆ M}n+l _{is closed, then}

pr(S) ⊆ Mn _{is closed.}

The structure M is quasi-compact (or just compact) if it is complete and satises: (QC) For any family {Ci| i ∈ I} of closed subsets of Mn with the nite intersection

property, Ti∈ICi 6= ∅.

Equivalently, M is quasi-compact if and only if every open cover of Mn _{admits a nite}

sub-cover.

The topological structure M is said to be Noetherian if the following holds:

(DCC) Descending chain condition: if S1 ⊇ S2 ⊇ · · · ⊇ Si ⊇ . . . is a descending

chain of closed subsets of Mn_{, then there exists i such that S}

j = Si for all j ≥ i.

Clearly, the notions just dened can be referred to closed subsets of Mn _{as well,}

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2.1. Topological structures and dimension notion 19 Remark 2.1.4. Noetherianity of the topological structure M implies that for any closed subset S ⊆ Mn_{, there exists a nite decomposition}

S = S1∪ S2∪ · · · ∪ Sk

where Siare distinct closed irreducible sets, called the irreducible components of S. This

decomposition is unique up to a permutation of the Si.

Generally speaking, a topological structure M will verify all properties of a topological space that descends from the axioms we just nished to enumerate.

To develop the theory of Zariski structures, a notion of dimension on non-empty projective sets of a topological structure M is needed. We are now going to dene a good dimension notion on such sets; the axioms will remind us very closely some properties of Krull dimension in the case of the classical Zariski topology.

Denition 2.1.5. Suppose that each projective S ⊆ Mn _{has been associated with a}

non-negative integer dim S, the dimension of S. This is a good notion of dimension if it satises:

(DP) Dimension of a point: the dimension of any singleton is 0; (DU) Dimension of unions: dim(S1∪ S2) = max{dim(S1), dim(S2)};

(SI) Strong irreducibility: for any irreducible locally closed S ⊆ Mn _{and S}

1 (cl S,

then dim S1 < dim S;

(AF) Addition formula: for any irreducible locally closed S ⊆ Mn _{and projection}

map pr : Mn_{→ M}k_,

dim S = dim pr(S) + min

a∈pr(S)dim(pr

−1_{(a) ∩ S)}

holds;

(FC) Fibre condition: for any irreducible locally closed S ⊆ Mn_{and projection map}

pr : Mn→ Mk_{, there exists V ⊆}

oppr(S)relatively open in pr(S) such that

min

a∈pr(S)dim(pr

−1_{(a) ∩ S) = dim(pr}−1_{(v) ∩ S)}

for any v ∈ V .

We say that this is a good dimension notion for closed sets if properties (SI),(AF) and

(FC)hold for closed S ⊆ Mn_.

Remark 2.1.6. In Noetherian structures, the conditions (AF) and (FC) hold for closed subsets S ⊆ Mn _{even if we do not require S to be irreducible. Moreover, in Noetherian}

structures, for any locally closed (not necessarily irreducible) S ⊂ Mn _{with at least an}

irreducible component S0⊆ S of the same dimension, the inequality

dim S ≥ dim pr(S) + min

a∈pr(S)dim(pr

−1_{(a) ∩ S)}

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In the following, when S is a constructible set, we will denote by ¯S the closure of S, that is the closed set dened by

¯

S =\{S0| S0 is a closed set, S0 ⊇ S}. Let us dene a weakening of the property (P)of projection maps.

(SP) Semi-properness: for any irreducible closed S ⊆ Mn_{and projection map pr : M}n_→

Mk, there exists a proper closed subset F ⊂ pr(S) such that pr(S) \ F ⊆ pr(S). Denition 2.1.7. A Noetherian topological structure M with a good dimension notion satisfying(SP)is called Zariski structure.

The following lemma illustrates some properties of dimension in Zariski structures; the proofs are applications of the denitions or given by a simple reasoning by induction on the dimension of the considered closed sets.

Lemma 2.1.8. The good dimension notion on the closed sets of a Zariski structure M satises:

1. dim S > 0 for any innite closed set S; 2. dim Mn_{= n · dim M}_;

3. dim S ≤ dim Mn _{for every irreducible constructible S ⊆ M}n_.

Lemma 2.1.9. Let M be a compact Zariski structure. Consider a closed subset S ⊆ Mn+l _{and the projection map pr : M}n+l _{→ M}n _{such that pr(S) ⊆ M}n _{is irreducible and}

all the bres pr−1_{(a) ∩ S}_{, a ∈ pr(S), are irreducible. Then S is irreducible.}

Familiar examples of Zariski structures are algebraic varieties over an algebraically closed eld and compact complex manifolds; we will look more closely to algebraic vari-eties as Zariski structures in section 2.5. Other relevant examples of Zariski structures are born from dierentially closed elds of characteristic zero; these have been of high importance since they were used by Hrushovski to give a counterexample to the Zilber's Conjecture, conjecture 1.4.8 (see [6]) and to prove the Manin-Mumford Conjecture (see for instance the survey in [3]). We will deal with these types of Zariski structures in section2.6.

Example 2.1.10. Consider the structure (N, <); it follows from basic model theory that any formula in this structure with free variables x1, . . . , xn is equivalent to a boolean

combination of the formulas xi ≤ xj and the distance between xi and xj is less than

n. Dene closed subsets of Nk _{to be the subsets satisfying nite conjunctions of the}

formulas mentioned above; the dimension notion is built naturally from the request that dim Nk= k. It can be veried that this is a Zariski structure.

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2.2. Zariski geometries 21 Example 2.1.11. Consider an algebraically closed eld k, whose powers k, k2_{, . . .} _are

endowed with the classical Zariski topology, namely, closed sets of kn _{are given by}

zero-sets of polynomials in n variables. The notion of dimension attached to irreducible closed subsets is Krull dimension; the dimension of a (not necessarily irreducible) closed set is the maximum of the dimensions of its irreducible components, and the dimension of a constructible set is simply the dimension of its closure.

This obviously makes k and kn _{Zariski structures, denoted by A}1_(k) _{and A}n_(k)

re-spectively. It also follows from classical facts about Zariski topology that An_(k)×Am_{(k) =}

An+m(k) (where the product on the left is endowed with the topology of the bered product).

The same construction can be done on the projective spaces P1_(k) _{and P}n_(k)_{; we}

briey recall how Zariski topology works in his case. Closed Zariski-sets of Pn_(k) _are

given by zero-sets of homogeneous polynomials in n+1 variables. Closed sets of (Pn_(k))m

are obtained by means of successive applications of the Segre embedding Pl(k) × Pm(k) → Plm+l+m(k).

This is to remark that in the case of projective space the Zariski topology on the product Pn(k) × Pm(k) diers from the Zariski topology on Pn+m(k). Again, this constructions provides examples of Zariski structures.

Remark 2.1.12. In example 2.1.11, it is necessary that the eld is algebraically closed. For instance, the eld of real numbers R with the Zariski topology and Krull dimension is not a Zariski structure.

Indeed, consider the projection pr : R3 _{→ R given by (x, y, z) 7→ z and the}

Zariski-closed set S = V(x2_{+ y}2_{− z) ⊂ R}3_{. Both axioms (AF ) and (F C) fail in this case, for}

there is only a point in pr(S) (the origin of R) whose inverse image has dimension 0.

2.2 Zariski geometries

We will now state other important properties that will permit to develop a richer theory and that are satised by the most relevant examples of such structures. Let M be a Zariski structure.

(EU) Essential uncountability: if a closed set S ⊆ Mn_{is a union of countably many}

closed subsets {Si}i∈N, then there exist a nite set I ⊂ N such that S = Si∈ISi.

(PS) Presmoothness: M is presmooth if for any (non-empty) closed irreducible S1, S2 ⊆

Mn and any irreducible component S0 of S1∩ S1, the following relation between

the dimensions of the considered sets holds:

dim S0 ≥ dim S1+ dim S2− dim Mn.

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Denition 2.2.1. A constructible set A ⊆ Mn _{is presmooth with M (or simply}

pres-mooth, if it is clear which Zariski structure we are working with) if for any relatively closed irreducible S1, S2 ⊆ Ak× Mm and any irreducible component S0 of S1∩ S2, the

following holds:

dim S0≥ dim S1+ dim S2− dim(Ak× Mm).

We also take in consideration a strengthening of (PS):

(sPS) Strong presmoothness: M is strongly presmooth if any constructible irreducible A ⊆ Mn admits a relatively open presmooth subset A0⊆opA.

Denition 2.2.2. A Zariski geometry is a Zariski structure which satises the properties

(sPS)and (EU).

Using universal specializations (section 2.4) and a notion of local functions, we can get a nice result for one-dimensional Zariski structures.

Theorem 2.2.3. Any one-dimensional uncountable Zariski structure M satisfying(PS)

is a Zariski geometry.

Proof. The proof of this theorem in carried out in [26, Subsection 3.6.4].

Most of the time, we will deal with one-dimensional presmooth Zariski structures satisfying(EU); theorem2.2.3 tells us that in this case Zariski structure are more easier to work with, indeed (PS) alone assures us that the additional properties of Zariski geometries are satised.

2.3 Model theory of Zariski structures

We want to describe some model theoretic properties of Zariski structures. We will nd out that Zariski structures have quantier elimination and that under the hypothesis that the property(EU) holds, they are structures with nite Morley rank.

To begin with, we need a few preliminary lemmas on projection maps and construct-ible sets. In the following, M will always be a Zariski structure.

Lemma 2.3.1. For any closed subset S ⊆ Mn _{and projection map pr : M}n _{→ M}m_,

pr(S)is a constructible set.

Proof. We proceed inductively on dim S. In the case dim S = 0, it is clear then that S is made by a nite number of points, hence so will be its projection pr(S), which will result to be closed, and in particular constructible.

Suppose now that dim S > 0. Without loss of generality, we can assume that S is irreducible. Thus also pr(S) and pr(S) are irreducible. By (SP), there exists a proper closed subset F ( pr(S) such that the open subset U = pr(S) \ F is included in pr(S). Because constructible sets are closed under boolean operations, pr(S) is constructible if

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2.3. Model theory of Zariski structures 23 and only if pr(S)\U ⊆ F is constructible. The axioms of a good dimension notion imply that:

dim(S) ≥ dim pr(S) = dim pr(S) > dim F ≥ dim(pr(S) \ U ).

Hence we can apply induction hypothesis on pr(S) \ U and conclude that this is a con-structible subset of Mm_.

In the classical algebraic contest, when dealing with projection maps, a theorem of "semicontinuity of the bres" can be proven. Following these steps, a similar result can be achieved with Zariski structures.

Denition 2.3.2. Let M be a Zariski structure, S ⊆ Mn _{a closed irreducible subset}

and pr : Mn_{→ M}m _{a projection maps. Dene, for k ∈ N, the set}

P(S, k) = {a ∈ pr(S) | dim(S ∩ pr−1(a)) > k},

that is, the subset of pr(S) whose elements have bre of dimension greater than k. Lemma 2.3.3. For any closed irreducible subset S ⊆ Mn_{, any projection map pr : M}n_→

Mm and k ∈ N, the set P(S, k) is a constructible set. If k ≥ k0 = min

a∈pr(S)dim(pr −1

(a) ∩ S),

then there exists a proper relatively closed subset F ⊂cl pr(S)such that P(S, k) ⊆ F .

Proof. If k < k0, then P(S, k) = pr(S) and it is constructible by lemma2.3.1.

Suppose k ≥ k0. Apply an inductive reasoning on dim S. If dim S = 0, then the

result follows immediately, pr(S) being made by a nite number of points.

Consider then the case dim S > 0. By the bre condition(FC), there exists an open subset U ⊆ pr(S) such that for each u ∈ U, the bre dim(S ∩ pr−1_{(u)) = k}

0. Because

pr(S)is irreducible and constructible (lemma2.3.1), dim U = dim pr(S). Consider the set S0 ⊆opS given by

S0 =

[

u∈U

S ∩ pr−1(u)) = pr−1(U ) ∩ S;

it follows from the addition formula (AF) that dim S0 = dim S. The complement S1 =

S \ S0 is a proper relatively closed subset of S and by irreducibility of S, dim S1< dim S.

As already noticed in remark2.1.6, the bre condition(FC)is valid for any closed subset in a Zariski structure; this implies that P(S, k) = P(S1, k). By induction, P(S, k) is a

constructible set. Moreover,

P(S, k) = P(S1, k) ⊆ pr(S) \ U ⊂cl pr(S)

is included in a proper relatively closed subset of pr(S).

Zilber's theory of Zariski structures

Università di Pisa

Tesi di Laurea

Zilber's theory of

Zariski structures

22 settembre 2017

Contents

Introduction

Chapter 1

Model theoretic preliminaries

1.1 Basic model theoretic notions

1.2 Stability theory and Morley rank

1.3 Geometry of strongly minimal sets

1.4 The Weak Trichotomy Theorem and Zilber's conjecture

Chapter 2

Zariski structures

2.1 Topological structures and dimension notion

2.2 Zariski geometries

2.3 Model theory of Zariski structures