Incidence systems on Cartesian powers of algebraic curves

We show that a reduct of the Zariski structure of an algebraic curve which is not locally modular interprets a field, answering a question of Zilber's.


Introduction
In [1, §2] Artin describes the basic problem of classifying abstract plane geometries (viewed as incidence systems of points and lines) as follows "Given a plane geometry [...] assume that certain axioms of geometric nature are true [...] is it possible to find a field k such that the points of our geometry can be described by coordinates from k and lines by linear equations?". Zilber's trichotomy principle (to be described in more detail in the next section) can be viewed as an abstraction of the above problem, replacing the "axioms of geometric nature" with a well behaved theory of dimension (see, e.g., [33, §1]).
Conjectured in various forms by Zilber throughout the late 1970s, essentially every aspect of Zilber's trichotomy, in its full generality, was refuted by Hrushovski [14], [13] in the late 1980s. Due to Hrushovski's cornucopia of counterexamples the conjecture has never been reformulated. Yet, Zilber's principle remains a central and powerful theme in model theory: it has been proved to hold in many natural examples such as differentially closed fields of characteristic 0, algebraically closed fields with a generic automorphism, o-minimal theories and more (see [5,32,30,6,19]). Many of these special cases of Zilber's trichotomy had striking applications in algebra and geometry (see [15,16,35]). More recently, in [40], Zilber outlines a model theoretic framework for studying far reaching extensions of the Mordell-Lang conjecture. One of the key features of Zilber's strategy is the trichotomy theorem for Zariski Geometries [19].
The key to the classification of Desarguesian plane geometries (the fundamental theorem of projective geometry) is the reconstruction of the underlying field k as the ring of direction preserving endomorphisms of the group of translations. The reconstruction of a field out of abstract geometric data is also the essence of Zilber's trichotomy and is the engine in many of its applications. A relatively recent application of one such result is Zilber's model theoretic proof [39] of a significant strengthening of a theorem of Bogomolov, Korotiaev, and Tschinkel [3]. The model theoretic heart of Zilber's proof is Rabinovich's trichotomy theorem for reducts of algebraically closed fields [34]. In the concluding paragraph of the introduction to [39] Zilber writes: "It is therefore natural to aim for a new proof of Rabinovich' theorem, or even a full proof of the Restricted Trichotomy along the lines of the classification theorem of Hrushovski and Zilber [19], or by other modern methods [...]. This is a challenge for the model-theoretic community." The conjecture referred to in Zilber's text above can be formulated as follows 3,4 : Conjecture A. Let M be a strongly minimal reduct of the full Zariski structure on an algebraic curve M over an algebraically closed field K which is not locally modular. Then there exist M-definable L, E such that E ⊆ L × L is an equivalence relation with finite classes and L/E with the M-induced structure is a field Kdefinably isomorphic to K.
Rabinovich [34] proved Conjecture A in the special case where M = A 1 , and her result can be extended by general principles to any rational curve. In the present paper we prove Conjecture A. Our approach to the problem follows a, by now, well known strategy introduced by Zilber and Rabinovich and owes to [24]. Using a standard model theoretic technique, Hrushovski's field configuration (see Section 4.1 for details), the problem is reduced to showing that tangency is (up to a finite correction) reduct-definable in families.
To achieve this goal, we proceed in two steps. In the first step (carried out in Section 3) we study slopes of families of branches (at a given point) and their behaviour under composition of curves and, in case the ambient structure is an expansion of a group, under point-wise addition. This culminates in Proposition 3.15, which is the key to the definability of tangency, and in Lemma 4.19, providing us with the (algebraic) group which is the template allowing us to construct the group configuration.
Section 4, where Conjecture A is proved (Theorem 4.33), is dedicated, mainly, to verifying that the assumptions of the technical result of the previous section can be met in the reduct. In Section 4.3 we show that our definition of slope is meaningful in reduct-definable families of curves (in positive characteristic). In Section 4.4, where the main step towards proving Theorem 4.33 is carried out, the key difficulty to overcome is in the application of Proposition 3.15.
As already mentioned, the general scheme of our proof seems to have much in common with Rabinovich's original work, though we were unable to understand significant parts of her argument, which are highly technical. For that reason we cannot pinpoint the reason for the present work being more general, considerably shorter, and technically simpler.
Finally, it should be mentioned that the tools developed in the present paper seem to extend naturally to various other contexts. For example, one can envisage extending the results of [21] to positive characteristic, and any algebraic group and -possibly -even a full proof of the restricted trichotomy conjecture for structures definable in ACVF (at least modulo the problem of showing that the 1-dimensional group reconstructed by our methods embeds in an algebraic group: it can probably be shown that the group will always be isomorphic to either G a or to G m ).

Model-theoretic background
For readers unfamiliar with model theory we give a self contained exposition of Conjecture A. In order to keep this introduction as short as possible, we specialise our definitions to the setting in which they will be applied. We refer interested readers to [37, §1.1-2] for a more detailed discussion of structures and definable sets. Readers familiar with the basics of model theory are advised to skip the remainder of the present section.
Given an algebaraic curve M over an algebraically closed field k (reduced, but not necessarily irreducible, smooth or projective), the full Zariski structure on M , denoted M, is the set of k-rational points, M (k) equipped with the collection of all Boolean algebras of constructible sets on the Cartesian powers M n (k). The full Zariski structure on a curve M is an example of the model theoretic notion of a structure.
A first-order structure or simply a structure N is a non-empty set N (called the universe of N ) equipped with a collection, Def(N ), of Boolean algebras Def l (N ) ⊆ P(N l ) for all l > 0, such that Def l (N ) contains all diagonals ∆ l i,j := {(x 1 , . . . x l ) : x i = x j }, and such that Def(N ) is closed under finite cartesian products and projections of the form (x 1 , . . . , x n ) → (x 1 , . . . x n−1 ). Somewhat analogously to geometric terminology the tuples (x 1 , . . . , x n ) ∈ S ⊂ M l are called points of the definable set S. If A ⊆ N is any set, a subset X ⊆ N l is definable with parameters in A (or A-definable) if there exists a definable set Y ⊆ N n+m (some m ≥ 0) such that Y = Y a := {x ∈ N l : (x, a) ∈ Y } for some a ⊆ A.
Note that by Chevalley's theorem (see, e.g., [23,Corollary 3.2.8]), over an algebraically closed field k, the collection of constructible sets on cartesian powers of an algebraic curve M is closed under projections, and therefore the full Zariski structure, M, on M is, indeed, a structure in the above sense. It is a well known fact (e.g., [19]) that the field k can be reconstructed from M. Let us now explain more precisely what is meant by that.
A (partial) function f : N l → N is definable if its graph is. Thus, for example, we say that a group is definable in N , if there exists a definable set G ⊆ N l and a definable function p : G × G → G such that (G, p) is a group (note that the function x → x −1 is automatically definable if (G, p) is a group). The definability of a field in a structure N is defined analogously. It is not hard to check (and follows from the main result of [19]) that if M is the full Zariski structure on an algebraic curve M over an algebraically closed field k then a field F is definable in M (and F is isomorphic, definably in the standard field structure on k, to k).
But we need a somewhat subtler notion than definability. Consider, as a simple example, the structure C with universe C × {0, 1}, and whose definable sets are all those of the form {((x 1 , i 1 ), . . . , (x n , i n ) : (x 1 , . . . , x n ) ∈ D} where D is a constructible subset of C n and i j ∈ {0, 1} for all 1 ≤ j ≤ n. It is easy to verify that all functions definable in C are locally constant, and therefore there is no definable field in C. Consider, however, the equivalence relation x ∼ y (in C) defined by y ∈ (1, 0)·x (recalling the interpretation of multiplication in C, this is a C-definable way of saying that x and y have the same first coordinate). Then ∼ is a C-definable equivalence relation, and C/ ∼ is naturally isomorphic to the full Zariski structure on C.
In model-theoretic terms the structure C in the previous example interprets a field definably isomorphic to C. In general, if N is a structure, E a definable equivalence relation on N l and π : N l → N l /E is the natural projection, the induced structure on N l /E is the push-forward of the Boolean algebras on powers of N l via π. We say that N interprets a field if a field is definable in the structure induced on N l /E for some l and N -definable equivalence relation E on N l .
In the above example the universe C × {0, 1} of C is definable in the full Zariski structure on C, and every definable set in C is definable in C. But, as we have seen, C is not the full Zariski structure on C. The structure C is an example of a reduct of the full Zariski structure on C. Generally, if M is a structure whose universe is an algebraic curve M and every M-definable set is M-definable then M is a reduct of M.
Zilber's conjecture is concerned with the question of interpreting a field in a reduct, M, of the full Zariski structure, M, on an algebraic curve M . Assume that an infinite field F is interpretable in M. Then by [23, Theorem 3.2.20] the universe F of F can be identified with a constructible subset of k l for some l, and by [33,Theorem 4.15] k is definably isomorphic to F. Thus there is a definable finite-tofinite correspondence Ψ ⊆ F × M . It is easy to check that Ψ can be taken to be M-interpretable (e.g., if F is definable in M then Ψ can be taken to be the graph of a projection function, the general case is slightly more delicate and we skip the details). If we push the family of affine lines in F 2 via Ψ we obtain a 1-dimensional constructible subset U of M such that for any p, q ∈ U there is a curve C := Ψ(L) -for L an affine line in F 2 -with p, q ∈ C. We have thus verified that for M to interpret a field it is necessary that there exists a 2-dimensional constructible U ⊆ M 2 and a definable set X ⊆ M 2+l such that X t := {(x, y) : (x, y, t) ∈ X} is 1-dimensional (or empty) for all t ∈ M l and such that for all p, q ∈ U there exists t ∈ M l such that p, q ∈ X t . The main result of the present work, Theorem 4.33, states that this condition is, in fact sufficient.
In model-theoretic terms the existence of an ample family as above is equivalent, [23,Lemma 8.1.13], to non local modularity of the structure M.
If X is an ample family in M 2 we denote by (M, X) the smallest reduct of M containing X. We can thus formulate Conjecture A: Conjecture B (Zilber's restricted trichotomy in dimension 1). Let M be an algebraic curve over an algebraically closed field k. Let X ⊆ M 2 × T be the total space of an ample family in M 2 , then a field K is interpretable in M = (M, X).
In [1, §2.4] not only is the field recovered from the affine geometry, but also the geometry is recovered as the affine plane over that field. In the present setting, there are examples due to Hrushovski (see, e.g., [25]) showing that the full Zariski structure of the curve M cannot be recovered from M. This can probably be achieved if X is very ample in the sense of [19] (namely, if the set X in Definition 2.1 the separates points in M 2 ), but we do not study this question here.

Tangency
The reconstruction of the field is obtained in two steps. First, we reconstruct a 1-dimensional algebraic group, and then -using the group structure to sharpen the same arguments -we reconstruct the field. Roughly, the reconstruction of a group is obtained in three stages: first we identify a reduct definable family X → T of algebraic curves whose associated family of slopes at some point P = (a, a) ∈ M 2 is a 1-dimensional algebraic group under composition. The second, and most crucial part of the proof is commonly dubbed definability of tangency. In its cleanest form this consists in showing that, given families X → T and Y → S as above, the set of all (t, s) ∈ T × S such that X t is tangent (in an appropriate sense) to Y s at P is M-definable. Finally, the group is reconstructed by invoking the group configuration theorem, a well known model theoretic technique (to be described in more detail in the next section), using the results of the previous stages. In the next two subsections we take care of the two first stages of the this strategy.
Before starting with our set up on the technical level let us discuss some of the challenges that motivated the definitions to be shortly presented. In the implementation of the strategy outlined above two difficulties arise.
Firstly, if we consider only the first-order slopes, then due to inseparability issues in positive characteristic it becomes hard to find a 1-dimensional family of curves definable in the reduct such that its associated slopes at some point range in a 1dimensional set -such a family is needed to construct the first group configuration (Section 4.4). The solution is to consider tangency information up to any order n and pick the order so that there are enough slopes. Interestingly -and this was apparent already in [24] -in the presence of a group structure, the problem doesn't arise, which is a good coincidence, since the second group configuration (Section 4.5) has to be built using the first-order tangency information.
Secondly, we can't work only with smooth points to define the slope, since the operations of composition and point-wise addition that are used in the construction of the group configurations do not preserve smoothness. Our approach to this is to track a particular branch of a curve at a particular point as the operations of composition and point-wise addition are applied: one can then have control over the slope of a branch, appropriately defined. Note that we use the term branch (Definition 3.3) in a more restrictive sense than what is usually understood by it: in a way our branches are 'always smooth' (or more precisely 'alwaysétale over the first factor M of M 2 '), so that the notion of a slope always makes sense for them. For any curve in M 2 the projection either on the first or on the second factor M is going to be genericallyétale (Lemma 4.14), even in positive characteristic, and so there is going to be a unique branch at any general enough point on this curve (Lemma 3.4). This statement generalizes appropriately to families of curves too. By virtue of Propositions 3.7 and 3.9 the slopes of relevant branches can be tracked as the curves are composed and point-wise added. All curves and branches that we work with in Section 4 are obtained this way.

Slopes and operations on correspondences
First of all, our main objects of interest are definable subsets in a reduct of the full Zariski structure on a fixed curve M , we will adopt the following non-standard terminology: we will call a constructible subset Z ⊆ M n (for some n) of dimension 1 a curve even if it is reducible and if it has connected components of dimension 0. If a curve Z does not contain connected components of dimension 0 then we call Z a pure-dimensional curve. Clearly, every curve contains a maximal pure-dimensional curve. In the few situations when we refer to abstract algebraic curves (that is, algebraic varieties of pure dimension 1 over a fixed algebraically closed field) we will use the term algebraic curve. We will not distinguish notationally between subsets of M n definable in a reduct of the full Zariski structure on M and constructible subsets of the varieties (or even schemes) M n , and in particular between definable curves and their algebro-geometric counterparts.
Recall that any algebraic variety over a perfect field admits a dense Zariski open subset that is smooth (see, e.g., Corollary to Theorem 30.5 of [27]). Let Z ⊂ M 2 be a pure-dimensional curve, and a = (a 1 , a 2 ) ∈ Z be a smooth point of M 2 . Since the completion of the local ring of a smooth point of a variety is a formal power series ring [27,Theorem 29.7], we can pick some isomorphisms We call the truncation to the n-th order of the series g α the n-th order slope of a branch α of Z. Naturally, the slope of a branch of a pure-dimensional curve depends on the choice of the isomorphism O M 2 ,a ∼ = k[[x, y]], but this choice does not affect any of the properties of slopes we will be interested in. We view curves in M 2 as finite-to-finite correspondences between the two factors M . The purpose of the present section is to study the behaviour of slopes of branches with respect to two natural operations on correspondences: composition and "pointwise addition" (see Definition 3.8) when M has a structure of an algebraic group. We will show that if Z, W are two curves and α, β are two branches of Z, W at a = (a 1 , a 2 ), b = (b 1 , b 2 ) ∈ M 2 respectively, and a 2 = b 1 then the composition W • Z has a branch β • α at (a 1 , b 2 ) such that its slope is the composition of the n-th order slopes of α and β (as truncated polynomials) whenever the latter are defined. A similar statement can be made about the slopes of branches of curves that are 'point-wise added' if M has a structure of a group. Later we will construct a group configuration starting from a family of curves definable in a reduct of a full Zariski structure on M such that the set of its n-th order slopes at a given point coincides, up to a finite set, with a 1-dimensional algebraic subgroup of Aut(k[[x]]/(x n+1 )) (a truncated polynomial f corresponds naturally to the automorphism of k[[x]]/(x n+1 ) sending x to f ).
Since we will have to work with families of curves, we will also introduce notions of families of branches and slopes. When the characteristic of the base field is positive, we will often have to work with curves and families of curves in M × M (p n ) where M (p n ) is the pull-back of M by the Frobenius endomorphism on k (see Section 4.3). That is why we do not assume that factors of the ambient product variety are isomorphic in the definitions below.
Definition 3.1 (Families of curves). If X 1 , X 2 are two algebraic curves then by a family of pure-dimensional curves in X 1 × X 2 we will understand a finite union Z of pure codimension 1 locally closed subsets Z i ⊂ X 1 × X 2 × T , where T is a variety, such that Z t is a pure-dimensional curve for all t ∈ T . By a family of curves we understand a constructible subset Z ⊂ X 1 × X 2 × T where T is a constructible subset of a variety and such that Z t is a curve for all t ∈ T . If X 1 = X 2 = M and T ⊂ M l for some l and Z is definable in a reduct of the full Zariski structure on M we call it a definable family of curves.
While families of curves arise naturally in definable context, in order to apply the machinery of slopes we need to work with families of pure-dimensional curves. As long as T is a variety, a family of curves Z ⊂ X 1 × X 2 × T contains a unique maximal family of pure-dimensional curves. The total space Z of a family of puredimensional curves is not necessarily a variety; while this is a desirable property that will be important in Subsection 3.2, we do not include it in the definition so that it can be readily seen that the operations of composition and point-wise addition preserve the class of families of pure-dimensional curves. However, one can easily ensure that the total space is a variety at the cost of shrinking the parameter space.
Lemma 3.2. Let T be a constructible subset of a variety, W ⊂ X 1 × X 2 × T be a family of curves. Then there exists a Zariski dense subset T ′ ⊂ T which is a variety and a maximal locally closed W ′ ⊂ W × T T ′ which is a family of pure-dimensional curves. In particular, W ′ is a variety.
Proof. It is easily checked using Noetherian induction that any constructible set contains a Zariski dense locally closed subset. In particular, there exists a dense locally closed subset T 0 ⊂ T which is therefore a variety. Without loss of generality we may assume T 0 connected. Then W × T T 0 is a union of locally closed sets of the form W i \ Z i , i ∈ I where W i and Z i ⊂ W i are Zariski closed and distinct, and the index set I is finite. Let I ′ ⊂ I be the set of those indices i for which W i is codimension 1 in M 2 × T . Further shrinking T 0 we may assume that dim(W i ) t = 1, dim(Z i ) t = 0 for all t ∈ T 0 and all i ∈ I ′ (in particular, Z i = ∅). We put is locally closed, the statement of the Lemma then follows by induction on the size of I ′ .
We have It follows from an easy dimension computation that If T ′ is a dense open set in the complement of the projections then Given a family of pure-dimensional curves Z as above, we would like to be able to pick branches of the curves Z t depending algebraically on the parameter t ∈ T . In this case the local equation of Z in a formal neighbourhood of {a} × T may only exist locally on T , and in order to capture this idea we have to phrase the definition in terms of formal schemes (we refer the reader to [11,II.9] or any other standard algebraic geometry reference for the definition of formal schemes).

Definition 3.3 (Branches and families of branches).
Let Z ⊂ V := X 1 × X 2 × T be a family of pure-dimensional curves, a ∈ M 2 and assume that a ∈ Z t for all t ∈ T . LetX 1 be the formal completion of X 1 × T along {a 1 } × T , and letẐ be the formal completion of Z along {a} × T . A family of branches of Z at a is a closed formal subschemeẐ α such that the natural projectionẐ α →X 1 is an isomorphism. We will call local generators of the ideal sheaf that definesẐ α local equations of α. When Z ⊂ X 1 × X 2 is a single curve, we regard it as a family parametrized by a single point, and we call families of branches of Z just branches. Given a family of branches α we will denote α t the branch given by the fibresẐ αt for all t ∈ T .
Remark. If Z ⊂ X 1 × X 2 × T is a family of curves and T is a variety then in order to simplify the exposition we will refer to branches of Z meaning branches of a family of pure-dimensional curves Z 0 ⊂ Z.
Let X i be algebraic curves, and let a = (a 1 , . . . , a n ) ∈ X = X 1 × . . . × X n be a smooth point. We say that a local coordinate system at a is picked when an is picked for each a i ; in this case we understand that there exists an isomorphism O X,a ∼ = k[[x 1 , . . . , x n ]] induced by these isomorphisms. If local coordinate systems are picked at a = (a 1 , a 2 ) ∈ X 1 × X 2 , b = (b 1 , b 2 ) ∈ X 2 × X 3 , we understand without explicit mention that local coordinate systems are automatically picked at the points (a 1 , b 2 ) ∈ X 1 × X 3 , (b 2 , a 1 ) ∈ X 3 × X 1 which will be of interest to us later on. Similarly, if X 1 has a group structure and a local coordinate system is picked at a point a ∈ X 1 then we assume it picked at any point a ′ ∈ X 1 via translation. The next lemma gives a sufficient condition for the existence of a family of branches at a point.
Recall that a morphism of schemes f : X → Y is called quasi-finite if the fibres f −1 (y) are finite for all y ∈ Y . A quasi-finite morphism f : X → Y of locally Noetherian schemes is unramified if Ω X/Y = 0 (see [22, Ch. 6, Corollary 2.3]) where Ω X/Y is the module of Kähler differentials of the morphism f . A morphism f locally of finite type is calledétale if it is flat and unramified. Basic properties of these notions will be recalled in detail and with references in Section 3.2.
T is a family of pure-dimensional curves and the projection Z → X 1 isétale in a neighbourhood of {a} × T for some a ∈ X 1 × X 2 , then there exists a unique family of branches of Z at a.

Proof.
For any affine open Spec R ⊂ X 1 × T let Spec R ′ ⊂ Z be an affine openétale over Spec R, let I, I ′ be the ideals vanishing on {a 1 } × T , {a} × T respectively, R and R ′ their respective completions. Then by [36, Tag 0ALJ] ( R, I) is a Henselian pair and by [36,Tag 09XI] there exists a unique isomorphism R ′ → R that defines the unique family of branches.
Definition 3.5 (Slope). Let X 1 , X 2 be algebraic curves, Z ⊂ V := X 1 × X 2 a puredimensional curve, a ∈ V a smooth point, a ∈ Z, and I, m a ⊂ O V,a the ideals of functions that vanish on Z, {a}, respectively. Assume that a local coordinate system is chosen at a, so that lim Note that τ n (Z, α) depends on the choice of the local coordinate system at a, and that if an n-th order slope of Z at α is defined, then the slopes of all orders of Z at α are defined.

Remark.
(i) Let f, g be local equations of branches α, β at a point a of pure-dimensional curves In particular, if Z 1 , Z 2 are smooth at a and α, β are their unique respective branches at a, then the intersection multiplicity of Z 1 and Z 2 at a (as defined in, for example, [11, ex. I.5.4]) is n.
s. Let α, β be some families of branches of X and Y at a. Then it follows from Krull's maximal ideal theorem that there exists a maximal integer n such that For the benefit of the reader we explain what data in the Definitions 3.3 and 3.5 specifies families of branches and slopes, specialising the description in the language of formal schemes to an affine situation. Take Zariski open subsets U ⊂ X 1 × X 2 , W ⊂ T such that a ∈ U . Let S, R be the rings of regular functions on U, W and let J a , J ⊂ R ⊗ S be the ideals of regular functions that vanish on {a} × W , Z ∩ U × W , respectively. We fix a local coordinate system at a which gives an isomorphism lim Note that the notion of slope is invariant under extensions of the base field. Assume that all objects in the previous paragraph are defined over k, and let k ′ ⊃ k be a field extension. Then there exists a family of branches is defined by the polynomials with the same coefficients as the function In model-theoretic terms, this observation implies that once a point a ∈ M 2 , a local coordinate system at a, and a family of branches α of Z are fixed, the slope τ n (Z t , α t ) is definable in the language of fields over t.
Further if X 1 × . . . × X n is a product of k-varieties, we denote the natural projections denote the natural projections. Although the notion of the composition of correspondences is standard, we reintroduce it here to fix conventions.
be families of curves, and let p i 1 ...i k denote projections on products of the factors of the space Define the family W • Z of compositions of curves from the families W and Z to be Clearly, if Z, W are definable then so is Z • W ; on the level of points: If for all t ∈ T, s ∈ S all irreducible components of Z t , W s project dominantly on X 1 , X 2 , respectively, then W • Z is a family of curves parametrized by T × S.
We denote by Z −1 the image of Z under the morphism X 1 ×X 2 ×T → X 2 ×X 1 ×T that permutes the factors X 1 and X 2 , in both geometric and definable contexts. We regard the above definitions as applicable to individual curves Z, W by putting T = S to be a point.
Remark. If Z, W are families of pure-dimensional curves such that for all t ∈ T, s ∈ S all irreducible components of Z t , W s project dominantly on X 1 , X 2 , respectively, then W • Z is a family of pure-dimensional curves.
such that for all t ∈ T, s ∈ S and for all n > 0 where the operation "•" in the right hand side expression is composition of truncated polynomials.
Proof. The proof consists essentially in unraveling the definitions. The choice of coordinate systems induces the isomorphisms If the family of branches α is given Zariski locally around t ∈ T by an equation , and β is given by z − g, g ∈ y O S,s [[y]], then let the family of branches β • α be given by ]. Note that the composition g • f of the formal power series makes sense and has a zero constant term, since both f and g have this property. Now let h Z , h W be generators of the kernels of the maps , and in order to show that β •α is a family of branches of Y •X at this point, we need to check that , and for that it would suffice to check that Indeed, it is straightforward to check that for any n > 0 (z − g • f ) = I n := I/(x n , z n ) and since I is the inverse limit of I n , it follows that (z − g • f ) = I. Definition 3.8 (Point-wise addition of curves). Let G be a 1-dimensional algebraic group, and let X ⊂ G 2 × T , Y ⊂ G 2 × S be families of curves. Let a : G × G → G be the group law, let Γ a ⊂ G 3 be its graph, and denote p i 1 ...i k projections of G × G × G × G × T × S on the products of factors. We define the family of curves X + Y of sums of elements of the families X and Y to be Clearly, if X, Y are definable then so is the family X + Y ; on the level of points: Remark. If X and Y are families of pure-dimensional curves then so is X + Y . It may seem from the notation above that G is supposed to be commutative, even though the definition applies even if this is not the case. In this paper we will only consider the operation "+" for curves inside groups with the commutative connected component of the identity.
Let G be a one-dimensional algebraic group, then the formal group law of G is defined as the image of the topological generator of k induced by the group operation morhpism. The truncation to first order of a one-dimensional formal group law is x + y (see, for example, [20, I.2.4]). Proposition 3.9. Let G be a one-dimensional algebraic group over an algebraically closed field k. Let F be the formal group law of G, and let F n be its n-th order truncation. Let X ⊂ G × G × T , Y ⊂ G × G × S be families of pure-dimensional curves and let α, β be families of branches at a = (a 0 , a 1 ), b = (b 0 , b 1 ), where b 0 = a 0 , respectively. Then there exists a family of branches α + β of X + Y at (a 0 , a 1 + b 1 ) such that if τ n (X t , α t ) and τ n (Y s , β s ) are defined. In particular, if n = 1, Proof. As in the proof of Proposition 3.7, this statement follows from the unfolding of the definitions. Reasoning locally, assume that α is cut out by the equation ). Checking that this is indeed a branch of X + Y is straightforward and we leave it to the reader.

Flat families and definability of tangency
In the present section we prove the main technical result of the paper, Proposition 3.15, characterizing the tangency of two generic elements of two families of curves in terms of properties of the families definable in the reduct M. While we do not give a full definable characterization of the tangency, we prove a standard weakening of this result, which, as we will see in the concluding section of the paper, is sufficient for our needs.
The key preliminary step is the observation that if X ⊂ M 2 × T , Y ⊂ M 2 × S are families of pure-dimensional curves and M, T, S are smooth, then the 'family of intersections' X × M 2 Y → T × S is flat if restricted to the open subset of T × S over which it has finite fibres. We refer the reader to any standard exposition of flatness, such as [28, I. §2], for details, and quickly recall some of the key facts. All schemes in this section are assumed Noetherian, and by varieties we mean schemes of finite type over an algebraically closed field k. We identify closed scheme-theoretic points of varieties with geometric points (that is, morphisms Spec k → X).
First recall that a morphism f : In particular, flatness can be checked Zariski locally on the source:   (ii) a composition of flat morphisms is flat; (iii) let X → Y be a flat morphism and let Z → Y be a morphism. Then X × Y Z → Z is flat; (iv) let B be a flat A-algebra and consider b ∈ B. If the image of b in B/mB is not a zero divisor for any maximal ideal m of A then B/(b) is a flat A-algebra; (vi) if A is an algebra and I ⊂ A is an ideal, then the completion lim ← − A/I n is flat over A.
Lemma 3.12. Assume that the total space X of a family of pure-dimensional curves X ⊂ M 2 × T is a variety. Then X flat over T .
Proof. By definition of a family of pure-dimensional curves X is open inX, so by Proof. By Lemma 3.12, X is flat over T . Since M, T, S are smooth and since regular local rings are unique factorization domains, and Y is pure-dimensional, Y is cut out in M 2 × S by a principal ideal sheaf (see, for example, [26, §19, Theorems 47, 48]).
(ii) assume that f is flat. Then m is lower semi-continuous and w is upper semicontinuous, that is, the lower level sets of m and the upper level sets of w where the first and the third one are tautological, and the second morphism is an isomorphism becauseĴ is the completion of J in the I-adic topology. This proves claim (i). We can now formulate our main technical result. Roughly, it states that, in suitably chosen families of curves tangency of two curves is witnessed by a lower number of intersection points: Proposition 3.15. Keep notation and assumptions of Lemma 3.13 and assume further that there exists a ∈ M 2 such that X t , Y s pass through a for all t ∈ T, s ∈ S. Let α, β be families of branches at a of X, Y respectively, such that for all t ∈ T, s ∈ S the slopes of α t , β s are defined. Define be the natural projection and let Z = n i=0 Z i be the decomposition into irreducible components where Z 0 = q −1 (a). We will first show that whenever In order to do that we will show that the function The projection p : Z → U is flat by Lemma 3.13. By [36, Tag 04PW] the closed embedding Z red → Z, where Z red is Z endowed with the canonical reduced structure, is flat. Since the invariant we are interested in does not depend on the scheme structure, by Fact 3.11(ii) we may assume Z reduced. Furthermore, there exists an open embedding Z ֒→Z whereZ is flat and finite over U . Indeed, letM be a smooth proper algebraic curve that contains M as a dense subset and letX,Ȳ be the closures of X, Y inM 2 × T andM 2 × S. LetZ =X ×M2Ȳ ∩M 2 × U , let p be the natural projection on U and denotep its restriction toẐ = ∪ i =0Zi , wherē Z i is the irreducible component ofZ that contains Z i for each i. By Lemma 3.13p is flat. By Fact 3.11(v) the morphismp is locally free. It is readily seen that (p) * OẐ is locally free of rank one less than the rank of (p) Note that while Z 0 may have non-trivial scheme-theoretic structure, the restriction p| Z 0 : (Z 0 ) red → U is a homeomorphism, so denote r : U → Z 0 its set-theoretic inverse. Let w : Z → Z, w(z) = dim k O Z,z ⊗ k(p(z)). We claim that w is constant on the open set Z ′ = Z 0 \ i =0 Z i . By Fact 3.11(i) Z ′ is flat over U and by Fact 3.11(iii) it is flat over the open p(Z ′ ) ⊂ U . The restriction p| Z ′ : Z ′ → p(Z ′ ) is still a homeomorphism; we will show that it is a finite morphism.
Note that since U is dense in T × S it is integral, and since Z ′ is dense in U , it is integral too. The scheme Z ′ is of finite type over a field, so clearly quasicompact, and p is clearly separated (for example, because it is affine), so Zariski's Main Theorem (see [28, Ch. I, Theorem 1.8]) can be applied to p. Therefore, p factors into a composition of an open embedding i : Z ′ → Z ′′ and a finite morphism is not finite then the open embedding i is an isomorphism and p ′′ is not an isomorphism. Since passing to a closed subscheme preserves finiteness, we may assume Z ′ to be dense in Z ′′ . The subset of Z ′′ red where p ′′ and the identity morphism coincide is closed and contains Z ′ , so must be the whole of Z ′′ , which in turn contradicts p ′′ not being an isomorphism. Now by Lemma 3.14(ii) w is upper semi-continuous on Z(k) and in particular on Z 0 , but since Z ′ is flat and finite over U , w is constant on Z ′ (k). Therefore, w takes the value w min,0 = min x∈Z 0 w(x) on the latter, and if w(r(u)) > w min,0 for some u ∈ U then r(u) ∈ Z i ∩ Z 0 for some i = 0 and therefore #p −1 (u) is not maximal. It follows that It is left to prove that w(r(t, s)) > w min,0 for all t, s such that τ nmax+1 (X t , α t ) = τ nmax+1 (Y s , β s ).
To establish this, it is enough to prove the statement on an affine Zariski open subset Spec R ⊂ U × M 2 intersecting Z 0 non-trivially. Let f, g ∈ R be the equations of X × S ∩ Spec R, Y × T ∩ Spec R, respectively. Let I ⊂ R be the ideal of functions that vanish on q −1 (a) and let R = lim ← − R/I n . Let f = f 1 · . . . · f N , g = g 1 · . . . · g K be decompositions into pairwise coprime factors in R and let f α and g β be those factors that are local equations of α, β. Apply the Chinese Remainder Theorem (see [2, Ch. 9, ex. 9]) twice: first, to R/(f, g) to get the decomposition Both applications are justified: since f i are pairwise coprime in R (that is, (f i ) + (f k ) = R for i = k), the ideals (f i , g) are pairwise coprime in R/(g), similarly, for each i, the ideals (f i , g j ) are pairwise coprime in R/(f i , g).
Tensoring with k(u) and applying Lemma 3.14(i) we get Therefore, if u = (t, s) ∈ U (k) and τ nmax+1 (X t , α t ) = τ nmax+1 (Y s , β s ) then by the remark after Definition 3.5 takes a value strictly greater than the minimum it achieves on U (k). By Fact 3.11(vi) R is a flat R-algebra, and by applying Fact 3.11(iv) twice, as in the proof of Lemma 3.13, Spec R/(f i , g j ) is flat over U for all i, j. Since by Lemma 3.14(ii) for each pair of prime factors f i , g j the value is upper semicontinuous in u, it follows that w(r(t, s)) > w min,0 as soon as slopes of order (n max + 1) of α t and β s coincide.

Interpretation of the field
In the preset section we tie together the results obtained above to produce the main result of the paper. We start with some additional technicalities and reductions.

The group configuration
In stable theories -a model theoretic framework encompassing all structures considered in the present work -certain combinatorial configurations of elements are known to exist only in the presence of an interpretable group or -in a more restrictive setting -an interpretable field. It is by constructing such configurations -using the "definable intersection theory" developed in the previous sections -that the main theorem of the present paper is proved. Before describing these configurations in more detail we need some model theoretic preliminaries. As in Section 2, we will specialise the definitions to the setting in which they will be used. As above, we will be working in the full Zariski structure M on an algebraic curve M over an algebraically closed field k. We will be mostly concerned with a structure M := (M, X) where X ⊆ M 2 × T ⊆ M 2+l is the total space of an ample family. Throughout the text by definable we mean 'definable with parameters'.  Remark. An M-definable set D may be strongly minimal with respect to the structure M but not with respect to the strucutre M.
As we will see below, we can easily reduce the proof of our main result to the case where M is strongly minimal. Under this additional assumption we can finally introduce the group configuration: is a group configuration if there exists an integer n such that -all elements of the diagram are pairwise independent and dim(a, b, c, x, y, z) = 2n + 1; -all triples of tuples lying on the same line are dependent, and moreover, dim(a, b, c) = 2n, dim(a, x, y) = dim(b, z, y) = dim(c, x, z) = n + 1; Two group configurations G 1 , G 2 are called inter-algebraic if for any pair of tuples a ∈ G 1 , a ′ ∈ G 2 in the corresponding vertices, acl M (a) = acl M (a ′ ).
Assume that G is a connected M-definable group acting transitively on a strongly minimal definable set X, then one can construct a group configuration as follows: let g, h be independent generics in G and let u be a generic of X (we will justify the assumption that such generics exist later on), then ( g, h, g · h, u, g · u, g · h · u ) is a group configuration (associated with the action of G on X). Below (Lemma 4.19) we show that, for a suitably constructed M-definable family of curves passing through a fixed point, the set of n-th slopes of curves in the family coincide for some n with a one-dimensional algebraic group, H (viewed as acting on itself by multiplication). Proposition 3.15 will then allow us to 'pull back' a group configuration (in M) associated with this group H into a group configuration in M. This will, essentially, finish the proof since:  (Hrushovski). Let M be a strongly minimal structure and let G 1 = ( a, b, c, x, y, z ) be a group configuration. Then there exists a definable group G acting transitively on a strongly minimal set X.
This follows from the main theorem of [4] and the fact that infinitely definable groups in stable theories are intersections of definable groups (see, for example, [33,Theorem 5.18]) and the fact that any group definable in an algebraically closed field is (definably isomorphic to) an algebraic group (see [33,Theorem 4.13]). The original proofs of these statements are contained in [12].
To construct a field we will have to work a little harder. First, Remark. We will not go into the definition of canonical bases (see, e.g., [31, p.19]), but for the benefit of readers unfamiliar with this model theoretic notion we mention that: 1. The minimality condition is readily checked to be equivalent to the condition that whenever there are a ′ i ∈ acl M (a i ) such that ( a ′ 1 , a ′ 2 , a ′ 3 , x, y, z ) is still a group configuration then a i ∈ acl M (a ′ i ) for all i = 1, 2, 3.
2. By dimension considerations it follows from the previous remark that any group configuration (a 1 , a 2 , a 3 , x, y, z) gives rise to a minimal group configuration (a ′ 1 , a ′ 2 , a ′ 3 , x, y, z) with a ′ i ∈ acl(a i ) for all i. In particular, if dim(a i ) = 1 for all i then (a 1 , a 2 , a 3 , x, y, z) is a minimal configuration.

Roughly, Cb((x, y)/a) is the model theoretic analogue of the field of definition
of the locus of (x, y) over a.
4. For our purposes it will suffice to know that if X → T is a nearly faithful family of curves (see below) then t is (up to inter-algebraicity) a canonical base for x/t for any generic point of x, and if through x 1 , . . . , x k there is only one curve X t in X then t is (up to inter-algebraicity) a canonical base of (x 1 , . . . , x k ).
For minimal group configurations we have: . If the group configuration in the statement of Fact 4.4 is, additionally, assumed to be minimal then the action of the group G on X as provided above can be taken to be faithful and this group action has an associated group configuration G 2 = ( g, h, g · h, u, g · u, g · h · u ) inter-algebraic with G 1 . In particular, dim G = dim a.
This, finally, allows to obtain a field as follows: Fact 4.7 (Hrushovski, [12]). Let G be an M-definable group acting transitively and faithfully on a strongly minimal set X. Then either dim(G) = 1 or there exists a definable field structure on X and either dim(G) = 2 and G ∼ = G a ⋊ G m , or dim(G) = 3 and G ∼ = PSL 2 .
An exposition of the above fact can be found in [33] (Theorem 3.27). Establishing that G is isomorphic to G a ⋊ G m or to PSL 2 is the crucial point in the proof of Fact 4.7. In the present context, where G and X are definable in an algebraically closed field (rather, the full Zariski structure on an algebraic curve) this statement can be established using a simpler direct algebraic proof.

Some standard reductions
We make some standard simple reductions that will allow us to more easily use the results obtained in the previous sections as well as the group and field configurations described above.
Lemma 4.8. We may assume that k is of infinite transcendence degree (over the prime field).
Proof. Let K ≥ k be an algebraically closed field extension of infinite transcendence degree. We let M ′ := M (K), and for any D M-definable without parameters we let D ′ := D(K). We obtain a structure M ′ := (M ′ , X ′ ). By Hilbert's Nullstellensatz and Chevalley's Theorem (see, e.g., [23,Corollary 3 is an open set witnessing the fact that X is ample then U ′ witnesses that X ′ is). Note also that any set S M ′ -definable without parameters is of the form D ′ for some M-definable set, D.
Assume that a field is interpretable in M ′ . This means that there are D, E M ′definable (without parameters) and parametersā ∈ K l andb ∈ K n such that Eb is an equivalence relation (of the correct arity) and such that Dā/Eb is an infinite field. Let Lc and Ad be the graphs of multiplication and addition respectively, for L, A M ′ -definable without parameters.
Consider the set S of all parameters (x,ȳ,z,w) such that Eȳ is an equivalence relation on Dx and Lz, Aw turn Dx/Eȳ into an infinite field. We claim that S is M ′definable without parameters. This is easy since, if C ⊆ M r+s is any constructible set then the set {v ∈ M s : |C v | < ∞} is uniformly bounded, say by N . So C v is infinite if and only if |C v | > N , which is a definable property (of v). By Hilbert's Nullstellensatz and Chevalley's Theorem again S has a point in k, meaning that an infinite field is interpretable already in k.
In model-theoretic terms the above lemma only means that interpretability of an (infinite) field is a first order property, and therefore preserved under the passage to elementary substructures. The most useful -for our purposes -property of fields of infinite transcendence degree is the following consequence of Chevalley's theorem and the compactness theorem of first-order logic:    We can now show: Lemma 4.12. We may assume that M is a smooth curve and that M is strongly minimal.
The proof is well known (see, e.g., [31, Lemma IV.1.7] for a much more general statement). For the sake of completeness we outline a simple proof in the present context.
Proof. Clearly, if S is an M-definable set, S the structure with universe S and definable sets D ∩ S l (as D ranges over Def(M) and l ranges over N >0 ), then, if a field F is interpretable in S, then F is already interpretable in M. So it will suffice to show that there exists a strongly minimal set S ⊆ M that is not locally modular with respect to the induced structure S.
Let X ⊆ M 2 ×T be an ample family witnessed by an open U ⊆ M 2 . Reducing U , if needed, we may assume that if U i := p i (U ) are the projections of U onto the two M factors, then for all a, a ′ ∈ U 1 the fibre π −1 1 (a) ∩ U is infinite and (π −1 1 (a) ∩ U ) = (π −1 1 (a ′ ) ∩ U ) (up to a finite set) for all a, a ′ ∈ U 1 . Thus, setting Y = X • X −1 we immediately see that Y is ample witnessed by U 1 × U 1 .
Since U 1 is M-definable and one-dimensional there exists some strongly minimal is an ample family in M. It would be an ample family in the induced structure on M 0 if we could replace T with some T 0 ⊆ M r 0 (for some r). Recall that an irreducible algebraic curve of degree d is uniquely determined by any d + 1 generic enough points on the curve. A similar argument would show that any curve in Y ∩ (M 2 0 × T 2 ) is uniquely determined (up to a finite set) by finitely many generic enough points on the curve. More specifically, by the previous fact we may assume that X is nearly faithful. So if X t is a curve and p ∈ X t is generic then dim(p/t) = 1, so dim(p/t) < dim(p/∅) and by symmetry, dim(t/p) < dim(t/∅). Proceeding in a similar way, we get that dim(t/p 1 , . . . , p r ) = 0 for r = dim(T ) and p 1 , . . . , p r ∈ X t generic enough. So there are only finitely many t ′ such that p 1 , . . . , p r ∈ X t ′ and by adding enough points we can assure that X t is uniquely determined (up to a finite set) by some p 1 , . . . , p l ∈ X t . We leave it to the reader to verify that the number l of points determining the curve can be taken to be independent of t. Since any X t has infinitely many points in M 2 0 we get that M 0 , with the induced structure, is, indeed, strongly minimal and not locally modular. Replacing M 0 with M 0 ∩ M reg , the regular locus of M 0 , the conclusion follows.
To sum up: Corollary 4.13. To prove Conjecture A it suffice to prove: Let M be a strongly minimal reduct of the full Zariski structure M on a smooth algebraic curve M over an algebraically closed field K of infinite transcendence degree. Then either M is locally modular or M interprets a field K-definably isomorphic to K. Moreover, we may assume that the lack of local modularity of M is witnessed by a nearly faithful family of curves whose generic members are strongly minimal.

Generically unramified projections
In order to apply the machinery of slopes and tangency discussed in Section 3.1 we need to produce, definably in M, large enough families of curves where these notions are defined and carry information. Lemma 4.14 below guarantees the former requirement, namely that for any curve X ⊂ M 2 the slope is defined on a dense open subset of either X or X −1 (uniformly in parameters). The second requirement is more delicate, as pointed out for example in the concluding remarks of [24]. In more technical terms, the problem pointed out by Marker and Pillay is that if the projection p 2 : Z → M is everywhere ramified for a curve Z ⊂ M 2 (e.g. the curve cut out by the equation y = x p in A 1 × A 1 ) then even if p 2 is dominant, τ 1 (Z, α) = 0 for any branch α at any point of Z. In Lemma 4.16 and Lemma 4.15 we develop the tools allowing us to construct, definably in M, curves in M 2 whose projections on both factors M are generically unramified.
The following lemma ensures that at least one of the projections on a factor M of a family of curves is genericallyétale for a general element of the family, which by Lemma 3.4 implies existence of slopes for a generic element of the family. The fact that the support of the module of Kähler differentials is closed and Fact 3.10 imply that beingétale and being unramfied is open on the source. In particular, in order to check whether a dominant morphism f : X → Y isétale on a dense open subset of X it suffices to check if Ω k(X)/k(Y ) = 0, or equivalently (see [22,Exercise 6.2.9], also [22,Lemma 6.1.13]), if k(X) ⊃ k(Y ) is a separable extension. We refer the reader to any standard algebraic geometry reference (e.g. [22,Section 6], [11, Section II.8,IV.2]) for the details on Kähler differentials and ramification.
Lemma 4.14. Let M be an irreducible algebraic curve over a field of any characteristic. Let X ⊂ M 2 × T be a family of pure-dimensional curves, and assume that X and T are irreducible. Then there exists a dense open subset T ′ ⊆ T such that either p 1 : Proof. Let ξ be the generic point of T in the scheme-theoretic sense. Denote M ξ = M ⊗ k(ξ), X ξ = X ⊗ k(ξ). By slightly abusing notation, denote p 1 , p 2 : X ξ → M ξ the natural projections.
Let Ω M ξ /k(ξ) , Ω X ξ /k(ξ) be the sheaves of modules of Kähler differentials on the generic fibres M ξ = M ⊗ k k(ξ) and X ξ = X ⊗ k k(ξ), respectively. Since ι : X ξ → M 2 ξ is a closed embedding the pull-back is surjective. Taking stalks at the generic point χ of X ξ we get a surjective map of vector spaces over the field k(χ) = k(X) Each summand on the left is either trivial or one-dimensional. Since i * is surjective, it follows that at least one of the summands is mapped surjectively on the destination. Therefore, the stalk at k(χ) of either Ω X ξ /k(ξ) /p * 1 Ω M ξ /k(ξ) or Ω X ξ /k(ξ) /p * 2 Ω M ξ /k(ξ) vanishes, and we conclude.
Suppose we have a family of pure-dimensional curves X ⊂ M 2 × T such that for some a ∈ X t for all t ∈ T , and assume that for all t in the morphism p 1 : X t → M iś etale in some neighbourhood of a. Then by Lemma 3.4 there exists a unique branch α of X at a. It might be the case, though, that τ n (X t , α) vanishes for all n, for all t ∈ T , if p 2 is everywhere ramified on the component of X t that contains a. Below we show that in this case one can consider the family X • X −1 which does not have this pathology, and p 1 , p 2 are both generically unramified for any of its members.
Recall that if f : X → Y is a morphism of schemes over a field of characteristic p then Fr f : The natural projection Fr X/Y (X) → Y is given by (y, x 1 , . . . , x n ) → y.
Lemma 4.15. Let f : X → Y be a finite morphism of irreducible varieties over a field of characteristic p > 0 and let F = Fr f be the relative Frobenius morphism. Assume that f is everywhere ramified. Then there exists an n > 0 such that the natural projection F n (X) = X × f,Y,F n Y → Y is generically unramified.
Proof. Since f is everywhere ramified, the field extension k(Y ) ⊂ k(X) is inseparable. Let L be the separable closure of k(Y ) in k(X), then k(Y ) ⊂ L is a separable extension and L ⊂ k(X) is a purely insepearable extension. Since L ⊂ k(X) is a finite extension, there exists a smallest number n such that h p n ∈ L for any h ∈ k(X). We claim that k(F n (X)) ⊂ L, which will conclude the proof, as this shows that k(F n (X)) is a separable extension of k(Y ). To prove the above claim, let . . , g n ]/I (p n ) , and there is an embedding of rings k[X 0 ] ⊂ k(X 0 ). It is immediate from the definition of the relative Frobenius morphism that there exists an injection k[F n (X 0 )] ֒→ L sending g i to h p n i , so F n (X 0 ) is unramified over Y 0 and we conclude.
onto products of factors by r, resp. r ′ , with subscripts. After unravelling the definitions one observes that Also, Fr r 1345 (a, b, c, t, s) = (a, F (b), c, t, s) and

Interpretation of a one-dimensional group
In the present section we construct a group interpretable in M. As already explained, this will be done by constructing a group configuration in M. In order to construct this group configuration a one-dimensional algebraic group (Lemma 4.19) G associated with slopes is 'lifted', using Proposition 3.15, to a group configuration in M.
Throughout this section and until the end of this paper we fix an algebraic curve M over an algebraically closed field K of infinite transcendence degree, and a reduct M of the full Zariski structure M on M . We assume that the reduct is not locally modular. By default the term definable will refer to definability in M. Unless explicitly stated otherwise, by definable families we mean stationary nearly faithful ample families of curves. Where a family X → T is stationary if every definable open subset of T is dense.
Before we proceed, we need a couple of easy observations: Proof. Fix (t, s) generic. Since any curve in X • Y intersecting X t • Y s in an infinite set must contain (up to a finite set) a strongly minimal component of X t • Y s , and since only finitely many such components exists, it will suffice to show that any such component is contained in finitely many members of X • Y .
Let E ⊆ X t • Y s be strongly minimal. By (the proof of) [8, Lemma 3.20] either M interprets a one-dimensional group or dim M (Cb M (E/∅)) = 2 (the latter notation can be interpreted, equivalently, as: there exist an M-definable nearly faithful family of curves defined over a two-dimensional parameter set and E is generic in that family). We may assume the latter case occurs. So, by obvious dimension considerations s, t ∈ acl M (Cb M (E/∅)). So there are only finitely many (t ′ , s ′ ) such that E ⊆ X t ′ • Y s ′ , which is what we had to show.
Remark. Recall that our aim in this section is to interpret in M a strongly minimal group G. It follows from the previous lemma that one way of achieving this is to find X → T and Y → S one-dimensional definable families of strongly minimal subsets of M 2 with the property that In order not to overload the formulation of the sequel we will tacitly assume that, whenever Lemma 4.18 is invoked, this is not the case -as otherwise we have found our group, and we can move on to the next section.
We now proceed to finding the 1-dimensional algebraic group of slopes needed for the construction of the group configuration: There exists a nearly faithful definable family Y ⊂ M 2 × S with S strongly minimal, a locally closed irreducible set S 0 ⊂ S, a point a = (a 1 , a 2 ) ∈ M 2 , a 1 = a 2 , such that a ∈ Y s for all s ∈ S 0 , and a family of branches β of Y × S S 0 at a such that for some n > 0 the locally closed set Similarly, we may assume that if a = (a 1 , a 2 ) is a zero-dimensional component of X t then a 1 , a 2 / ∈ acl M (∅). Thus, we may assume that both (a 1 , a 2 ) are M-generic over ∅. The same is, of course, true of Y .
Denoting X 0 t , Y 0 s the zero-dimensional components of X t , Y s and noting that s we get that for s, t ∈ T independent generics any isolated point of X t • Y s is generic over ∅.
We have thus shown: There exists a 1 ∈ M , a natural number n > 0 and a one-dimensional definable family of curves that satisfies property (a, n) for a = (a 1 , a 1 ). (1) and (2) of the definition of property (a, n) are achieved by taking a family as provided by Lemma 4.19. Condition (4) is provided by Lemma 4.18, and condition (3) is obtained by removing finitely many points common to all generic independent curves in the resulting family.

Proof. Clauses
The same proofs give also: Corollary 4.23. If X → T is a family that satisfies property (a, n), then up topossibly -finitely many corrections, X • X and X • X −1 also satisfy property (a, n).
Note however that in the above corollary if X is one-dimensional then the families X • X and X • X −1 will not be one-dimensional. It follows, however, that if t ∈ T is generic then the one-dimensional families X • X t and X • X −1 t will satisfy property (a, n). The following is a strengthening of the above observation that we will need later on for technical reasons: Lemma 4.24. Let X → T be a family that satisfies property (a, n). Let H be the group of slopes of X at a (associated with some family of branches). Then there exists a one-dimensional nearly faithful family of strongly minimal sets Z → L such that a ∈ Z l for all l, and there exists a family of branches γ at Q such that τ n (Z l , β l ) = 1 ∈ H for all l ∈ L.
Proof. There exists an M-irreducible component W ⊆ T such that τ n (X t , β t ) ∈ H for all t ∈ W (and in particular the slope is defined). Let t 0 ∈ W be generic. So there exists some t 1 ∈ T such that τ n (X t 1 , Let Z → L be the M-definable family whose generic member is Z l . So there is an M-generic sub-family of Z → L with the property that τ n (Z l ′ , γ l ′ ) = 1 (see Proposition 3.7) for a family γ of branches of Z at a and for all l ′ in that subfamily.
We are finally ready to prove the main result of this section: Proof. We prove the theorem by constructing a group configuration. By Corollary 4.13 we may assume that M is smooth and we identify M with M (K) for some algebraically closed field K of infinite transcendence degree. We will freely use the remark after Definition 3.3 and Lemma 3.2, referring to branches of suitable pure-dimensional subfamilies of definable families of curves when we speak about branches of definable families of curves.
Let X → T be a one-dimensional definable family of curves that satisfies property (a, n) for some point a = (a 1 , a 1 ) as provided by Corollary 4.22, H the associated group of slopes for the family of branches β. Absorbing into the language the parameters needed to define X, we may assume that it is ∅-definable.
We fix a standard group configuration H := {g, h, k, gh, gk, h −1 k} associated with the action of H on itself by multiplication. By Lemma 4.19 there exists an irreducible component W ⊆ T such that τ n (X t , β t ) ∈ H for all generic t ∈ W . Identifying (up to a finite set) {τ n (X t , β t ) : t ∈ W } with elements of H ≤ Aut(k[ε]/(ε n+1 )) we get that τ n (X t , β t ) is M-inter-algebraic with t. At the price of replacing W with a (dense) open subset, we may assume that W is smooth.
Any M-independent points s, t ∈ W generic over all the data are, in particular, generic and independent in the sense of the reduct M. Let u ∈ T be such that τ n (X u , β u ) = τ n (X s , β s )τ n (X t , β t ) = τ n (X s • X t , β s • β t ).
Such a u exists, since the relative slopes of X t and X s are generic in H, which is one-dimensional. Since the product of two independent generic elements of H is again generic in H, we can find such a u.
Getting back to our group configuration H the above construction gives us a subset of T , such that for every s ∈ H we have τ n (X ts , β ts ) = s. Our goal is to show that T H is a group configuration in the sense of M.
We have to verify the three sets of conditions appearing in Definition 4.3. That the elements of T H are pairwise M-independent follows from the fact that for all s ∈ H also s ∈ acl M (t s ) and the elements of H are M-independent. That all elements in T H have dimension 1 follows from the fact that T is strongly minimal and the elements of T H are generic in T . So it remains only to verify the third set of conditions, namely, that every collinear triple of elements in the following diagram is M-dependent: The rest of the proof will be dedicated to that end. Since the situation is symmetric, it will suffice to show that if s, t ∈ W are generic independent τ (X u , β u ) = τ n (X s , β s )τ (X t , β t ) then u ∈ acl M (s, t). Note that since W is M-strongly minimal, u ∈ acl M (s, t).
To achieve our goal, we would like to apply Proposition 3.15 to the familyẼ → R given by X • X and the family X → T , in order to show that the curve X u intersects the curve X s • X t in a smaller than generic number of points. The problem is that neitherẼ → R nor X → T can be assumed to be pure-dimensional families of curves, which is a crucial assumption in the statement of the proposition. To circumvent this problem, we will show that X u ∩(X s •X t ) contains no zero-dimensional components of either curve, allowing us to apply the proposition with the pure-dimensionalẼ 1 → R and X 1 → T without changing the number of intersection points.
For technical reasons that will be made clear later on, we need to slightly twist the familyẼ → R that we are working with. Let Z → L be a nearly faithful family of curves that satisfies property (a, m) for m > n, let γ be a family of branches at a, all as provided by Lemma 4.24. We let E → R beẼ • Z l 0 for some l 0 ∈ L M-generic and independent over all the data. Note that, by Proposition 3.7 and the choice of the family Z → L, we have that τ n (X s • X t , β s • β t ) = τ n (X s • X t • Z l 0 , β s • β t • γ l 0 ) whenever both sides of the equations are defined. For the sake of clarity we let α be the family of branches of E at a. Namely, α = β • β • γ l 0 .
It will be convenient to already note at this stage the following slight strengthening of Lemma 4.18: Claim 1 We may assume that if r ∈ R is generic then |{r ′ :Ẽ r ∩Ẽ r ′ }| = ∞ is finite.
Proof. The claim would follow from Lemma 4.18, if the members of X were strongly minimal. In the general case, if r ∈ R is generic andẼ r = X s • X t then any strongly minimal F r ⊆Ẽ r is contained in C s • D t for some strongly minimal C s ⊆ X s and D t ⊆ X t . By Lemma 4.18 applied to the families {D t : t ∈ T } and {C s : s ∈ S}, we get that s, t ∈ acl M (Cb(F r )]). Since Cb(F r ) ∈ acl(Cb(E r )) we conclude that s, t ∈ acl(Cb(E r )), which is what we needed. Claim 1 .
Note that the fact thatẼ is the composition of two copies of X did not play any role in the proof above, and we could invoke Lemma 4.18 with X and (X • Z l 0 ) to get the same conclusion for the family E := X • (X • Z l 0 ).
Let us fix some additional notation. We let R(u) be the set of all r ∈ R such that E r is tangent to X u at a, i.e., τ n (E r , α r ) = τ n (X u , β u ). Let E(u) := {E r : r ∈ R(u)}. In other words, R(u) is the parameter set of all curves in the family E → R ntangent to X s • X t at a and E(u) is the subfamily of E over the parameter set R(u). So E(u) → R(u) is an M-definable subfamily of E of dimension 1. Fix, once and for all, r = (s, t) ∈ R. So r ∈ R(u) and it is M-generic as such. Replacing, if needed, R(u) with the M-definable strongly minimal component of R(u) containing r we may assume that R(u) is strongly minimal.
The following is the main step in the proof: Proof. First, note that x i / ∈ acl M (∅), because otherwise, since u ∈ T is generic, we would get that dim(T x i ) = 1, contradicting clause (3) of Definition 4.20. Note that the exact same argument shows that x i / ∈ acl M (∅). Next, as r ∈ acl M (s, t) and since X is an M-strongly minimal family, our negation assumption implies that u is Mgeneric over r and by Lemma 4.18 (and the remark following it) applied to the acl(∅)definable strongly minimal subsets of E r we get that dim(Cb(E r )/∅) = dim(r/∅) = 2. Now assume that x 1 is not M-generic in X u . Since r is M-generic in R(u) (and thus also in R), it follows that dim(R(u) x 1 ) = 1. Indeed, since dim M (x 1 /u) = 0 (by assumption), it follows that dim M (r/ux 1 ) = dim M (r/u), so r is generic in R(u) over x 1 , and the strong minimality of R(u) implies that x 1 ∈ E r ′ for all generic r ′ ∈ R(u). Thus, in fact R(u) is a generic subset of R x 1 . Recall, moreover, that there exists a family α of branches of all curves in E(u) at a such that τ n (E(u), α r ) = τ (X u , β u ).
We will show that this gives the desired conclusion. We split the argument into cases according to dim M (x 1 /∅). The case x 1 ∈ acl M (∅) has already been discarded. If x 1 is non-M-generic in M 2 then there exists a curve F , M-definable over ∅ such that x 1 is generic in F . So u is contained in the set of all u ′ such that F ∩X u ′ ∩E r = ∅. Because |E r ∩ F | < ∞ and condition (3) of Definition 4.20, there are only finitely many such u ′ . So u is M-algebraic over r -contradicting our assumption.
Thus, we may assume that x 1 is M-generic in M 2 . We will now focus on the family E → R. Since x 1 is M-generic in M 2 , for any r 1 , r 2 ∈ R x 1 independent generic, m := |E r 1 ∩ E r 2 | is obtained on an M-generic subset of parameters of R × R. Consider the M-definable family E 1 → R of pure-dimensional curves associated with E → R. Note that for M-generic independent u, w ∈ R we have |E u ∩ E w | = |E 1 u ∩ E 1 w | = m. On the other hand, Lemma 3.13, and hence Proposition 3.15, is applicable to two copies of the family E 1 → R, possibly after shrinking R so as to ensure, using Fact 3.10, that E 1 → R is flat and that R is smooth. So, by Lemma 3.13 and Proposition 3.15 there is a dense open set W 0 ⊆ R defined over ∅ such that (keeping the above notation) for all (v, w) ∈ W : We now show that this must imply that E 0 w ∩ E r = ∅ for generic w. Indeed, since W 0 is dense in R and ∅-definable, M-genericity of r in R implies that it is also generic in W 0 . Since R(u) x 1 is generic in R x 1 (in the sense that it contains an open subset of R x 1 ) we can find some w ∈ R(u) ∩ W 0 M-generic and M-independent from r (over all the data gathered so far) so that (r, w) is M-generic in W . Moreover, by definition of R(u) we know that τ n (E 1 r , β r ) = τ n (E 1 w , β w ), and by what we have just said, this must imply that #(E 1 w ∩ E 1 r ) < m. Because x 1 is M-generic in M 2 and w, r ∈ R x 1 are M-independent generics, they are, in fact, independent generic in R 2 over ∅. So #(E w ∩ E r ) = m, implying that E w ∩ E 0 r = ∅. Finally, since w was M-independent from r and M-generic in R(u), and since E 0 r ⊆ acl M (r), we get -precisely as above -that there is some c ∈ E 0 r such that R c contains R(u), up to a finite set. This implies that dim M (c/u) = 0, and therefore dim M (c/∅) ≤ dim M (u/∅) = 1. This contradicts Corollary 4.23 (specifically, clause (4) of Definition 4.20).
It follows from Claim 2 that X 0 u ∩ E r = ∅. We also need to show that X u does not meet E r in an isolated point of the latter. It is here that the twist of the familỹ E → R by a generic curve from Z → L plays its role: Proof. Recall that E r = X s • X t • Z l 0 . Assume that there exists some x i ∈ X u ∩ (X s • X t • Z l 0 ) 0 . By Lemma 4.21 applied toẼ(u) → R(u) and Z → L, if r ′ ∈ R(u) is generic and l ∈ L is generic independent from r ′ , then any x i ∈ (E r ′ • Z l ) 0 is either M-generic over ∅ or contained in one of finitely many sets of the form {a} × M and contradicting the previous claim.

Claim 3
The conclusion of the discussion, up to this point, is that if u / ∈ acl M (s, t) then E r ∩ X u = E 1 r ∩ X 1 u . This allows us to conclude that, in fact: LetT ⊆ T be as provided by Lemma 4.19. By Lemma 3.13 and Proposition 3.15, the set of parameters w ∈T such that τ n (X 1 w , β w ) = τ n (X 1 t , β t ) • τ n (X 1 s , α s ) is contained in By strong minimality of T the set {w : is finite, and moreover, W is M-definable. Our assumption that u / ∈ acl M (s, t) allows us to apply Claim 2 combined with Claim 3 to get that u . Since u ∈ W 1 it follows that u ∈ W , proving that, in fact u ∈ acl(s, t).

Claim 4
Claim 4 shows that, indeed, H T is an M-group configuration, and the desired conclusion is obtained by applying Fact 4.4.
The next proposition can be proved in greater generality (and follows, essentially, from [18,Section 3]), but we only need the following elementary result: Proposition 4.26. In the notation of the previous proof, assume that the group H almost coinciding with {τ n (Y t , α t ) : t ∈S} is isomorphic to G a . Then the connected component of the identity of the group from the conclusion of the theorem is not M-isomorphic to G m .
Proof. In this proof we will be working solely in M. It suffices to show that if { a, b, a + b, x, x + a, x + b } and G 2 = { e, f, e · f, y, e · y, f · y } are group configurations for the groups G a and G m , respectively, and acl(a) = acl(e), acl(b) = acl(f ), acl(x) = acl(y) then G 1 and G 2 are not inter-algebraic.
Indeed, dim(a, b, e + f, ef ) = dim(a, a + b, ef ), since ef is inter-algebraic with b over a. On the other hand, dim(a, a + b, ef ) = dim(a + b, ef ) since a + b is inter-algebraic with a over ef .

Interpretation of the field and proof of the main theorem
In this section we interpret the field K in the reduct M, concluding the proof of the main theorem of this paper. The results of the previous subsection allow us replace M with an algebraic group, G, interpretable in M (we only have to verify that the induced structure is non-locally modular). As in the previous subsection, the interpretation of the field boils down to the construction of a field configuration. The construction of the field configuration will depend on whether the (connected component of the) group G is isomorphic (in K) to G a , G m or to an elliptic curve. The question to address is how to find an M-definable strongly minimal Z ⊆ G 2 whose set of slopes {τ 1 (Z, z) : x ∈ Z} (see below) is infinite. The easiest is the case of an elliptic curve: Lemma 4.27. Let E be an elliptic curve and Z be a closed one-dimensional irreducible subset of G = E 2 . Identify T g G with T 0 G via the isomorphism dλ g : T 0 G → T g G, for λ g (x) = g · x. Suppose that for any z ∈ Z the tangent space T z Z ⊂ T 0 G is constant. Then Z is a coset of a closed subgroup of G.
Proof. Since Z is a projective curve with a trivial tangent bundle, it is an elliptic curve itself by the Riemann-Roch formula. Since any morphism between Abelian varieties with finite fibres which preserves the identity automatically preserves the group structure by the Rigidity Lemma (see [29, p. 43]), Z is a coset of an Abelian subvariety of G.
Let M be an algebraic curve, and consider a curve Z ⊂ M 2 . For every point z ∈ Z such that p 1 isétale in a neighbourhood of z, there exists by Lemma 3.4 a unique branch at z, call it α z . We will use the notation τ n (Z, z) := τ n (Z, α z ). For any group (G, ·) with identity e ∈ G, for any a = (a 1 , a 2 ) ∈ G 2 define the maps t a : ) and for any one-dimensional locally closed subset Z ⊂ G 2 define the set s n (Z) Also for any c ∈ k define u c : We can consider the family of translatesZ ⊂ M 2 × Z such thatZ a = t a (Z). If Z is a pure-dimensional curve thenZ is a family of pure-dimensional curves, and if the projection on the first coordinate is genericallyétale then for a dense subset Z 0 ⊂ Z there is a unique family of branchesᾱ of the familyZ| Z 0 at (0, 0) such thatᾱ a = α a for all a ∈ Z 0 .
Lemma 4.28. Let Z ⊂ G 2 a be an irruducible pure-dimensional curve that is not contained in a coset of a subgroup of G 2 a . Then either Z − t x Z is not contained in a subgroup of G 2 a for some x ∈ Z, or the closure of Z is cut out by an equation of the form y − ax 2 = 0, when the characteristic p of the ground field is 0, or, when p > 0, p = 2, by an equation y − ax 2p n = 0, for some non-negative integer n and a constant a ∈ k.
Proof. We may assume that (0, 0) ∈ Z, and that the projection on the first coordinate in G 2 a isétale in a neighbourhood of (0, 0) ∈ Z. Assume that Z − t x Z is contained in a subgroup of G 2 a for all x ∈ Z, and let the closure of Z be cut out by a polynomial equation f (x, y). Let Z ′ ⊂ G 4 a be the set cut out by the equation f (u, v), and let Z ′′ ⊂ Z ′ be cut out in Z ′ by the equation f (x − u, y − v) where x, y, u, v are the coordinates. A fibre of Z ′′ over (u, v) ∈ G 2 a is thus the closure of Z − t (u,v) Z. We identify the completion of the local ring of Z ′ at (0, 0, 0, 0) with k[[x, y, u]] via the projection on the first three coordinates. If the local equation of Z in G 2 a at (0, 0) is y − g(x), g ∈ k[[x]] then one readily sees that the local equation of Z ′′ in Z ′ at (0, 0, 0, 0) is If Z − t x Z is contained in a subgroup for all x ∈ Z then the above expression must be the local equation of the subgroup, that is, it is of the form ay p m − bx p n for some non-negative integers n, m. Then one sees immediately that h can only be of the form y − bx p n , so g can only be of the form g = ax 2 , when p = 0, or g = ax 2p n otherwise. In particular, if p = 2 then Z is a subgroup of G 2 a .
Lemma 4.29. Let G be an algebraic group such that the connected component of the identity G 0 is isomorophic to G a . Let Z ⊂ G 2 be a curve that is not a Boolean combination of cosets of subgroups of G 2 . Then there exists a pure-dimensional curve W ⊂ G 2 definable in (G, Z), and an irreducible component W ′ ⊂ W ∩ G 2 0 of dimension 1, such that dim s 1 (W ′ ) = 1. In characteristic 0, one can take W = Z.
Proof. Replacing Z by a shift, we may assume that there exists a pure-dimensional irreducible curve Z 0 ⊂ Z such that (0, 0) ∈ Z 0 , such that Z 0 is not contained in a coset, and such that the projection on the first coordinate in G 2 a isétale in a neighbourhood of (0, 0) ∈ Z.We will construct a finite sequence of curves W i ⊂ G 2 a , W 0 = Z 0 that are irreducible components of curves definable in (G, Z), such that N (W i ) := min{ n ≥ 1 | dim s n (W i ) = 1 } is strictly decreasing.
We pick the local coordinate systems at all points c ∈ G a in a uniform way, with k[[x]] ∼ = O Ga,c given by x → x − c. Clearly, τ n (Z, x) = τ n (t b (Z), t b (x)) for any b ∈ G 2 a and u c (Z) = Z + L c where L c is the line x 2 = −c · x 1 . By Proposition 3.9 τ 1 (u c (Z), (0, 0)) = τ 1 (Z, (0, 0)) − c.
If N (W i ) > 1, then, again by Proposition 3.9 for all x ∈ W i and since W i was not a coset, by Lemma 4.28, either W i − t x (W i ) is not a subgroup for some x = 0, or it is cut out by an equation of the form y − ax 2 or y − ax 2p n for some non-negative integer n, a ∈ k. In the latter case by Lemma 4.16 we have that N (W i+1 ) = 1 for W i+1 = W i • (t x (W i )) −1 for some x ∈ W i . In the former case, pick an x such that W i − t x (W i ) is not a subgroup of G 2 a and define and since Y i is not a coset, p 2 restricted to u c (Y i ) is dominant and everywhere ramified. In particular, since the latter is impossible when the characteristic of the base field is 0, it follows in that case that N (W 0 ) = 1.
By Lemma 4.15, there exists a number m such that p 2 restricted to Y ′ i = Fr m p 2 (Y i ) is generically unramified. Let a ∈ Y i be a point such that p 1 isétale over M 0 in some neighbourhoods of a, and let the local equation of Y i at a, b be x 2 − f p m (x 1 ), respectively. Then the local equation of Y ′ i at the point Fr m p 2 (a) is for all points a = (a 1 , a 2 ), b = (b 1 , b 2 ) ∈ Y i such that a 1 = b 1 and such that the righthand side makes sense. In particular, p 2 restricted to W i+1 is generically unramified and it follows that N (W i+1 ) = N (W i )/p m < N (W i ). Therefore for some finite l, N (W l ) = 1.
Lemma 4.30. Let G be a one-dimensional algebraic group such that the connected component of the identity G 0 is isomorphic to G m . Let Z ⊂ G 2 be a curve that is not a Boolean combination of cosets of subgroups of G 2 . Then either there exists an irreducible pure-dimensional curve Z 0 ⊂ Z such that dim s 1 (Z 0 ) = 1, or there exists a group definable in (G, Z) such that its connected component of the identity is not isomorphic to G m .
Proof. Pick a local coordinate systems on G m , uniformly, as in the proof of Lemma 4.29. Assume that dim s 1 (Z) = 0, and so s 1 (Z i ) is a singleton for each onedimensional irreducible component Z i ⊂ Z. Let Z 0 be one of the irreducible components of Z that is not contained in a coset, then there exists a smallest n > 1 such that dim s n (Z 0 ) = 1. Then, by the same reasoning as in the proof of Lemma 4.19, we may consider the family Y ⊂ G 2 × Z by putting Y z = t z (Z) • (t z 0 (Z)) −1 for some z 0 ∈ Z 0 , so that {τ 1 (Y z , (0, 0)) | z ∈ Z 0 } almost coincides with The definable family Y can be used to construct a group configuration as in the proof of Theorem 4.25, and therefore a group is interpretable in (G, Z). By Proposition 4.26, the connected component of the identity of this group is not isomorphic to G m .
We can finally interpret the field: Theorem 4.31. Let G be a one-dimensional algebraic group over an algebraically closed field, Z ⊂ G 2 be a one-dimensional constructible subset that is not a Boolean combination of cosets. Then (G, ·, Z) interprets a field.
Proof. Let G 0 be the connected component of the identity e of G. If G 0 = G a or G 0 is an elliptic curve then, by Lemmas 4.27, 4.29, there exists a definable family Y ⊂ G 2 ×S of curves, S strongly minimal, and an irreducible locally closed set S 0 ⊂ S such that there is a unique family of branches α of Y 0 = Y ∩ S 0 at (e, e) ∈ G 2 , and such that τ 1 (Y s , α s ) is not constant as s ranges in S 0 . By Lemma 4.30' either such a family exists, or a definable one-dimensional group G ′ with the connected component of the identity not isomorphic to G m (and therefore isomorphic to either G a or to an elliptic curve) is interpretable in (G, ·, Z), and we may prove the theorem for the structure induced on G ′ . We, therefore, may continue with the assumption that such a family exists. Clearly, Y 0 → S 0 and we may assume that S 0 is smooth at the price of possibly shrinking S 0 . Further shrinking S 0 we can ensure Y 0 → S 0 to be flat (by Fact 3.10). Let K be a field of infinite transcendence degree over the base field k. We identify first order slopes, which are truncated polynomials in k[ε]/(ε 2 ) divisible by ε, with K, and we will use multiplicative notation for composition. We will freely use the remark after Definition 3.3 and Lemma 3.2, referring to branches of suitable pure-dimensional subfamilies of definable families of curves when we speak about branches of definable families of curves. Take a 1 , a 2 , b 1 , b 2 , u ∈ S 0 (K) generic and pairwise independent. Let c 1 , c 2 ∈ S 0 (K) be such that This is possible, since the image of the function s → τ 1 (Y s , α s ) for s ranging in S 0 is of dimension 1, and the values of slopes on the right hand side of the equations above are generic in End(k[x]/(x 2 )) for generic pairwise independent values of parameters. Therefore τ 1 (Y a 1 , α a 1 )τ 1 (Y b 1 , α b 1 ) and τ 1 (Y a 2 , α a 2 )τ 1 (Y b 1 , α b 1 ) + τ 1 (Y b 2 , α b 2 ) are generic, and c 1 , c 2 as required can be found in S 0 (K). Let z, v be such that τ 1 (Y z , α z ) = τ 1 (Y a 1 , α a 1 )τ 1 (Y u , α u ) + τ 1 (Y a 2 , α a 2 ), By a similar reasoning, z, v are generic. It also follows from the way c 1 , c 2 , z, v were defined that τ 1 (Y z , α z ) = τ 1 (Y c 1 , α c 1 )τ 1 (Y v , α v ) + τ 1 (Y c 2 , α c 2 ).
We will now show that (c 1 , c 2 ) is algebraic over (a 1 , a 2 ) and (b 1 , b 2 ) in the sense of (G, ·, Z). By Propositions 3.7 and 3.9, for a 1 , a 2 , b 1 , b 2 , c 1 , c 2 ∈ S 0 generic and independent. Since the number of intersection points is an (G, ·, Z)-definable property, it does not matter what particular parameters a i , b i , c i we take as long as they are generic and independent (in the sense of (G, ·, Z)). By Lemma 3.13 and Proposition 3.15, the (M, X)-definable set { w ∈ S 0 | dim(Y w ∩ (Y a 1 • Y b 1 )) = 1 or #(Y w ∩ (Y a 1 • Y b 1 )) < l 1 } contains c 1 and by definition of l 1 is finite. By Lemma 3.13 and Proposition 3.15 again, the (M, X)-definable set contains c 2 and by definition of l 2 is finite.
Arguing in a similar fashion, by application of Lemma 3.13 and Proposition 3.15, we deduce that for all lines in the diagram z u v (a 1 , a 2 ) each vertex is in the algebraic closure of two other collinear vertices, and so this constitutes a group configuration. Therefore, by Fact 4.4, there exists a two-dimensional group definable in (G, ·, Z) that acts transitively on a one-dimensional set. The conditions of the Fact 4.6 are verified as well: for instance, for the uppermost line, B = {τ 1 (Y a 1 , α a 1 ), τ 1 (Y a 2 , α a 2 )} is by construction a canonical base of the type tp(τ 1 (Y z , α z ), τ 1 (Y u , α u )/B) in the full Zariski structure. Since the natural morphism S 0 → Aut(k[ε]/(ε 2 )), s → τ 1 (Y s , α s ) has finite fibres, a canonical base of tp(z, u/a 1 , a 2 is inter-algebraic with {a 1 , a 2 } in the full Zariski structure. Since passing to the reduct can only enlarge a canonical base, the canonical base of tp(z, u/a 1 , a 2 ) is inter-algebraic with {a 1 , a 2 }. The same argument applies to tp(u, v/b 1 , b 2 ) and tp(z, v/c 1 , c 2 ).
By Fact 4.7, the group G is isomorphic to the affine group G a (k) ⋊ G m (k) of an infinite definable field k.
In order to apply the above results we need the following, which is well known model theoretic folklore. We give a short proof specialised to the case where we need it: Lemma 4.32. Let G be a strongly minimal group interpretable in M. Then there exists a strongly minimal Z ⊆ G 2 that is not a finite boolean combination of cosets of definable subgroups.
Proof. To simplify the discussion let us call subsets of G that are finite boolean combinations of cosets of G n (any n) affine. By strong minimality G is in finiteto-finite correspondence with M (this follows, in general, from the fact that M is unidimensional. In the present setting G can be assumed to have been obtained from Theorem 4.4 using a 1-dimensional group configuration, so the existence of a finite-to-finite correspondence follows from the statement). It follows that G is not locally modular, as the image of any ample family of 1-dimensional subsets of M 2 under this finite-to-finite correspondence is an ample family in G 2 .
Since G is not locally modular it admits, by [8, Proposition 3.21] a nearly faithful ample family of generically strongly minimal curves X → T ⊆ G 2 × T of dimension 3 (i.e. dim(T ) = 3). Let G 0 denote the M-connected component of G. let t ∈ T and x 0 ∈ G 0 be independent M-generics. Let y 0 be such that (x 0 , y 0 ) ∈ X t and assume that y 0 ∈ gG 0 for some g ∈ G (that we can choose independent from (x 0 , y 0 )). Then gX t := {(x, y) : (x, gy) ∈ (G 0 ) 2 ∩ X t } is an M-definable curve in (G 0 ) 2 and gX := {gX t : t ∈ T } is a definable family of curves in (G 0 ) 2 . Since G/G 0 is finite and X is nearly faithful the correspondence X t → gX t is finite-to-one and on a generic subset of T . Therefore gX is readily checked to be a 3-dimensional, nearly faithful ample family of curves in (G 0 ) 2 . Moreover, if X t is affine (for some t ∈ T ) then so is gX t . So it will suffice to show that X can be chosen so that gX t is not affine for generic t ∈ T .
If G 0 is M-definably isomorphic to either G m or to an elliptic curve, E, then for generic t ∈ T gX t is not affine, since there are no definable families of subgroups of G 2 m or of E 2 . Let us elaborate: assume towards a contradiction that for generic t ∈ T the curve X t is affine. So gX t is also affine for all such t. In this setting there are finitely many ∅-definable 1-dimensional subgroups H 1 , . . . H k of (G 0 ) 2 such that gX t coincides, up to a finite set, with a union of cosets of the H i . Since (G 0 ) 2 /H i is 1-dimensional for all i, near faithfulness of gX implies that gX is, at most, 1dimensional, a contradiction.
So we are reduced to the case where G 0 is M-definably isomorphic to G a . It is an easy exercise to verify that the definable subgroups of G 2 a are precisely closed subsets cut out by linear equations in {x p n } ∞ n=0 and {y p m } ∞ m=0 (for p = char(K)). Thus, if we choose X whose projections on (G 0 ) 2 are generically unramified (as provided by the combination of Lemma 4.15 and Lemma 4.16) then also gX has this property (for a suitable choice of g), so if gX t is affine for generic t gX t contains (up to a finite set) the graph of a linear function x → ax + b. As a above, near faithfulness of X implies near faithfulness of gX, and therefore gX can be at most 2-dimensional, again, a contradiction.

Remark.
1. It follows from [17, Theorem 4.1(b)] that there is some definable Z ⊆ G n (some n) that is not affine. Reducing n to be 2 requires a little more effort.
2. In the above proof it is not hard to see that if we obtain a 2-dimensional nearly faithful family X → T such that each X t contains (up to a finite set) a curve of the form a t x+b t then X can be used directly to construct a field configuration.
We can now sum up everything to obtain the main result of this paper: