JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 9, Number 1, January 1996 ZARISKI GEOMETRIES EHUD HRUSHOVSKI AND BORIS ZILBER 1. Introduction Let k be an algebraically closed field. The set of ordered n-tuples from k is viewed as an n-dimensional space; a subset described by the vanishing of a polynomial, or a family of polynomials, is called an algebraic set,oraZariski closed set. Algebraic geometry describes the behavior of these sets. The goal of this paper is a converse. We start with a set X, together with a collection of subsets of the powers of X, satisfying certain axioms of a geometric nature. We then show that the points of X can be identified with a curve over an algebraically closed field k,insuchaway that the given collection of sets is precisely the family of Zariski closed ones. This project is of course familiar if one allows only subsets of kn defined by linear equations. The reader is referred to E. Artin’s Geometric algebra, Introduction to Chapter II [AE], paraphrased above, which could serve as a better introduction to the present paper. As customary we use topological language to describe our axioms for the alge- braic sets. We recall some pertinent terms (see e.g. [Ha]). A topological space is Noetherian if it has the descending chain condition on closed subsets. A closed set is irreducible if it is not the union of two proper closed subsets. If X is Noetherian, then every closed set can be written as a finite union of irreducible closed sets. These are uniquely determined (provided no one is a subset of the other), and are called the irreducible components of the given set. We say that X has dimension n if n is the maximal length of a chain of closed irreducible sets C0 C2 Cn (proper inclusions). The dimension of a closed set is its dimension⊂ as a subspace⊂···⊂ of X.Amapfis closed if the image of a closed set is closed. If X is irreducible and a property P holds for all elements of X outside of a proper closed subset, we say that P holds generically on X, or that a generic point satisfies P .IfC E Y and a E,weletC(a)= y Y :(a, y) C . ⊆ × ∈ { ∈ ∈ } Definition. A Zariski geometry on a set X is a topology on Xn for each n,satis- fying: (Z0) Let fi be a constant map (fi(x1,...,xn)=c) or a projection n m (fi(x1,...,xn)=xj(i)). Let f(x)=(f1(x),...,fm(x)). Then f : X X is n → continuous. The diagonals xi = xj of X are closed. Received by the editors November 11, 1992 and, in revised form, September 1, 1993. 1991 Mathematics Subject Classification. Primary 03C68, 14A99; Secondary 03C65, 14E10. The first author was supported by National Science Foundation grant 9106711-DMS. c 1996 American Mathematical Society 1 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 2 EHUD HRUSHOVSKI AND BORIS ZILBER (Z1) Let C be a closed irreducible subset of Xn,andletπbe the projection to Xk. Then there exists a proper closed subset F of cl(πC) such that πC cl(πC) F. (Z2) X is irreducible and uniformly one-dimensional:ifC Xn⊇ Xis closed,− then for some m, for all a Xn, C(a)=Xor C(a) m. ⊆ × (Z3) (Dimension theorem)∈ dim(Xn) n.Let| U|≤be a closed irreducible subset n ≤ of X ,andletTij be the diagonal xi = xj. Then every component of U Tij has dimension dim(U) 1. ∩ ≥ − Comments. (Z0) is needed in order to relate the various topologies; it replaces the classical understanding that the topology on Xn is the product topology. X is called complete if all projection maps are closed. Note that (Z1) then follows trivially. We prefer not to assume completeness axiomatically, since we wish not to exclude affine models. From a model theoretic point of view (Z1) is just “quantifier elimination”; this will be explained in 2. (Z2) states that X is one dimensional,§ i.e., every proper closed set is finite, but in a way uniform for families of closed subsets. (Z3) is the key structural condition. It plays the dual role of giving a basic dimension theory for closed sets, and of allowing later a theory of specializations, leading up to what is implicitly a construction of a tangent space. (This suggests that (Z3) can be viewed as a smoothness condition on X; see the remark following 1.1 below.) The axioms we have so far are valid also in the linear situation, where k is identified with a field k, and a closed subset of kn is one defined by linear equations. In the linear case, k need not be algebraically closed or even commutative; we refer to the resulting structure as the linear Zariski geometry over k. To rule out this class we assert the existence of a rich enough family of plane curves. By a plane curve over X we mean an irreducible one-dimensional subset of X2. A family of plane curves consists of a closed irreducible set E Xn (parametrizing the family), and a closed irreducible C E X2, such that ⊆C(e) is a plane curve for generic e E. ⊆ × ∈ Definition. A Zariski geometry X is very ample if there is a family C E X2 of plane curves such that: ⊆ × (i) For generic (a, b) X2 there exists a curve C(e) passing through a, b. (ii) For any a, b X2∈there exists e E such that C(e) passes through just one of a, b. ∈ ∈ If only (i) holds, X is called ample. Any smooth algebraic curve C can be viewed as a Zariski geometry. One takes the closed subsets to be the Zariski closed subsets of Cn for each n.Ourmain result is the converse statement. Theorem A. Let X be a very ample Zariski geometry. Then there exists a smooth curve C over an algebraically closed field F , such that X, C are isomorphic as Zariski geometries. Here C is not necessarily complete (but is complete in the sense of algebraic geometry iff X is complete as a Zariski geometry). It is worth noting the following complement, showing that C, F are unique. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ZARISKI GEOMETRIES 3 Proposition 1.1. Let C be a smooth curve over an algebraically closed field F , and C0 be a smooth curve over a field F 0.Supposeh:C C0is an isomorphism of Zariski geometries (the induced map on Cn is a homeomorphism→ for each n). Then there exists an isomorphism of fields hF : F F 0.IfF, F 0 are identified via → hF ,thenhbecomes an isomorphism of algebraic varieties. Remark. Let C be a curve, and suppose that the collection of Zariski closed subsets of Cn makes it a Zariski geometry. Then C is nonsingular, or at worst has only cusps for singularities, in the sense that the map f : C C from the normalization of C to C is bijective. This follows from 4.7, 4.8 to be→ proved below. (Resolution of cusps is irrelevant from the point of view of the Zariskie structure; it is natural to consider only curves for which bijective coverings are isomorphisms, and then the Zariski condition (Z3) implies normality, hence smoothness.) An arbitrary Zariski geometry can also be closely analyzed. If X is ample, but not very ample, it can be shown to be in a certain sense a finite cover of the projective line over an algebraically closed field (Theorem B). However no analog of the Riemann existence theorem is valid here; there exist finite covers of the projective line which do not arise from algebraic curves (Theorem C). If X is not ample, but in a certain technical sense nondegenerate, then there exists a division ring k canonically associated with X,andXis closely related to the linear Zariski geometry over K. One shows that X interprets an Abelian group A ([Hr2]), and that the constructible subsets of An are Boolean combinations of cosets ([HP]). A precise structure theorem is then obtained for A ([HL]). (This paper uses the constructible category, described in 2, but the result can be converted to the Zariski framework, since one can identify§ the closed sets among the constructible ones: they are the finite unions of cosets of constructible subgroups of An.) X is called degenerate if the only nonconstant families of plane curves consist of curves a X and X a ; the degenerate Zariski geometries have not been studied in{ detail.}× In this paper×{ } we restrict ourselves to the ample case. One can also define Zariski geometries of dimension higher than one (see 3.10). Then any compact complex analytic manifold can be viewed as a Zariski geometry, the closed sets being the closed analytic subvarieties. The theorems proved here are geared for such higher-dimensional applications. (This is part of the reason for insisting on removing the completeness assumption.) We refer the reader to [HZ] for some statements of this type. In higher dimensions, one may have degen- erate, ample, and nonample parts of the same geometry; they are not in general easily separable. Here we will consider higher-dimensional objects only insofar as they arise from, and are needed for the analysis of, one-dimensional ones. (Model theoretically, these are called “almost strongly minimal”; they are characterized by the existence of enough multi-valued maps into a one-dimensional geometry to separate points.