## Zariski geometries (1996)

Venue: | Journal of the American Mathematical Society |

Citations: | 44 - 2 self |

### BibTeX

@ARTICLE{Hrushovski96zariskigeometries,

author = {Ehud Hrushovski and Boris Zilber},

title = {Zariski geometries},

journal = {Journal of the American Mathematical Society},

year = {1996},

volume = {9},

pages = {1--56}

}

### Years of Citing Articles

### OpenURL

### Abstract

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.

### Citations

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Algebraic Geometry
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(Show Context)
Citation Context ...bove, which could serve as a better introduction to the present paper. As customary we use topological language to describe our axioms for the algebraic 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, th... |

1354 |
Principles of algebraic geometry
- Griffiths, Harris
- 1994
(Show Context)
Citation Context ...Thus M is a finite covering of the complex torus S and hence is itself a complex torus. Theorem 3 resembles Chow’s theorem that a closed analytic subvariety of projective space must be algebraic (see =-=[GH]-=-). Indeed a somewhat more general result can easily be deduced from the theorem. Proposition 2. Let X be a complex algebraic variety. View X as a complex analytic space (as in [HA, Appendix 1]). Then ... |

450 |
Model Theory
- Chang, Keisler
- 1990
(Show Context)
Citation Context ... some basic facts concerning first-order structures. However in certain sections we will require more familiarity with these notions than can probably be attained here. We refer the reader to [FJ] or =-=[CK]-=- for the general notions, and to [Pi1] for results connected with Morley’s dimension for definable sets. Structures and maps. A structure is a set M together with a collection of subsets of M n for ea... |

152 | Algebraic Geometry - Harris - 1992 |

115 |
Geometric Algebra. Interscience
- Artin
- 1957
(Show Context)
Citation Context ...ariski 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 algebraic sets. We recall some pertinent t... |

39 |
A new strongly minimal set
- Hrushovski
- 1993
(Show Context)
Citation Context ...gly minimal sets not satisfying it are well understood). See [Z, HL]. However, it is not clear how to state (Z3) in terms of constructible sets alone. In the absence of such an axiom, it was shown in =-=[Hr1]-=- that the analog of Theorem 2 is false and in [Hr2] that the analog of Theorem 3 also fails. We note that Macintyre (see [Mac]) characterized the strongly minimal fields as the algebraically closed fi... |

35 |
Categoricity in Power
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(Show Context)
Citation Context ...tures, ones fully described by their cardinality and first-order theory. (Thus the complex field is determined by its cardinality, and being an algebraically closed field of characteristic 0.) Morley =-=[Mo]-=- showed the existence of a dimension theory for the definable sets in uncountable categorical structures; Baldwin and Lachlan [BL] discovered that such structures are controlled by strongly minimal se... |

28 |
Introduction to Algebraic Geometry. Interscience
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(Show Context)
Citation Context ...cible, then it is Tbirreducible. Proof. We use the following algebraic facts; all refer to Ta. The reader is referred to any book on algebraic geometry, e.g. Lang’s Introduction to algebraic geometry =-=[L]-=-. (i) There exists a smooth curve C ′ ⊆ P 3 and a morphism from C ′ onto C. (ii) If C is a smooth curve, there exists an integer g associated with C,thegenus of C, with the following properties.s38 EH... |

15 |
Weakly normal groups, in: Logic Colloquium 85
- Hrushovski, Pillay
- 1987
(Show Context)
Citation Context ...h 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 A n 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 iden... |

14 |
The structure of models of uncountably categorical theories
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(Show Context)
Citation Context ...le sets in uncountable categorical structures; Baldwin and Lachlan [BL] discovered that such structures are controlled by strongly minimal sets within them. It was suggested by the second author (see =-=[Z]-=-) that examples from algebraic geometry have some importance in this context. A dividing line was introduced among the strongly minimal sets: Definition. A strongly minimal set D is locally modular if... |

12 |
On algebraic groups of transformations
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(Show Context)
Citation Context ...G be a definable group in a Zariski geometry D. Then G may be endowed with a manifold structure, in such a way that multiplication G 2 → G and inversion G → G are morphisms. Proof. This is similar to =-=[W]-=-. We first find a definable map g0 : V0 → G, where V0is a regular subset of some special sort, g0 is injective, and g0V0 has the same Morley rank and degree as G. Wemaytakeg0to be the identity. By red... |

11 |
Groups of small Morley rank
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- 1979
(Show Context)
Citation Context ... (fijk = ai(cj + bk)) but also from an Abelian group structure (fijk = ai + bj + ck). Probably these are the only possibilities. Proof of 6.3. We prove 6.3(c). The other clauses follow using [Hr6] or =-=[Ch]-=-. We will use the following general remark. The same proof is valid in any homogeneous geometry, which has been sufficiently localized. Claim 1. Suppose rk(c/d1d2) =rk(c/d1) =rk(c/d2). Then there exis... |

8 |
Field arithmetic
- Fried, Jarden
- 1986
(Show Context)
Citation Context ...ist here some basic facts concerning first-order structures. However in certain sections we will require more familiarity with these notions than can probably be attained here. We refer the reader to =-=[FJ]-=- or [CK] for the general notions, and to [Pi1] for results connected with Morley’s dimension for definable sets. Structures and maps. A structure is a set M together with a collection of subsets of M ... |

8 | Unimodular minimal structures - Hrushovski - 1992 |

4 | Algebraic spaces - Artin - 1971 |

4 | Projective planes in algebraically closed fields
- Evans, Hrushovski
- 1991
(Show Context)
Citation Context ...entsZARISKI GEOMETRIES 27 true of M will also be true of M ′ . It turns out that the simplest such statement suffices to determine M ′ as a pseudo-action. The following result is from [Hr5]; see also =-=[EH]-=- and [B]. For the rest of this section we assume D is a strongly minimal set, or just a minimal set within some saturated structure, in which dcl(∅)=acl(∅) is infinite. In other words for every defina... |

3 |
On ! 1 -categorical theories of fields, Fund
- Macintyre
- 1971
(Show Context)
Citation Context ..., it must be closed (since it is dense open in its closure). As in (a) we see it must be irreducible. Thus h is an isomorphism of manifolds. (c) Infinite definable fields are automatically connected (=-=[Ma]-=-), additively and multiplicatively. Now let F be a definable field. Applying 5.11 to the additive group structure of F, wemaytakeFto have a manifold structure, so that the graph of addition is closed.... |

2 |
On strongly minimal sets,J.SymbolicLogic36
- Baldwin, Lachlan
- 1971
(Show Context)
Citation Context ... and being an algebraically closed field of characteristic 0.) Morley [Mo] showed the existence of a dimension theory for the definable sets in uncountable categorical structures; Baldwin and Lachlan =-=[BL]-=- discovered that such structures are controlled by strongly minimal sets within them. It was suggested by the second author (see [Z]) that examples from algebraic geometry have some importance in this... |

2 |
Fundamentals of forking, Ann
- Harnik, Harrington
- 1984
(Show Context)
Citation Context ...the family is the Morley rank P ∗ . 3.9. There exists C ∗ ⊆ P ∗ × Y such that for a ∈ P, C(a) ≡k C ∗ (a/ ≡). Thus any given family can be replaced with a normal one, for which ≡ is the identity. (Cf. =-=[HH]-=-, normalization theorem.) Z-structures. 3.10. Let M be a structure of finite Morley rank, together with a distinguished generation family of definable sets, referred to as closed. WecallMaZ-structure ... |

2 |
An introduction to stability theory, Oxford Univ
- Pillay
- 1983
(Show Context)
Citation Context ...der structures. However in certain sections we will require more familiarity with these notions than can probably be attained here. We refer the reader to [FJ] or [CK] for the general notions, and to =-=[Pi1]-=- for results connected with Morley’s dimension for definable sets. Structures and maps. A structure is a set M together with a collection of subsets of M n for each n (the “0-definable sets”) closed u... |

2 |
Missionary Mathematics
- Poizat
- 1983
(Show Context)
Citation Context ...pretable over C ′ . Now since K ′ is interpretable over C ′ and C ′ over F ′ ,K ′ is also interpretable over F ′ , perhaps by a different formula than the one interpreting K over F. At all events (by =-=[Po]-=-) there exists an F ′ -definable field isomorphism g ′ : F ′ → K ′ . Let p be the characteristic exponent, and let Fr(x) =x p be the Frobenius map (on any of the fields involved). Extend h to F by the... |

1 |
Hrushovski’s group configuration, The Model Theory of Groups
- Bouscaren
- 1989
(Show Context)
Citation Context ...KI GEOMETRIES 27 true of M will also be true of M ′ . It turns out that the simplest such statement suffices to determine M ′ as a pseudo-action. The following result is from [Hr5]; see also [EH] and =-=[B]-=-. For the rest of this section we assume D is a strongly minimal set, or just a minimal set within some saturated structure, in which dcl(∅)=acl(∅) is infinite. In other words for every definable subs... |

1 |
Locally modular strongly minimal sets
- Hrushovski, Loveys
(Show Context)
Citation Context ...ver K. One shows that X interprets an Abelian group A ([Hr2]), and that the constructible subsets of A n 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... |

1 |
On strongly minimal sets, Symbolic Logic 36
- Baldwin, Lachlan
- 1971
(Show Context)
Citation Context ... The motivation was the importance in model theory of certain structures, called strongly minimal sets. (In particular they form the backbone of any structure categorical in an uncountable power (see =-=[BL]-=-).)s4 EHUD HRUSHOVSKI AND BORIS ZILBER Definition 3. A structure is an infinite set D together with a collection of subsets of D n (n = 1, 2, . . .) closed under intersections, complements, projection... |

1 |
Embeddings of matroids in fields of prime characteristic
- Evans, Hrushovski
(Show Context)
Citation Context ...own that given a trace of associativity, there exists a group G such that f is conjugate (by a multivalued correspondence between the given set and G) to the single-valued function xy −1 z of G. (See =-=[EH]-=-.) Applying the theorem to the two-dimensional group of affine transformations of a field F, we obtain the following higher-dimensional analog: Proposition 4. Let cij (i, j = 1, 2, . . .) be a symmetr... |