Model Theory and Algebraic Geometry pp 107-128 | Cite as

# Zariski geometries

## Abstract

Zariski geometries were introduced by Hrushovski and Zilber in [HrZi 96], [HrZi 93] and [Zil]. From a technical point of view this work provides a class of strongly minimal sets where Zilber’s conjecture holds (see [Zie, end of section 5]. It also provides the answer to two metamathematical questions. *How do you characterize the topological spaces that arise from the Zariski topology of an algebraic curve? Can you recover the field from the topological spaces?* The answer to these questions is provided by Theorem 3.3 below. This result plays a key role in Hrushovski’s proof of the Mordell-Lang conjecture for function fields.

## Keywords

Irreducible Component Algebraic Curve Transcendence Degree Zariski Topology Quantifier Elimination## Preview

Unable to display preview. Download preview PDF.

## References

- [AtMc]M. F. Atiyah and I. G. Macdonald,
*Introduction to Commutative Algebra*, Addison-Wesley, 1969.Google Scholar - [Bous2 89]E. Bouscaren,
*The group configuration-after E. Hrushovski*, in The Model Theory of Groups, A. Nesin and A. Pillay ed., Notre Dame University Press, 1989.Google Scholar - [Bous]E. Bouscaren,
*Proof of the Mordell-Lang conjecture for function fields*, this volume.Google Scholar - [Da]V.I. Danilov,
*Algebraic Varieties and Schemes*, in Algebraic Geometry I, I.R. Shafarevich ed., EMS 23, Springer, 1994.Google Scholar - [De]F. Delon,
*Separably closed fields*, this volume.Google Scholar - [Gr]P. Griffiths,
*Introduction to Algebraic Curves*, Translations of Math. Mon. 76, AMS (1989).Google Scholar - [Ha]R. Hartshorne,
*Algebraic Geometry*, Springer, 1977.Google Scholar - [Hr 96]E. Hrushovski,
*The Mordell-Lang conjecture for function fields*, J.AMS 9 (1996), 667–690.zbMATHMathSciNetGoogle Scholar - [HrSo]E. Hrushovski and Z. Sokolovic,
*Strongly minimal sets in differentially closed fields*, preprint.Google Scholar - [HrZi 93]E. Hrushovski and B. Zilber,
*Zariski Geometries*, Bulletin AMS 28 (1993), 315–323.zbMATHCrossRefMathSciNetGoogle Scholar - [HrZi 96]E. Hrushovski and B. Zilber,
*Zariski Geometries*, Journal AMS 9 (1996), 1–56.zbMATHMathSciNetGoogle Scholar - [Mar 96]D. Marker,
*Model Theory of Differential Fields*, Model Theory of Fields, Lecture Notes in Logic 5, Springer, 1996.Google Scholar - [MMP]D. Marker, M. Messmer and A. Pillay,
*Model Theory of Fields*, Lecture Notes in Logic 5, Springer, 1996.Google Scholar - [MaPi]D. Marker and A. Pillay,
*Reducts of*(ℂ, +,·)*which contain*+, J. Symbolic Logic 55 (1990), 1243–1251.zbMATHCrossRefMathSciNetGoogle Scholar - [Pi1 96]A. Pillay,
*Differential algebraic groups and the number of countable differentially closed fields*, Model Theory of Fields, Lecture Notes in Logic 5, Springer, 1996.Google Scholar - [Pi2 96]A. Pillay,
*Geometrical Stability Theory*, Oxford University Press, 1996.Google Scholar - [Pi1]A. Pillay,
*Model theory of algebraically closed fields*, this volume.Google Scholar - [Rab]E. Rabinovich,
*Interpreting a field in a sufficiently rich incidence system*, QMW Press (1993).Google Scholar - [Sha]I.R. Shafarevich,
*Basic Algebraic Geometry*, Springer, 1977.Google Scholar - [Wo]C. Wood,
*Differentially closed fields*, this volume.Google Scholar - [Zie]M. Ziegler,
*Introduction to stability theory and Morley rank*, this volume.Google Scholar - [Zil]B. Zilber,
*Lectures on Zariski type structures*, lecture notes (1992).Google Scholar