Model Theory and Algebraic Geometry: An introduction to E. Hrushovski’s proof of the geometric Mordell-Lang conjecture

Elisabeth Bouscaren (auth.), Elisabeth Bouscaren (eds.)3540648631, 9783540648635

This introduction to the recent exciting developments in the applications of model theory to algebraic geometry, illustrated by E. Hrushovski’s model-theoretic proof of the geometric Mordell-Lang Conjecture starts from very basic background and works up to the detailed exposition of Hrushovski’s proof, explaining the necessary tools and results from stability theory on the way. The first chapter is an informal introduction to model theory itself, making the book accessible (with a little effort) to readers with no previous knowledge of model theory. The authors have collaborated closely to achieve a coherent and self- contained presentation, whereby the completeness of exposition of the chapters varies according to the existence of other good references, but comments and examples are always provided to give the reader some intuitive understanding of the subject.

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Table of contents :
Front Matter….Pages I-XV
Introduction to model theory….Pages 1-18
Introduction to stability theory and Morley rank….Pages 19-44
Omega-stable groups….Pages 45-59
Model theory of algebraically closed fields….Pages 61-84
Introduction to abelian varieties and the Mordell-Lang conjecture….Pages 85-100
The model-theoretic content of Lang’s conjecture….Pages 101-106
Zariski geometries….Pages 107-128
Differentially closed fields….Pages 129-141
Separably closed fields….Pages 143-176
Proof of the Mordell-Lang conjecture for function fields….Pages 177-196
Proof of Manin’s theorem by reduction to positive characteristic….Pages 197-205
Back Matter….Pages 207-216