Model Theory and Algebraic Geometry: An introduction to E. Hrushovski's proof of the geometric Mordell-Lang conjecture: 1696 (Lecture Notes in Mathematics) - Softcover

Softcover

ISBN 10: 3540648631 ISBN 13: 9783540648635

Verlag: Springer, 2009

Inhaltsangabe

Introduction Model theorists have often joked in recent years that the part of mathemat­ ical logic known as "pure model theory" (or stability theory), as opposed to the older and more traditional "model theory applied to algebra" , turns out to have more and more to do with other subjects ofmathematics and to yield gen­ uine applications to combinatorial geometry, differential algebra and algebraic geometry. We illustrate this by presenting the very striking application to diophantine geometry due to Ehud Hrushovski: using model theory, he has given the first proof valid in all characteristics of the "Mordell-Lang conjecture for function fields" (The Mordell-Lang conjecture for function fields, Journal AMS 9 (1996), 667-690). More recently he has also given a new (model theoretic) proof of the Manin-Mumford conjecture for semi-abelian varieties over a number field. His proofyields the first effective bound for the cardinality ofthe finite sets involved (The Manin-Mumford conjecture, preprint). There have been previous instances of applications of model theory to alge­ bra or number theory, but these appl~cations had in common the feature that their proofs used a lot of algebra (or number theory) but only very basic tools and results from the model theory side: compactness, first-order definability, elementary equivalence...

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Reseña del editor

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.

Details zur Ausgabe

• Verlag: Springer
• Erscheinungsdatum: 2009
• ISBN 10: 3540648631
• ISBN 13: 9783540648635
• Einband: Tapa blanda
• Sprache: Englisch
• Auflage: 1
• Anzahl der Seiten: 228
• Herausgeber: Bouscaren Elisabeth
• Kontakt zum Hersteller: Nicht verfügbar