Model Theory and Algebraic Geometry

An introduction to E. Hrushovski's proof of the geometric Mordell-Lang conjecture

Buch | Softcover

XV, 216 Seiten

1998 | 1st ed. 1998. Corr. 2nd printing 1998
Springer Berlin (Verlag)
978-3-540-64863-5 (ISBN)

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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...

to model theory.- to stability theory and Morley rank.- Omega-stable groups.- Model theory of algebraically closed fields.- to abelian varieties and the Mordell-Lang conjecture.- The model-theoretic content of Lang's conjecture.- Zariski geometries.- Differentially closed fields.- Separably closed fields.- Proof of the Mordell-Lang conjecture for function fields.- Proof of Manin's theorem by reduction to positive characteristic.

Erscheint lt. Verlag	17.9.1998
Reihe/Serie	Lecture Notes in Mathematics
Zusatzinfo	XV, 216 p.
Verlagsort	Berlin
Sprache	englisch
Maße	155 x 235 mm
Gewicht	311 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie
	Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
	Mathematik / Informatik ► Mathematik ► Logik / Mengenlehre
Schlagworte	Abelian varieties • abelian variety • Algebraic Geometry • Algebraische Geometrie • Modelltheorie • model Theory • Proof
ISBN-10	3-540-64863-1 / 3540648631
ISBN-13	978-3-540-64863-5 / 9783540648635
Zustand	Neuware