Read e-book online Birationally Rigid Varieties: Mathematical Analysis and PDF

By Aleksandr Pukhlikov

ISBN-10: 0821894765

ISBN-13: 9780821894767

Birational tension is a notable and mysterious phenomenon in higher-dimensional algebraic geometry. It seems that definite ordinary households of algebraic forms (for instance, three-d quartics) belong to an analogous category kind because the projective area yet have greatly various birational geometric homes. particularly, they admit no non-trivial birational self-maps and can't be fibred into rational forms by means of a rational map. The origins of the idea of birational tension are within the paintings of Max Noether and Fano; even though, it used to be purely in 1970 that Iskovskikh and Manin proved birational superrigidity of quartic three-folds. This publication provides a scientific exposition of, and a entire creation to, the idea of birational tension, providing in a uniform approach, rules, ideas, and effects that to this point may perhaps in basic terms be present in magazine papers. the new speedy development in birational geometry and the widening interplay with the neighboring parts generate the transforming into curiosity to the rigidity-type difficulties and effects. The e-book brings the reader to the frontline of present examine. it really is basically addressed to algebraic geometers, either researchers and graduate scholars, yet is usually obtainable for a much wider viewers of mathematicians acquainted with the fundamentals of algebraic geometry

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The first attempts to apply this method to varieties of higher degree (the complete intersection V2·3 ⊂ P5 of a quadric and a cubic [I80]) and singular varieties (three-dimensional quartics with a double point) were not completed; see Notes and references for Chapter 2. At the same time, from the geometric point of view, varieties of higher degree are more interesting; there are not too many Fano varieties of small degree. It seemed for a long time that the method developed in [IM] makes it possible to obtain isolated results of exceptional type only [I80], whereas the majority of Fano varieties (the more so, in higher dimensions) are out of reach for this approach.

1. CANONICAL ADJUNCTION 43 Proof. This is obvious: we have just listed the possible cases.

The Sarkisov theorem showed once again that the very rationality problem needs to be modified to develop an adequate higher-dimensional theory, and confirmed the direction, in which this generalization was to be sought. 3. The Sarkisov theorem on conic bundles. Let S be a smooth projective variety of dimension dim S ≥ 2, ρ : E → S an (algebraic) vector bundle of rank 3, ρ : P(E) → S its projectivization, that is, a locally trivial P2 -bundle over S. A hypersurface V ⊂ P(E), equipped with the natural projection π : V → S, π = ρ | V , is called a conic bundle over S, if every fibre π −1 (s) ⊂ P2 = ρ−1 (s) is a conic in P2 .

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Birationally Rigid Varieties: Mathematical Analysis and Asymptotics by Aleksandr Pukhlikov


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