By Aleksandr Pukhlikov

ISBN-10: 0821894765

ISBN-13: 9780821894767

Birational stress is a remarkable and mysterious phenomenon in higher-dimensional algebraic geometry. It seems that sure normal households of algebraic forms (for instance, third-dimensional quartics) belong to an identical class variety because the projective area yet have appreciably assorted birational geometric houses. particularly, they admit no non-trivial birational self-maps and can't be fibred into rational kinds through a rational map. The origins of the speculation of birational tension are within the paintings of Max Noether and Fano; despite the fact that, it was once in basic terms in 1970 that Iskovskikh and Manin proved birational superrigidity of quartic three-folds. This booklet provides a scientific exposition of, and a entire creation to, the speculation of birational stress, featuring in a uniform means, rules, recommendations, and effects that to this point may purely be present in magazine papers. the new speedy growth in birational geometry and the widening interplay with the neighboring components generate the growing to be curiosity to the rigidity-type difficulties and effects. The booklet brings the reader to the frontline of present learn. it's basically addressed to algebraic geometers, either researchers and graduate scholars, yet can be available for a much broader viewers of mathematicians accustomed to the fundamentals of algebraic geometry

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**Extra info for Birationally Rigid Varieties: Mathematical Analysis and Asymptotics**

**Sample text**

Using the methods of [Pukh98a], the following claim was proved in [Sob02]. 3. (i) There are no structures of a conic bundle on V , that is to say, RC1 (V ) = ∅. (ii) Every pencil of rational surfaces on V can be transformed by a map χ ∈ Bir V into the pencil of ﬁbres of the projection π: RC 2 (V ) = {π}. (iii) The group of birational self-maps is the free product, Bir V = Bir(V /P1 ) ∗ τ . Therefore, modulo the ﬁbrewise birational equivalence there is a unique rationally connected structure on V , the projection π.

Another example is given by iterated double covers [Pukh03]. 3 are realized as Fano complete intersections in weighted projective spaces. 1. 1. A non-singular Fano complete intersection of index one and dimension ≥ 4 in a weighted projective space is birationally rigid, of dimension ≥ 5 birationally superrigid. To conclude, let us recall that for a primitive Fano variety birational (super)rigidity immediately implies the absence of non-trivial structures of a rationally connected ﬁbre space. 2.

On a rationally connected variety there are no diﬀerential forms, but one can say a lot more: such varieties satisfy the classical condition of termination of canonical adjunction: for any divisor D the 0, since the canonical class is negative linear system |D + N KV | is empty for N on every family of rational curves sweeping out V . Starting with the work of the Italian school, it was understood that of crucial importance is the parameter that measures the moment of termination of adjunction, in the modern terminology, the threshold of canonical adjunction (discussed in detail below).

### Birationally Rigid Varieties: Mathematical Analysis and Asymptotics by Aleksandr Pukhlikov

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