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

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

Show description

Read Online or Download Birationally Rigid Varieties: Mathematical Analysis and Asymptotics PDF

Similar algebraic geometry books

Download e-book for iPad: Algebraic spaces by Michael Artin

Those notes are in keeping with lectures given at Yale collage within the spring of 1969. Their item is to teach how algebraic features can be utilized systematically to strengthen sure notions of algebraic geometry,which tend to be handled by means of rational services through the use of projective tools. the worldwide constitution that is ordinary during this context is that of an algebraic space—a area received through gluing jointly sheets of affine schemes via algebraic features.

Get Topological Methods in Algebraic Geometry PDF

Lately new topological equipment, particularly the idea of sheaves based through J. LERAY, were utilized effectively to algebraic geometry and to the idea of features of numerous advanced variables. H. CARTAN and J. -P. SERRE have proven how primary theorems on holomorphically whole manifolds (STEIN manifolds) could be for­ mulated by way of sheaf conception.

Download PDF by William Fulton: Introduction to Intersection Theory in Algebraic Geometry

This e-book introduces a few of the major rules of contemporary intersection concept, strains their origins in classical geometry and sketches a couple of commonplace purposes. It calls for little technical history: a lot of the cloth is on the market to graduate scholars in arithmetic. A large survey, the e-book touches on many subject matters, most significantly introducing a strong new technique constructed through the writer and R.

Read e-book online Rational Points on Curves over Finite Fields: Theory and PDF

Rational issues on algebraic curves over finite fields is a key subject for algebraic geometers and coding theorists. right here, the authors relate an enormous software of such curves, particularly, to the development of low-discrepancy sequences, wanted for numerical equipment in varied parts. They sum up the theoretical paintings on algebraic curves over finite fields with many rational issues and talk about the purposes of such curves to algebraic coding idea and the development of low-discrepancy sequences.

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).

Download PDF sample

Birationally Rigid Varieties: Mathematical Analysis and Asymptotics by Aleksandr Pukhlikov


by Jeff
4.2

Rated 4.28 of 5 – based on 48 votes