Cohomological and Geometric Approaches to Rationality Problems

Rationality problems link algebra to geometry. The difficulties involved depend on the transcendence degree over the ground field, or geometrically, on the dimension of the variety. A major success in 19th century algebraic geometry was a complete solution of the rationality problem in dimensions one and two over algebraically closed ground fields of characteristic zero. These advances have led to many interdisciplinary applications of algebraic geometry.

This comprehensive text consists of surveys and research papers by leading specialists in the field. Topics discussed include the rationality of quotient spaces, cohomological invariants of finite groups of Lie type, rationality of moduli spaces of curves, and rational points on algebraic varieties.

This volume is intended for research mathematicians and graduate students interested in algebraic geometry, and specifically in rationality problems.

I. Bauer

C. Böhning

F. Bogomolov

F. Catanese

I. Cheltsov

N. Hoffmann

S.-J. Hu

M.-C. Kang

L. Katzarkov

B. Kunyavskii

A. Kuznetsov

J. Park

T. Petrov

Yu. G. Prokhorov

A.V. Pukhlikov

Yu. Tschinkel

Résumé

Rationality problems link algebra to geometry, and the difficulties involved depend on the transcendence degree of $K$ over $k$, or geometrically, on the dimension of the variety. A major success in 19th century algebraic geometry was a complete solution of the rationality problem in dimensions one and two over algebraically closed ground fields of characteristic zero. Such advances has led to many interdisciplinary applications to algebraic geometry.

This comprehensive book consists of surveys of research papers by leading specialists in the field and gives indications for future research in rationality problems. Topics discussed include the rationality of quotient spaces, cohomological invariants of quasi-simple Lie type groups, rationality of the moduli space of curves, and rational points on algebraic varieties.

This volume is intended for researchers, mathematicians, and graduate students interested in algebraic geometry, and specifically in rationality problems.

Contributors: F. Bogomolov; T. Petrov; Y. Tschinkel; Ch. Böhning; G. Catanese; I. Cheltsov; J. Park; N. Hoffmann; S. J. Hu; M. C. Kang; L. Katzarkov; Y. Prokhorov; A. Pukhlikov

Contenu
The Rationality of Certain Moduli Spaces of Curves of Genus 3.- The Rationality of the Moduli Space of Curves of Genus 3 after P. Katsylo.- Unramified Cohomology of Finite Groups of Lie Type.- Sextic Double Solids.- Moduli Stacks of Vector Bundles on Curves and the King#x2013;Schofield Rationality Proof.- Noether#x2019;s Problem for Some -Groups.- Generalized Homological Mirror Symmetry and Rationality Questions.- The Bogomolov Multiplier of Finite Simple Groups.- Derived Categories of Cubic Fourfolds.- Fields of Invariants of Finite Linear Groups.- The Rationality Problem and Birational Rigidity.