Cohomological and Geometric Approaches to

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

Preface	p. vii
The Rationality of Certain Moduli Spaces of Curves of Genus 3	p. 1
The Rationality of the Moduli Space of Curves of Genus 3 after P. Katsylo	p. 17
Unramified Cohomology of Finite Groups of Lie Type	p. 55
Sextic Double Solids	p. 75
Moduli Stacks of Vector Bundles on Curves and the King-Schofield Rationality Proof	p. 133
Noether's Problem for Some p-Groups	p. 149
Generalized Homological Mirror Symmetry and Rationality Questions	p. 163
The Bogomolov Multiplier of Finite Simple Groups	p. 209
Derived Categories of Cubic Fourfolds	p. 219
Fields of Invariants of Finite Linear Groups	p. 245
The Rationality Problem and Birational Rigidity	p. 275
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Cohomological and Geometric Approaches to Rationality Problems

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