Mathematics Colloquia and Seminars

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We will discuss some questions related to the stable rationality of quotient varieties $V/G$ where $G$ is an algebraic group and $V$ is a faithful complex linear representation of $G$. The first examples of nonrational and even nonstably rational varieties $V/G$ were obtained by showing that a birational invariant, the so-called unramified Brauer group, is nontrivial for some series of groups. In fact, this invariant coincides with the cohomological (or Grothendieck's) Brauer group of a smooth projective model for $V/G$. However, the unramified point of view enables one to dispense with the construction of an explicit smooth model, and even with the existence of such a model. This obstruction is the first of the series of birational invariants of $V/G$ constructed via group cohomology. Some recent results when $G$ is a finite simple group will be presented.