Algebraic Geometry Seminar: Alex Duncan (UBC)

Informally, the essential dimension of a finite group is the minimal number of parameters required to describe any of its actions. It has connections to Galois cohomology and several open problems in algebra. I will discuss how one can use techniques from birational geometry to compute this invariant and indicate some of its applications to the Noether Problem, inverse Galois theory, and the simplification of polynomials.

Building on I. Dolgachev and V. Iskovskikh's recent work classifying finite subgroups of the plane Cremona group, I will classify all finite groups of essential dimension 2. In addition, I show that the symmetric group of degree 7 has essential dimension 4 using Yu. Prokhorov's classification of all finite simple groups with faithful actions on rationally connected threefolds.