Essential dimensions of algebraic groups and a resolution theorem for G-varieties

Abstract

Let G be an algebraic group and let X be a generically free G-variety. We show that X can be transformed, by a sequence of blowups with smooth G-equivariant centers, into a G-variety X′ with the following property: the stabilizer of every point of X′ is isomorphic to a semidirect product 17 ⋊ A of a unipotent group U and a diagonalizable group A. As an application of this result, we prove new lower bounds on essential dimensions of some algebraic groups. We also show that certain polynomials in one variable cannot be simplified by a Tschirnhaus transformation.