Essential Dimensions of Algebraic Groups and a Resolution Theorem for G-Varieties

Abstract

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 ʹ with the following property: the stabilizer of every point of X ʹ is isomorphic to a semidirect product U × 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.