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

Abstract

AbstractLet 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 Xʹ with the following property: the stabilizer of every point of Xʹ 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.