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

TL;DR

This paper introduces a resolution theorem for G-varieties involving blowups, and applies it to establish new lower bounds on the essential dimensions of algebraic groups, also discussing polynomial transformations.

Contribution

It provides a novel resolution method for G-varieties and derives new bounds on the essential dimensions of algebraic groups.

Findings

Resolved G-varieties via blowups with smooth G-equivariant centers.

Established new lower bounds on essential dimensions of certain algebraic groups.

Showed limitations of Tschirnhaus transformations on polynomial simplification.

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 of a unipotent group U and a diagonalizable group A. As an application of this and related results, 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.