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

http://dx.doi.org/10.4153/CJM-2000-043-5
Canad. J. Math. 52(2000), 1018-1056

  Published:2000-10-01
 Printed: Oct 2000

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 $U \sdp A$ 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.