Essential Dimensions of Algebraic Groups and a Resolution Theorem for $G$-Varieties
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
14L30 - Group actions on varieties or schemes (quotients) [See also 13A50, 14L24, 14M17]
14E15 - Global theory and resolution of singularities [See also 14B05, 32S20, 32S45]
14E05 - Rational and birational maps
12E05 - Polynomials (irreducibility, etc.)
20G10 - Cohomology theory