  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.
