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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 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.

No abstract

We prove that the essential dimension of the spinor group Spin n grows exponentially with n and use this result to show that quadratic forms with trivial discriminant and Hasse-Witt invariant are more complex, in high dimensions, than previously expected.

We define and study a numerical invariant of an algebraic group action which we call the canonical dimension. We then apply the resulting theory to the problem of computing the minimal number of parameters required to define a generic hypersurface of degree d in P n−1 .