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

Abstract

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.

Essential dimension of central simple algebras

“…As every central simple algebra A of degree p is cyclic over a finite field extension of degree prime to p, A can be given by two parameters (see Section 2.1). In fact, ed p (Alg p ) = 2 by [13,Lemma 8.5.7]. By Albert's theorem, every algebra in Alg 4 ,2 is biquaternion and hence can be given by 4 parameters.…”

Splitting fields of G-varieties

“…By [RY,Proposition 7.1], X is birationally isomorphic to a complete G-variety. (Note that the proof of [RY,Proposition 7.1] is based on Sumihiro's equivariant completion theorem.) Thus we may assume without loss of generality that X is complete.…”

“…Moreover, such points survive under dominant rational G-equivariant maps and under certain G-equivariant covers; see [RY,Section 5 and Appendix]. We review and further extend these results in Section 2; see Proposition 2.2 and Theorems 2.5, 2.6 and 2.…”

Untitled

“…Along the way we prove an equivariant form of Chow's lemma (Proposition 2), generalizing a theorem of Sumihiro ([Su,Theorem 2]). For applications of Theorem 1, see [RY1,Remark 5.4] and [RY2,Remark 6.3].…”