Splitting fields of G-varieties

Abstract

Galois extension then Gal(L/K) is called a splitting group of X.We prove a lower bound on the size of a splitting field of X in terms of fixed points of nontoral abelian subgroups of G. A similar result holds for splitting groups. We give a number of applications, including a new construction of noncrossed product division algebras.

Resolving G -torsors by abelian base extensions

“…By our construction F G as G-varieties (here G is viewed as a G-variety with respect to the left G-action). We conclude that H has a fixed point in G. Since G has G as a G-invariant dense open subset, it is split as a G-variety (i.e., it represents the trivial class in H 1 (k, G)), [RY2,Lemma 4.3] now tells us that H is toral. This shows that α has trivial fixed point obstruction, thus completing the proof of Proposition 8.1.…”

“…Assume the contrary: gv = v for some v ∈ V . By [RY2,Theorem 9.3] (with s = 1 and H 1 = g ), after performing a sequence of blowups with smooth G-invariant centers on V , we may assume that the fixed point locus V g of g contains a divisor D ⊂ V . If R = O X,π(D) is the local ring of the divisor π(D) in X then α does not lie in the image of the natural morphism H 1 (R, G) → H 1 (K, G), a contradiction.…”

“…[RY3,Introduction]. Note that after birationally modifying V , we may assume that V is smooth and complete (or even projective, see, e.g., [RY2,Proposition 2.2]), and that the fixed point obstruction can be detected on any such model. In other words, if V and V are smooth complete birationally isomorphic Gvarieties then V H = ∅ if and only if (V ) H = ∅ for any diagonalizable subgroup H ⊂ G; see [RY1,Proposition A2].…”

“…If α is split (i.e., α = 1 in H 1 (K, G)) then by [RY2,Lemma 4.3] α has trivial fixed point obstruction. We will now extend this result as follows.…”

Pseudo-reflection groups and essential dimension

“…. , p r be the prime divisors of |G| and d = max rank(G pi ), as i ranges from 1 to r. By Reichstein and Youssin [37, Theorem 8.6], there exists a faithful primitive d-dimensional G-variety Y with smooth k-points y 1 , . .…”