Birational isomorphisms between twisted group actions

摘要

Let X be an algebraic variety with a generically free action of a connected algebraic group G. Given an automorphism phi:G -> G, we will denote by X-phi the same variety X with the G-action given by g:x -> phi(g) (.) x. We construct examples of G-varieties X such that X and X-phi are not Gequivariantly isomorphic. The problem of whether or not such examples can exist in the case where X is a vector space with a generically free linear action, remains open. On the other hand, we prove that X and X-phi are always stably birationally isomorphic, i.e., X x A(m) and X phi x A(m) are G-equivariantly birationally isornorphic for a suitable m >= 0.