G\'eom\'etrie birationnelle \'equivariante des grassmanniennes
Abstract
Let k be a field, and A a finite-dimensional k-algebra. Let d be an integer. Denote by Gr(d,A) the Grassmannian of d-subspaces of A (viewed as a k-vector space), and by GL1(A) the algebraic k-group whose points are invertible elements of A. The group GL1(A) acts naturally on Gr(d,A) (by the formula g.E=gE). The aim of this paper is to study some birational properties of this action. More precisely, let r be the gcd of d and dim(A). Under some hypothesis on A (satisfied if A/k is \'etale), I show that the variety Gr(d,A) is birationally and GL1(A)-equivariantly isomorphic to the product of Gr(r,A) by a projective space (on which GL1(A) acts trivially). By twisting, this result has some corollaries in the theory of central simple algebras. For instance, let B and C be two central simple algebras over k, of coprime degrees. Then the Severi-Brauer variety SB(B C) is birational to the product of SB(B) × SB(C) by an affine space of the correct dimension. These corollaries are in the spirit of Krashen's generalized version of Amitsur's conjecture.