G\'eom\'etrie birationnelle \'equivariante des grassmanniennes

TL;DR

This paper investigates the birational properties of Grassmannians associated with finite-dimensional algebras, revealing their structure and implications for central simple algebras and Severi-Brauer varieties.

Contribution

It establishes a birational and equivariant isomorphism of Grassmannians to a product involving a smaller Grassmannian and a projective space, with applications to central simple algebras.

Findings

Grassmannian varieties are birationally isomorphic to products involving smaller Grassmannians.

Results apply to Severi-Brauer varieties of tensor products of central simple algebras.

Provides insights related to Krashen's generalization of Amitsur's conjecture.

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 GL_1(A) the algebraic k-group whose points are invertible elements of A. The group GL_1(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 GL_1(A)-equivariantly isomorphic to the product of Gr(r,A) by a projective space (on which GL_1(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 \otimes C) is birational to the product of SB(B) \times 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.