草的几何学双原理
Géométrie birationnelle équivariante des grassmanniennes
摘要：
设k是域，A是有限维k-代数.设d为整数。记Gr（d，A）为A的d-子空间的Grassmannian，记GL_1（A）为点为A的可逆元的代数k-群。群GL_1（A）自然地作用于Gr（d，A）（通过公式g.E=gE）。本文的目的是研究这个作用的一些双有理性质。更精确地说，设r是d和dim（A）的gcd。在对A的某些假设下（若A/k为零则满足），证明了簇Gr（d，A）与Gr（r，A）在一个射影空间上的乘积（GL_1（A）在该空间上作用平凡）是双有理和等变同构的.通过扭曲，这个结果在中心单代数理论中有一些推论。例如，设B和C是两个互素度的中心单代数。则Severi-Brauer簇SB（B \otimes C）对于SB（B）\times SB（C）与正确维数的仿射空间的乘积是双有理的。这些推论是在Krashen的Amitsur猜想的广义版本的精神。
英文摘要：
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 étale), 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.
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