Canonical Dimension Of Orthogonal Groups

Abstract

We prove Berhuy{Reichstein's conjecture on the canonical dimension of orthogonal groups showing that for any integer n> 1, the canonical dimension of SO2n+1 and of SO2n+2 is equal to n(n + 1)=2. More precisely, for a given (2n + 1)-dimensional quadratic form dened over an arbitrary eld F of characteristic 6 2, we establish a certain property of the correspondences on the orthogonal grassmannian X of n-dimensional totally isotropic subspaces of , provided that the degree over F of any nite splitting eld of is divisible by 2 n ; this property allows us to prove that the function eld of X has the minimal transcendence degree among all generic splitting elds of . 1. Results Let F be an arbitrary eld of characteristic 6 2, a nondegenerate (2n + 1)- dimensional quadratic form over F (with n > 1), X the orthogonal grassmannian of n-dimensional totally isotropic subspaces of . The variety X is projective, smooth, and geometrically connected; dim X = n(n + 1)=2. We write d(X) for the greatest common divisor of the degrees of all closed points on X. In this paper, a eld extension E=F is called a splitting eld of if the Witt index (see (10) for the denition of the Witt index of a quadratic form) of the form E is maximal (i.e., equal to n). Note that a eld extension E=F is a splitting eld of if and only if the set X(E) is nonempty. We write d( ) for the greatest common divisor of the degrees of all nite splitting elds of . Clearly, d( ) = d(X). Moreover, this integer is a power of 2 not exceeding 2 n . The equality d( ) = 2 n holds if, for example, the even Cliord algebra C0( ) of the