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 φ defined over an arbitrary field 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 φ, provided that the degree over F of any finite splitting field of φ is divisible by 2n; this property allows us to prove that the function field of X has the minimal transcendence degree among all generic splitting fields of φ. 1. Results Let F be an arbitrary field of characteristic 6 = 2, φ a nondegenerate (2n + 1)-dimensional quadratic form over F (with n> 1), X the orthogonal grassmannian of n-dimensional totally isotropic subspaces of φ. 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 field extension E/F is called a splitting field of φ if the Witt inde