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 SO_2n+1 and of SO_2n+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 ≠ 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.