Minimal canonical dimensions of quadratic forms

Abstract

Canonical dimension of a smooth complete connected variety is the minimal dimension of image of its rational endomorphism. The $i$-th canonical dimension of a non-degenerate quadratic form is the canonical dimension of its $i$-th orthogonal grassmannian. The maximum of a canonical dimension for quadratic forms of a fixed dimension is known to be equal to the dimension of the corresponding grassmannian. This article is about the minima of the canonical dimensions of an anisotropic quadratic form. We conjecture that they equal the canonical dimensions of an excellent anisotropic quadratic form of the same dimension and we prove it in a wide range of cases.