The Hilbert-Kunz density functions of quadric hypersurfaces

Abstract

We show that the Hilbert-Kunz density function of a quadric hypersurface of Krull dimension n+1 is a piecewise polynomial on a subset of [0,n], whose complement in [0,n] has measure zero. Our explicit description of the Hilbert-Kunz density function confirms a conjecture of Watanabe-Yoshida on the lower bound of the Hilbert-Kunz multiplicity of the quadric of dimension n+1, provided the characteristic is at least n−1. We also show that the Hilbert-Kunz multiplicity of a quadric of fixed dimension is an eventually strictly decreasing function of the characteristic confirming a conjecture of Yoshida.The main input comes from the classification of Arithmetically Cohen-Macaulay bundles on the projective variety defined by the quadric via matrix factorizations.