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.

51 pages. Final version, accepted for publication in Advances in Mathematics