The Hilbert-Kunz density functions of quadric hypersurfaces

TL;DR

This paper explicitly describes the Hilbert-Kunz density function for quadric hypersurfaces, confirming conjectures on lower bounds of Hilbert-Kunz multiplicity and its behavior with respect to characteristic.

Contribution

It provides an explicit piecewise polynomial description of the Hilbert-Kunz density function for quadrics and proves related conjectures on multiplicity bounds and characteristic dependence.

Findings

Hilbert-Kunz density function is piecewise polynomial on a subset of [0, n]

Confirms Watanabe-Yoshida conjecture on lower bounds of multiplicity in certain characteristics

Shows Hilbert-Kunz multiplicity decreases with characteristic for fixed dimension

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.