The lower bound on the HK multiplicities of quadric hypersurfaces

TL;DR

This paper establishes a lower bound for the Hilbert-Kunz multiplicity of odd-dimensional quadric hypersurfaces in positive characteristic, advancing the understanding of a long-standing conjecture by Watanabe-Yoshida.

Contribution

It proves a lower bound on HK multiplicities for quadrics of odd dimension, using HK density functions and ACM bundle classification, partially confirming Watanabe-Yoshida's conjecture.

Findings

Lower bound of 1 + m_d for HK multiplicity of quadrics

Use of HK density function in bounding multiplicities

Application of matrix factorizations for ACM bundle classification

Abstract

Here we prove that the Hilbert-Kunz mulitiplicity of a quadric hypersurface of dimension $d$ and odd characteristic $p\geq 2d-4$ is bounded below by $1+m_d$, where $m_d$ is the $d^{th}$ coefficient in the expansion of $\mbox{sec}+\mbox{tan}$. This proves a part of the long standing conjecture of Watanabe-Yoshida. We also give an upper bound on the HK multiplicity of such a hypersurface. We approach the question using the HK density function and the classification of ACM bundles on the smooth quadrics via matrix factorizations.