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arXiv · 2109.11784

The Hilbert-Kunz density functions of quadric hypersurfaces

Vijaylaxmi Trivedi
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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.

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Vijaylaxmi Trivedi. 2021-09-24. The Hilbert-Kunz density functions of quadric hypersurfaces. https://arxiv.org/abs/2109.11784

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