Towards Hilbert-Kunz density functions in Characteristic $0$

TL;DR

This paper establishes a connection between Hilbert-Kunz density functions in characteristic zero and their reductions modulo p, providing new methods to prove the existence of Hilbert-Kunz multiplicities in characteristic zero.

Contribution

It introduces a criterion linking the convergence of HK density functions in characteristic p to their limits in characteristic zero, enabling new cases to be analyzed.

Findings

Proves equivalence between limits of HK multiplicities and lengths of Frobenius powers.

Shows convergence of HK density functions in characteristic p implies convergence in characteristic zero.

Establishes existence of HK multiplicities in new geometric cases, such as Segre products of curves.

Abstract

For a pair $(R, I)$, where $R$ is a standard graded domain of dimension $d$ over an algebraically closed field of characteristic $0$ and $I$ is a graded ideal of finite colength, we prove that the existence of $\lim_{p\to \infty}e_{HK}(R_p, I_p)$ is equivalent, for any fixed $m\geq d-1$, to the existence of $\lim_{p\to \infty}\ell(R_p/I_p^{[p^m]})/p^{md}$. This we get as a consequence of Theorem 1.1: As $p\rightarrow \infty $, the convergence of the HK density function $f{(R_p, I_p)}$ is equivalent to the convergence of the truncated HK density functions $f_m(R_p, I_p)$ (in $L^{\infty}$ norm) of the {\it mod $p$ reductions} $(R_p, I_p)$, for any fixed $m\geq d-1$. In particular, to define the HK density function $f^{\infty}(R, I)$ in characteristic 0, it is enough to prove the existence of $\lim_{p\to \infty} f_m(R_p, I_p)$, for any fixed $m\geq d-1$. This allows us to prove the existence of $e_{HK}^{\infty}(R, I)$ in many new cases, {\em e.g.}, when $\mbox{Proj~R}$ is a Segre product of curves, for example.