Towards Hilbert-Kunz density functions in Characteristic $0$

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 $m\ge d-1$, 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 $p\to \infty$, 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 $m\ge d-1$. 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 $m\ge d-1$. 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.

23 pages, one section added

References in corpus (3)

• The limit as p -> infinity of the Hilbert-Kunz multiplicity of sum(x_i^(d_i))
• Hilbert-Kunz density function and Hilbert-Kunz multiplicity
• Arithmetic behaviour of Frobenius semistability of syzygy bundles for plane trinomial curves

Cited by in corpus (2)

• Hilbert-Kunz density function and Hilbert-Kunz multiplicity
• Density function for the second coefficient of the Hilbert-Kunz function