Asymptotic socle behaviors for cones over curves in positive characteristic

Abstract

This paper studies the distribution of socle degrees of $R/I^{[p^e]}$ when $e$ is large, for a homogeneous ideal $I$ in a two-dimensional standard-graded normal domain $R$ in positive characteristic $p$. We prove that the distribution is very much related to the asymptotic slopes of the syzygy bundle Syz$ I $, which have been known to determine the Hilbert-Kunz multiplicity of $I$.