Asymptotic socle behaviors for cones over curves in positive characteristic

Abstract

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