Asymptotic socle behaviors for cones over curves in positive characteristic

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 $I$.

Keywords:

Frobenius power, socle, diagonal $F$-threshold, semistability, strong semistability, syzygy bundle Hilbert-Kunz multiplicity,

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