On the (LC) conjecture

TL;DR

This paper proves the (LC) conjecture in specific cases, applies it to analyze generalized Hilbert-Kunz functions of curves, explores stability of sheaves, and connects these findings to broader algebraic properties.

Contribution

It establishes the (LC) conjecture in new cases and links it to stability theory and $F$-thresholds, providing new insights and applications in algebraic geometry and commutative algebra.

Findings

Proof of (LC) conjecture in certain non-trivial cases

Reproof of numerical evidence for Hilbert-Kunz functions of smooth curves

Analysis of stability and semistability of sheaves on Klein's quartic

Abstract

We prove the (LC) conjecture of Hochster and Huneke in some non-trivial cases. This has several applications. Recently, Brenner and Caminata answered a numerical evidence due to Dao and Smirnov on the shape of generalized Hilbert-Kunz functions of smooth curves. As applications, we first reprove this by a short argument. Then we give a proof of second numerical evidence predicted by Dao and Smirnov on the shape of generalized Hilbert-Kunz functions of nodal curves. Thirdly, we answer a question posted by Vraciu on the (LC) property of a proposed ring. Inspiring with the (LC) property, we present a connection to the stability theory. This leads us to investigate the stability and the strong semistability of the sheaf of relations on $\{x^2,y^2,z^2\}$ over the Klein's quartic curve. This answers questions of Brenner. After presenting a connection from (LC) to the $F$-threshold, we answer a question posted by Huneke et al. Additional applications and examples are given.