A Direct Limit for Limit Hilbert-Kunz Multiplicity for Smooth Projective Curves

TL;DR

This paper investigates whether a direct limit approach can effectively determine the limit Hilbert-Kunz multiplicity for smooth projective curves, providing new estimates and connecting to characteristic zero analogues.

Contribution

It establishes an affirmative answer for graded ideals in smooth projective curves and refines bounds on cohomology dimensions related to the limit Hilbert-Kunz multiplicity.

Findings

Confirmed the existence of a direct limit for limit Hilbert-Kunz multiplicity in smooth projective curves.

Provided bounds on cohomology dimensions of syzygy bundles as characteristic p approaches infinity.

Discussed implications for maximal ideals in diagonal hypersurfaces based on unpublished results.

Abstract

This paper concerns the question of whether a more direct limit can be used to obtain the limit Hilbert-Kunz multiplicity, a possible candidate for a characteristic zero Hilbert-Kunz multiplicity. The main goal is to establish an affirmative answer for one of the main cases for which the limit Hilbert-Kunz multiplicity is even known to exist, namely that of graded ideals in the homogeneous coordinate ring of smooth projective curves. The proof involves more careful estimates of bounds found independently by Brenner and Trivedi on the dimensions of the cohomologies of twists of the syzygy bundle as the characteristic p goes to infinity and uses asymptotic results of Trivedi on the slopes of Harder-Narasimham filtrations of Frobenius pullbacks of bundles. In view of unpublished results of Gessel and Monsky, the case of maximal ideals in diagonal hypersurfaces is also discussed in depth.