Semistability and Hilbert–Kunz multiplicities for curves

2004/02/29 by V. Trivedi · 2 citations
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #Polynomial and algebraic computation #math.AC #math.AG #msc:13D40

paper · pdf · doi:10.1016/j.jalgebra.2004.10.016

Latex2e, 12 pages, final version, revised as per the referee's suggestions, to appear in Journal of Algebra

openalex publication_date 2004/12/12 · arxiv created 2004/12/13 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study Hilbert-Kunz multiplicity of non-singular curves in positive characteristic. We analyse the relationship between the Frobenius semistability of the kernel sheaf associated with the curve and its ample line bundle, and the HK multiplicity. This leads to a lower bound, achieved iff the kernel sheaf is Frobenius semistable, and otherwise to formulas for the HK multiplicity in terms of parameters measuring the failure of Frobenius semistability. As a byproduct, an explicit example of a vector bundle on a curve is given whose n-th iterated Frobenius pullback is not semistable, while its (n-1)-th such pullback is semistable, where n>0 is arbitrary.

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• Hilbert-Kunz series, F-signature series, and weak p-fractals
• On the (LC) conjecture

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