Hilbert-Kunz theory for nodal cubics, via sheaves

TL;DR

This paper extends the understanding of Hilbert-Kunz functions for nodal cubic curves in characteristic p, expressing the function in terms of vector bundle classification data and Frobenius pull-back effects.

Contribution

It generalizes previous results by expressing the Hilbert-Kunz function for nodal cubics using vector bundle classification data and Frobenius morphism effects.

Findings

Derived a formula for Hilbert-Kunz functions involving vector bundle data.

Connected Frobenius pull-back effects to classification of vector bundles.

Provided explicit descriptions of parameters based on curve classification.

Abstract

Suppose B=F[x,y,z]/h is the homogeneous coordinate ring of a characteristic p degree 3 irreducible plane curve C with a node. Let J be a homogeneous (x,y,z)-primary ideal and n -> e_n be the Hilbert-Kunz function of B with respect to J. Let q=p^n. When J=(x,y,z), Pardue (see R. Buchweitz, Q. Chen. Hilbert-Kunz functions of cubic curves and surfaces. J. Algebra 197 (1997). 246-267) showed that e_n=(7q^2)/3-q/3-R where R=5/3 if q is congruent to 2 (3), and is 1 otherwise. We generalize this, showing that e_n= (mu q^2) + (alpha q) - R where R only depends on q mod 3. We describe alpha and R in terms of classification data for a vector bundle on C. Igor Burban (I. Burban. Frobenius morphism and vector bundles on cycles of projective lines. 2010. arXiv 1010.0399) provided a major tool in our proof by showing how pull-back by Frobenius affects the classification data of an indecomposable vector bundle over C. We are also indebted to him for pointing us towards Y. A. Drozd, G.-M. Greuel, I. Kashuba. On Cohen-Macaulay modules on surface singularities. Mosc. Math. J. 3 (2003). 397-418, 742, in which h^0 is described in terms of these classification data.