Semistability and Hilbert-Kunz multiplicities for curves

Details Semistability and Hilbert-Kunz multiplicities for curves Semistability and Hilbert-Kunz multiplicities for curves

2004-02-15

arxiv.org/abs/math/0402245v4

Mathematics

Commutative Algebra

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

Scientific paper

10.1016/j.jalgebra.2004.10.016

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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