Irrational Hilbert-Kunz multiplicities

Abstract

We interpret Hilbert-Kunz theory of a graded ring of positive characteristic in terms of Frobenius asymptotic of cohomology of vector bundles on projective varieties. With this method we show that for almost all prime numbers there exist three-dimensional quartic hypersurface domains and modules of finite length with irrational Hilbert-Kunz multiplicity. From this we deduce that also the Hilbert-Kunz multiplicity of a local noetherian domain might be an irrational number.

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References in corpus (4)

• Semistability and Hilbert-Kunz multiplicities for curves
• The limit as p -> infinity of the Hilbert-Kunz multiplicity of sum(x_i^(d_i))
• Transcendence of some Hilbert-Kunz multiplicities (modulo a conjecture)
• Hilbert-Kunz functions of 2 x 2 determinantal rings

Cited by in corpus (8)

• On generalized Hilbert-Kunz function and multiplicity
• Frobenius Splitting in Commutative Algebra
• Hilbert-Kunz functions of surface rings of type ADE
• Local Okounkov bodies and limits in prime characteristic
• Globalizing F-invariants
• $F$-singularities: applications of characteristic $p$ methods to singularity theory
• Length of local cohomology of powers of ideals
• On the (LC) conjecture