Lower bounds on Hilbert--Kunz multiplicities and maximal F-signatures

Abstract

Hilbert-Kunz multiplicity and F-signature are numerical invariants of commutative rings in positive characteristic that measure severity of singularities: for a regular ring both invariants are equal to one and the converse holds under mild assumptions. A natural question is for what singular rings these invariants are closest to one. For Hilbert--Kunz multiplicity this question was first considered by the last two authors and attracted significant attention. In this paper, we study this question, i.e., an upper bound, for F-signature and revisit lower bounds on Hilbert--Kunz multiplicity.

arXiv commentAn updated version with several general improvements

References in corpus (6)

• F-signature exists
• Fundamental groups of $F$-regular singularities via $F$-signature
• Noncommutative resolution, F-blowups and D-modules
• Conic divisorial ideals of Hibi rings and their applications to non-commutative crepant resolutions
• The limit as p -> infinity of the Hilbert-Kunz multiplicity of sum(x_i^(d_i))
• Generalized F-signatures of Hibi rings