Hilbert-Kunz multiplicity and F-signature can disagree

2025/08/27 by Lee, Seungsu, Pande, Suchitra, Simpson, Austyn
#Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.2508.19985

Abstract

We compute the F-signature function of the ample cone of any nontrivial ruled surface over ℙ¹_k where k is an algebraically closed field of prime characteristic. As an application, we construct a Noetherian F-finite strongly F-regular ring R of prime characteristic admitting two maximal ideals \mathfrakn₁,\mathfrakn₂∈ Spec R at which the Hilbert-Kunz multiplicity and F-signature measure different singularities; that is, e_HK(R_\mathfrakn₁)<e_HK(R_\mathfrakn₂) and s(R_\mathfrakn₁)

Citations

• Hilbert-Kunz Multiplicity of Fibers and Bertini Theorems
• Continuity of Hilbert-Kunz multiplicity and F-signature
• Korff F-signatures of Hirzebruch surfaces
• Hilbert-Kunz functions of a Hirzebruch surface
• Lower bounds for Hilbert-Kunz multiplicities in local rings of fixed dimension
• Two theorems about maximal Cohen-Macaulay modules
• The F-signature and strong F-regularity

Related

• Lower bounds on Hilbert--Kunz multiplicities and maximal F-signatures
• Continuity of Hilbert-Kunz multiplicity and F-signature
• A function determined by a hypersurface of positive characteristic
• Perfectoid signature, perfectoid Hilbert-Kunz multiplicity, and an application to local fundamental groups
• Limits of F-invariants and Riemann-Stieltjes integral