Multiplicities of graded families of linear series and ideals

2013/01/23 by Steven Dale Cutkosky, Cutkosky, Steven Dale · 5 citations
Mathematics · #13D40 #14C40 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #FOS: Mathematics #Meromorphic and Entire Functions #math.AC #math.AG #msc:13D40 #msc:14C40

paper · pdf · doi:10.48550/arxiv.1301.5613

33 pages. In this version, the applications to epsilon type multiplicities in the final section are extended to hold on arbitrary unramified local rings

openalex publication_date 2013/01/23 · arxiv created 2013/01/31 · arxiv updated 2013/02/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We give simple necessary and sufficient conditions on projective schemes over a field k for asymptotic limits of the growth of all graded linear series of a fixed Kodaira-Iitaka dimension to exist. We also give necessary and sufficient conditions on a local ring R of dimension d for the limit lim_i→∞(ℓ_R(R/I_n))/(n^d) to exist whenever \I_n\ is a graded family of m_R-primary ideals. We give a number of applications.

Citations

• The asymptotic growth of graded linear series on arbitrary projective schemes
• Multiplicities Associated to Graded Families of Ideals

Cited by

• Some computations of generalized Hilbert-Kunz function and multiplicity
• On generalized Hilbert-Kunz function and multiplicity
• Teissier's problem on inequalities of nef divisors over an arbitrary field
• Volumes on Complex Analytic Spaces
• Multiplicities Associated to Graded Families of Ideals
• Asymptotic Multiplicities of Graded Families of Ideals and Linear Series

Related

• The asymptotic growth of graded linear series on arbitrary projective schemes
• Growth of multiplicities of graded families of ideals
• Asymptotic Multiplicities
• Multiplicities Associated to Graded Families of Ideals
• Multiplicities of weakly graded families of ideals