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Asymptotic Multiplicities of Graded Families of Ideals and Linear Series · arXivDesk

1508.01938Aug 8, 201545 pages. This paper is a compilation of the two arXiv preprints "The asymptotic growth of graded linear series on arbitrary projective schemes", arXiv:1212.6186, and "Multilicities of graded families of linear series", arXiv:1301.5613, with some additional material

Asymptotic Multiplicities of Graded Families of Ideals and Linear Series

Steven Dale Cutkosky

Commutative Algebra Algebraic Geometry

Abstract

We find simple necessary and sufficient conditions on a local ring $R$ of dimension $d$ for the limit $\lim_{i\rightarrow\infty}\frac{\ell_R(R/I_n)}{n^d}$

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to exist whenever

\{I_n\}

is a graded family of

m_R

-primary ideals, and give a number of applications. We also 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.

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