Asymptotic Multiplicities of Graded Families of Ideals and Linear Series

TL;DR

This paper establishes simple criteria for the existence of asymptotic limits of lengths of graded families of ideals and growth of graded linear series in algebraic geometry, with various applications.

Contribution

It provides necessary and sufficient conditions for the existence of asymptotic limits in local rings and projective schemes, advancing understanding of asymptotic algebraic properties.

Findings

Criteria for limits of lengths of ideals in local rings

Conditions for growth limits of linear series on schemes

Applications to algebraic geometry and commutative algebra

Abstract

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