Asymptotic multiplicities of graded families of ideals and linear series

We find simple necessary and sufficient conditions on a local ring R of dimension d for the limitlimn→∞⁡ℓR(R/In)nd to exist whenever {In}{In} is a graded family of mRmR-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.