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 limit lim(n ->infinity) l(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. (C) 2014 Elsevier Inc. All rights reserved.