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.
Categories: math.AC math.AG