Asymptotic Multiplicities of Graded Families of Ideals and Linear Series

الأصل · EN

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