Multigraded algebras and multigraded linear series

This paper is devoted to the study of multigraded algebras and multigraded linear series. For an Ns$\mathbb {N}s$-graded algebra A$A$, we define and study its volume function FA:N+s -> R$F_A:\mathbb {N}_+s\rightarrow \mathbb {R}$, which computes the asymptotics of the Hilbert function of A$A$. We relate the volume function FA$F_A$ to the volume of the fibers of the global Newton-Okounkov body Delta(A)$\Delta (A)$ of A$A$. Unlike the classical case of standard multigraded algebras, the volume function FA$F_A$ is not a polynomial in general. However, in the case when the algebra A$A$ has a decomposable grading, we show that the volume function FA$F_A$ is a polynomial with nonnegative coefficients. We then define mixed multiplicities in this case and provide a full characterization for their positivity. Furthermore, we apply our results on multigraded algebras to multigraded linear series. Our work recovers and unifies recent developments on mixed multiplicities. In particular, we recover results on the existence of mixed multiplicities for (not necessarily Noetherian) graded families of ideals and on the positivity of the multidegrees of multiprojective varieties.