Multigraded algebras and multigraded linear series

TL;DR

This paper investigates multigraded algebras and linear series, defining a volume function that captures asymptotic behavior, and explores its properties, especially in decomposable grading cases, with applications to mixed multiplicities.

Contribution

It introduces the volume function for multigraded algebras, characterizes its polynomial nature in decomposable cases, and applies these results to multigraded linear series and mixed multiplicities.

Findings

Volume function relates to Newton-Okounkov body fibers.

In decomposable grading, the volume function is polynomial.

Provides criteria for positivity of mixed multiplicities.

Abstract

This paper is devoted to the study of multigraded algebras and multigraded linear series. For an $\mathbb{N}^s$-graded algebra $A$, we define and study its volume function $F_A:\mathbb{N}_+^s\to \mathbb{R}$, which computes the asymptotics of the Hilbert function of $A$. We relate the volume function $F_A$ to the volume of the fibers of the global Newton-Okounkov body $\Delta(A)$ of $A$. Unlike the classical case of standard multigraded algebras, the volume function $F_A$ is not a polynomial in general. However, in the case when the algebra $A$ has a decomposable grading, we show that the volume function $F_A$ is a polynomial with non-negative 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.