2021/04/12 by Yairon Cid‐Ruiz, Fatemeh Mohammadi, Cid-Ruiz, Yairon +3
Mathematics · Physics and Astronomy · #13A18 #13H15 #14M25 #52B20 #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.2104.05397
openalex publication_date 2021/04/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper is devoted to the study of multigraded algebras and multigraded linear series. For an ℕs-graded algebra A, we define and study its volume function FA:ℕ+s→ ℝ, which computes the asymptotics of the Hilbert function of A. We relate the volume function FA to the volume of the fibers of the global Newton-Okounkov body Δ(A) of A. Unlike the classical case of standard multigraded algebras, the volume function FA is not a polynomial in general. However, in the case when the algebra A has a decomposable grading, we show that the volume function FA 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.