Multigraded algebras and multigraded linear series
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https://hdl.handle.net/10037/34987Date
2024-03-07Type
Journal articleTidsskriftartikkel
Peer reviewed
Abstract
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 𝐹�𝐴� ∶ ℕ𝑠� + → 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 Δ(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 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.
Publisher
WileyCitation
Cid-Ruiz, Mohammadi, Monin. Multigraded algebras and multigraded linear series. Journal of the London Mathematical Society. 2024;109(3)Metadata
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