2020
DOI: 10.1016/j.aim.2020.107382
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When are multidegrees positive?

Abstract: be a multiprojective space over k, and X ⊆ P be a closed subscheme of P. We provide necessary and sufficient conditions for the positivity of the multidegrees of X. As a consequence of our methods, we show that when X is irreducible, the support of multidegrees forms a discrete algebraic polymatroid. In algebraic terms, we characterize the positivity of the mixed multiplicities of a standard multigraded algebra over an Artinian local ring, and we apply this to the positivity of mixed multiplicities of ideals. … Show more

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Cited by 18 publications
(30 citation statements)
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“…denote the projection of P n onto ∈ P . The main result of [4] is that the support of can be computed from dim ( ). The above result can also be generalized to non-irreducible varieties as for a pure dimensional multiprojective variety with decomposition [4,Corollary 3.13] for the exact statement.…”
Section: The Multidegree and The Support Of A Multiprojective Varietymentioning
confidence: 99%
“…denote the projection of P n onto ∈ P . The main result of [4] is that the support of can be computed from dim ( ). The above result can also be generalized to non-irreducible varieties as for a pure dimensional multiprojective variety with decomposition [4,Corollary 3.13] for the exact statement.…”
Section: The Multidegree and The Support Of A Multiprojective Varietymentioning
confidence: 99%
“…We say that R is standard Z p -graded if the total degree of each variable x i is equal to one (i.e., for each 1 i n, we have deg(x i ) = e k i ∈ Z p with 1 k i p). The study of standard multigraded algebras is of utmost importance as they correspond with closed subschemes of a product of projective spaces (see, e.g., [4] and the references therein). Since the coefficients of the multidegree polynomial are non-negative in the standard multigraded case, it becomes natural to address the positivity of these coefficients.…”
Section: A Short Recap On Multidegreesmentioning
confidence: 99%
“…Proof. We embed X = MultiProj(R) as a closed subscheme of a multiprojective space P : for each J ⊆ [p], where r : 2 [p] → N is the rank function r(J) := dim MultiProj R (J) /I (J) (see [4,Proposition 5.1]). Finally, we can check that s :…”
Section: Theorem 22 ([4]mentioning
confidence: 99%
See 1 more Smart Citation
“…By using Proposition 6. For any p > 0, [5,Theorem 4.4] implies that the inequality e (d 0 ,d) m | J(j 1 ) p , . .…”
Section: Application To Graded Families Of Idealsmentioning
confidence: 99%