When are multidegrees positive?

Let k be an arbitrary field, P=Pkm1×k⋯×kPkmp 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 multideg...

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Bibliographic Details
Published in:Advances in mathematics (New York. 1965) 2020-11, Vol.374, p.107382, Article 107382
Main Authors: Castillo, Federico, Cid-Ruiz, Yairon, Li, Binglin, Montaño, Jonathan, Zhang, Naizhen
Format: Article
Language:English
Subjects:
Hilbert polynomial
Mixed multiplicities
Multidegrees
Multiprojective scheme
Polymatroids
Positivity
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Summary:Let k be an arbitrary field, P=Pkm1×k⋯×kPkmp 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. Furthermore, we use our results to recover several results in the literature in the context of combinatorial algebraic geometry.
ISSN:0001-8708
1090-2082
DOI:10.1016/j.aim.2020.107382