When are multidegrees positive?

Federico Castillo, Yairon Cid Ruiz, Binglin Li, Jonathan Montaño, and Naizhen Zhang

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Submission date: 19. May. 2020
Bibtex
MSC-Numbers: 14C17, 13H15, 52B40, 13A30
Link to arXiv: See the arXiv entry of this preprint.

Abstract:
Let k be an arbitrary field and X be a multiprojective scheme over k. 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.