When are multidegrees positive?

TL;DR

This paper establishes necessary and sufficient conditions for the positivity of multidegrees of subschemes in multiprojective spaces, linking algebraic and combinatorial properties and unifying various results in algebraic geometry.

Contribution

It provides a complete characterization of multidegree positivity and connects it to algebraic polymatroids and mixed multiplicities, advancing understanding in algebraic and combinatorial geometry.

Findings

Characterization of multidegree positivity conditions

Support of multidegrees forms a discrete algebraic polymatroid for irreducible X

Applications to positivity of mixed multiplicities of ideals

Abstract

Let $k$ be an arbitrary field, $P = P_k^{m_1} \times_k \cdots \times_k P_k^{m_p}$ be a multiprojective space over $k$, and $X \subseteq P$ 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.