When are multidegrees positive?

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.

This is a generalized and expanded version of arXiv:1612.00154. Final version, to appear in Advances in Mathematics

References in corpus (1)

• Integral closure of ideals