Multidegrees, prime ideals, and non-standard gradings

TL;DR

This paper investigates properties of multihomogeneous prime ideals in multigraded settings, revealing special characteristics of their generic initial ideals, and extending key results on multidegrees and multiplicity behavior.

Contribution

It introduces a comprehensive study of multidegrees in multigraded contexts and extends the concept of Cartwright-Sturmfels ideals through a standardization approach.

Findings

Multigraded generic initial ideals of primes have radical Cohen-Macaulay properties.

Extended the notion of Cartwright-Sturmfels ideals to positive multigraded environments.

Provided a multidegree version of Hartshorne's upper semicontinuity of arithmetic degree.

Abstract

We study several properties of multihomogeneous prime ideals. We show that the multigraded generic initial ideal of a prime has very special properties, for instance, its radical is Cohen-Macaulay. We develop a comprehensive study of multidegrees in arbitrary positive multigraded settings. In these environments, we extend the notion of Cartwright-Sturmfels ideals by means of a standardization technique. Furthermore, we recover or extend important results in the literature, for instance: we provide a multidegree version of Hartshorne's result stating the upper semicontinuity of arithmetic degree under flat degenerations, and we give an alternative proof of Brion's result regarding multiplicity-free varieties.