Cohomological support varieties of certain monomial ideals

TL;DR

This paper explores the structure of cohomological support varieties of monomial ideals, providing examples, computational methods, and classifications that challenge previous assumptions about their geometric nature.

Contribution

It introduces new examples of monomial ideals with support varieties not unions of linear subspaces and develops an efficient computational procedure for their analysis.

Findings

Existence of support varieties not unions of linear subspaces.

An improved computational method for support variety calculation.

Classification of support varieties for homogeneous monomial ideals with 6 generators over .

Abstract

Building on work of Briggs, Grifo and Pollitz arXiv:2506.10827, we give an example of two cohomological support varieties of monomial ideals which are not unions of linear subspaces. We provide a procedure for the computation of the cohomological support varieties of certain other monomial ideals - including those with homogeneous generators - with improved computational efficiency, leading to a computer-assisted verification of the existence of a third support variety of a monomial ideal which is not a union of linear subspaces and a computer-assisted proof of a classification of cohomological support varieties of homogeneous monomial ideals over $\mathbb{Q}$ with 6 generators.