Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Cohomological support varieties of certain monomial ideals

Michael Gintz

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.

Cohomological support varieties of certain monomial ideals

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 with 6 generators.
Paper Structure (14 sections, 30 theorems, 80 equations)

This paper contains 14 sections, 30 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Preliminary notions
  4. Cohomological support varieties
  5. Constructing VR(R) for R a monomial ideal
  6. Decomposing T(f) tensor k-bar
  7. Our decomposition
  8. A natural double complex structure on T(f) for f homogeneous
  9. Edge ideals of cycles
  10. Simplicial complexes concerning edge ideals
  11. The case d=6
  12. The case d=10
  13. Computational implementations
  14. Future work

Key Result

Theorem 1.1

Let $Q/I$ be a minimal regular presentation of $R$ and let $\bm{f}$ be a minimal generating set of $I$ consisting of $n\leq5$ monomials. Then $\operatorname{V}_R(R)$ is either a coordinate subspace of $\mathbb{A}^5_k$ or a union of two hyperplanes.

Theorems & Definitions (59)

  • Theorem 1.1: embdef
  • Theorem A
  • Theorem B: cf. embdef
  • Example C: Example \ref{['code1']}
  • Theorem D: Theorem \ref{['code2']}
  • Definition 2.1
  • Remark 2.2
  • Theorem 2.3: pollitz21
  • Theorem 2.4: bounds
  • Theorem 2.5: embdef
  • ...and 49 more