Green-Lazarsfeld index of square-free monomial ideals and their powers

TL;DR

This paper investigates the Green-Lazarsfeld index of square-free monomial ideals and their powers, providing combinatorial characterizations for ideals generated in degree 3 and analyzing the index behavior for small numbers of variables.

Contribution

It offers new combinatorial characterizations of degree 3 square-free monomial ideals with specific Green-Lazarsfeld index properties and examines the index behavior of their powers.

Findings

Characterization of degree 3 square-free monomial ideals with index > 1

Conditions for ideals with index > 1 and their squares having index 1

Index > 1 for all powers when n ≤ 5 and initial index > 1

Abstract

Let $\mathbb{K}$ be a field and $I$ be a square-free monomial ideal in the polynomial ring $\mathbb{K}[x_1, \ldots, x_n]$. The Green-Lazarsfeld index, $\mathrm{index}(I)$, counts the number of steps to reach to a syzygy minimally generated by a nonlinear form in a graded minimal free resolution of $I$. In this paper, we study this invariant for $I$ and its powers from a combinatorial point of view. We characterize all square-free monomial ideals $I$ generated in degree $3$ such that $\mathrm{index}(I)>1$. Utilizing this result, we also characterize all square-free monomial ideals generated in degree $3$ such that $\mathrm{index}(I)>1$ and $\mathrm{index}(I^2)=1$. In case $n\leq5$, it is shown that $\mathrm{index}(I^k)>1$ for all $k$ if $I$ is any square-free monomial ideal with $\mathrm{index}(I)>1$.