Powers of componentwise linear ideals: The Herzog--Hibi--Ohsugi Conjecture and related problems

TL;DR

This paper surveys the properties of componentwise linear ideals, focusing on the Herzog-Hibi-Ohsugi conjecture regarding powers of cover ideals of chordal graphs, and explores related problems about symbolic powers.

Contribution

It reviews progress over the past decade on the Herzog-Hibi-Ohsugi conjecture and related issues about symbolic powers of cover ideals.

Findings

Progress made on the Herzog-Hibi-Ohsugi conjecture for specific classes of graphs.

Characterization of when symbolic powers of cover ideals are componentwise linear.

Identification of cases where componentwise linearity is preserved under powers.

Abstract

In 1999 Herzog and Hibi introduced componentwise linear ideals. A homogeneous ideal $I$ is componentwise linear if for all non-negative integers $d$, the ideal generated by the homogeneous elements of degree $d$ in $I$ has a linear resolution. For square-free monomial ideals, componentwise linearity is related via Alexander duality to the property of being sequentially Cohen-Macaulay for the corresponding simplicial complexes. In general, the property of being componentwise linear is not preserved by taking powers. In 2011, Herzog, Hibi, and Ohsugi conjectured that if $I$ is the cover ideal of a chordal graph, then $I^s$ is componentwise linear for all $s \geq 1$. We survey some of the basic properties of componentwise linear ideals, and then specialize to the progress on the Herzog-Hibi-Ohsugi conjecture during the last decade. We also survey the related problem of determining when the symbolic powers of a cover ideal are componentwise linear.