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

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.