Homological Properties of Ideals Generated by Fold Products of Linear Forms

Abstract

Given $\mathrm{\Sigma }\subset \mathit{R}:=\mathbb{K}[{\mathit{x}}_1,\dots ,{\mathit{x}}_{\mathit{k}}]$, where $\mathbb{K}$ is a field, any finite collection of linear forms, some possibly proportional, and any $1\le \mathit{a}\le |\mathrm{\Sigma }|$, we prove that ${\mathit{I}}_{\mathit{a}}(\mathrm{\Sigma })$, the ideal generated by all a-fold products of Σ, has linear graded free resolution, and we exhibit a combinatorial primary decomposition for ${\mathit{I}}_{\mathit{a}}(\mathrm{\Sigma })$. The linear graded free resolution allows us to determine a generating set for the defining ideal of the Orlik–Terao algebra of the second order of a line arrangement in ${\mathbb{P}}_{\mathbb{K}}^2$, and to conclude that for the case $\mathit{k}=3$, and Σ defining such a line arrangement, the ideal ${\mathit{I}}_{|\mathrm{\Sigma }|-2}(\mathrm{\Sigma })$ is of fiber type. Finally, with the help of the primary decomposition, we prove several conjectures of symbolic powers for the defining ideals of star configurations of any codimension c, a special class of ideals generated by a-fold products of linear forms.