Ideals generated by $a$-fold products of linear forms have linear graded free resolution

Abstract

Given $Σ\subset R:=\mathbb K[x_1,\ldots,x_k]$, where $\mathbb K$ is a field of characteristic 0, any finite collection of linear forms, some possibly proportional, and any $1\leq a\leq |Σ|$, we prove that $I_a(Σ)$, the ideal generated by all $a$-fold products of $Σ$, has linear graded free resolution. This 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 $k=3$, and $Σ$ defining such a line arrangement, the ideal $I_{|Σ|-2}(Σ)$ is of fiber type. We also prove several conjectures of symbolic powers for defining ideals of star configurations of any codimension $c$.