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

Abstract

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

16 pages. In this new version, with appropriate new title, we prove in its full generality the conjecture that any ideal generated by all a-fold products of linear forms has linear graded free resolution. We also prove various conjectures that involve symbolic powers of star configurations of any codimension