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

TL;DR

This paper proves that ideals generated by all a-fold products of linear forms have linear free resolutions, leading to new insights into Orlik-Terao algebras and symbolic powers of star configurations.

Contribution

It establishes linear graded free resolutions for a broad class of ideals and applies these results to line arrangements, Orlik-Terao algebras, and star configurations.

Findings

Ideals generated by a-fold products of linear forms have linear free resolutions.

Determined generators for the defining ideal of the second order Orlik-Terao algebra.

Proved several conjectures regarding symbolic powers of star configurations.

Abstract

Given $\Sigma\subset R:=\mathbb K[x_1,\ldots,x_k]$, where $\mathbb K$ is a field of characteristic 0, any finite collection of linear forms, some possibly proportional, and any $1\leq a\leq |\Sigma|$, 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 $c$.