# On the Geramita-Harbourne-Migliore conjecture

## TL;DR

This paper proves a conjecture about the linearity of resolutions for ideals generated by products of linear forms, leading to results on primary decompositions and resurgence of ideals of star configurations.

## Contribution

It establishes the Geramita-Harbourne-Migliore conjecture on linear resolutions of certain ideals and applies this to primary decompositions and resurgence of star configuration ideals.

## Key findings

- Ideals generated by all $a$-fold products have linear graded free resolutions.
- Confirmed the Geramita-Harbourne-Migliore conjecture for these ideals.
- Determined the resurgence of ideals of star configurations.

## Abstract

Let $\Sigma$ be a finite collection of linear forms in $\mathbb K[x_0,\ldots,x_n]$, where $\mathbb K$ is a field. Denote ${\rm Supp}(\Sigma)$ to be the set of all nonproportional elements of $\Sigma$, and suppose ${\rm Supp}(\Sigma)$ is generic, meaning that any $n+1$ of its elements are linearly independent. Let $1\leq a\leq |\Sigma|$. In this article we prove the conjecture that the ideal generated by (all) $a$-fold products of linear forms of $\Sigma$ has linear graded free resolution. As a consequence we prove the Geramita-Harbourne-Migliore conjecture concerning the primary decomposition of ordinary powers of defining ideals of star configurations, and we also determine the resurgence of these ideals.