On the Geramita-Harbourne-Migliore conjecture

Let $Σ$ be a finite collection of linear forms in $\mathbb K[x_0,\ldots,x_n]$, where $\mathbb K$ is a field. Denote ${\rm Supp}(Σ)$ to be the set of all nonproportional elements of $Σ$, and suppose ${\rm Supp}(Σ)$ is generic, meaning that any $n+1$ of its elements are linearly independent. Let $1\leq a\leq |Σ|$. In this article we prove the conjecture that the ideal generated by (all) $a$-fold products of linear forms of $Σ$ 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.

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