On the Geramita-Harbourne-Migliore conjecture

Tohǎneanu, Ştefan; Xie, Yu

doi:10.1090/tran/8351

On the Geramita-Harbourne-Migliore conjecture
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by Ştefan O. Tohǎneanu and Yu Xie PDF

Trans. Amer. Math. Soc. 374 (2021), 4059-4073 Request permission

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 $\mathrm {Supp}(\Sigma )$ to be the set of all nonproportional elements of $\Sigma$, and suppose $\mathrm {Supp}(\Sigma )$ 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.

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