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.

中文翻译：

---

关于 Geramita-Harbourne-Migliore 猜想

令 $\Sigma$ 是 $\mathbb K[x_0,\ldots,x_n]$ 中线性形式的有限集合，其中 $\mathbb K$ 是一个域。表示 ${\rm Supp}(\Sigma)$ 是 $\Sigma$ 的所有非比例元素的集合，并假设 ${\rm Supp}(\Sigma)$ 是泛型的，这意味着任何 $n+1$其元素线性无关。让 $1\leq a\leq |\Sigma|$。在本文中，我们证明了由 $\Sigma$ 的线性形式的（所有）$a$-fold 乘积产生的理想具有线性分级自由分辨率的猜想。因此，我们证明了 Geramita-Harbourne-Migliore 猜想，该猜想涉及定义恒星配置理想的普通权力的初级分解，并且我们还确定了这些理想的复兴。