  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.
