  Given $\Sigma\subset R:=\mathbb K[x_1,\ldots,x_k]$, where $\mathbb K$ is a
field of characteristic 0, any finite collection of linear forms, some possibly
proportional, and any $1\leq a\leq |\Sigma|$, we prove that $I_a(\Sigma)$, the
ideal generated by all $a$-fold products of $\Sigma$, has linear graded free
resolution. This allows us to determine a generating set for the defining ideal
of the Orlik-Terao algebra of the second order of a line arrangement in
$\mathbb P_{\mathbb{K}}^2$, and to conclude that for the case $k=3$, and
$\Sigma$ defining such a line arrangement, the ideal $I_{|\Sigma|-2}(\Sigma)$
is of fiber type. We also prove several conjectures of symbolic powers for
defining ideals of star configurations of any codimension $c$.
