  We study the Fano scheme of $k$-planes contained in the hypersurface cut out
by a generic sum of products of linear forms. In particular, we show that under
certain hypotheses, linear subspaces of sufficiently high dimension must be
contained in a coordinate hyperplane. We use our results on these Fano schemes
to obtain a lower bound for the product rank of a linear form. This provides a
new lower bound for the product ranks of the $6\times 6$ Pfaffian and $4\times
4$ permanent, as well as giving a new proof that the product and tensor ranks
of the $3\times 3$ determinant equal five. Based on our results, we formulate
several conjectures.
