Fano Schemes for Generic Sums of Products of Linear Forms

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 × 6 Pfaffian and 4 × 4 permanent, as well as giving a new proof that the product
and tensor ranks of the 3 × 3 determinant equal five. Based on our results, we formulate several conjectures.