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 6x6 Pfaffian and 4x4 permanent, as well as giving a new proof that the product and tensor ranks of the 3 x 3 determinant equal five. Based on our results, we formulate several conjectures.
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