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We study the Fano scheme of $k$-planes contained in the hypersurface cut outby a generic sum of products of linear forms. In particular, we show that undercertain hypotheses, linear subspaces of sufficiently high dimension must becontained in a coordinate hyperplane. We use our results on these Fano schemesto obtain a lower bound for the product rank of a linear form. This provides anew lower bound for the product ranks of the $6\times 6$ Pfaffian and $4\times4$ permanent, as well as giving a new proof that the product and tensor ranksof the $3\times 3$ determinant equal five. Based on our results, we formulateseveral conjectures.

Fano Schemes for Generic Sums of Products of Linear Forms