Fano Schemes for Generic Sums of Products of Linear Forms
Abstract
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.
 Publication:

arXiv eprints
 Pub Date:
 October 2016
 arXiv:
 arXiv:1610.06770
 Bibcode:
 2016arXiv161006770I
 Keywords:

 Mathematics  Algebraic Geometry
 EPrint:
 22 pages. v2: minor revisions to v1