Fano Schemes for Generic Sums of Products of Linear Forms

TL;DR

This paper investigates the geometry of certain hypersurfaces defined by sums of products of linear forms, deriving new bounds on tensor ranks and providing insights into classical algebraic problems.

Contribution

It introduces new geometric methods to analyze Fano schemes of linear subspaces in hypersurfaces, leading to bounds on product ranks of specific tensors.

Findings

Linear subspaces of high dimension are contained in coordinate hyperplanes under certain conditions.

New lower bounds for product ranks of the 6x6 Pfaffian and 4x4 permanent.

A new proof that the product and tensor ranks of the 3x3 determinant are five.

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.