Computations and Equations for Segre-Grassmann hypersurfaces

TL;DR

This paper investigates specific hypersurfaces related to secant varieties of Segre-Grassmann varieties, providing algebraic equations, confirming conjectures, and offering both numerical and non-numerical proofs.

Contribution

It confirms that an infinite family of these hypersurfaces is defined by known determinantal equations and provides explicit equations and degrees for these hypersurfaces.

Findings

Determined degrees of several hypersurfaces using Bertini algorithms

Provided representation-theoretic descriptions of hypersurface equations

Confirmed that these hypersurfaces are minimally defined by known determinantal equations

Abstract

In 2013, Abo and Wan studied the analogue of Waring's problem for systems of skew-symmetric forms and identified several defective systems. Of particular interest is when a certain secant variety of a Segre-Grassmann variety is expected to fill the natural ambient space, but is actually a hypersurface. Algorithms implemented in Bertini are used to determine the degrees of several of these hypersurfaces, and representation-theoretic descriptions of their equations are given. We answer Problem 6.5 [Abo-Wan2013], and confirm their speculation that each member of an infinite family of hypersurfaces is minimally defined by a (known) determinantal equation. While led by numerical evidence, we provide non-numerical proofs for all of our results.