Computations and equations for Segre-Grassmann hypersurfaces

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 [6] are used to determine the degrees of several of these hypersurfaces, and representation-theoretic descriptions of their equations are given. We answer ([3], Problem 6.5), 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.