Computations and Equations for Segre-grassmann Hypersurfaces

In 2013, Abo and Wan studied Waring’s problem for systems of skew-symmetric forms and identified several defective systems. The cases of particular interest occur when a certain secant variety of a Segre-Grassmann variety is expected to fill the natural ambient space, but is actually a hypersurface. In these cases, one aims to obtain both a defining polynomial for these hypersurfaces along with a representation theoretic description of the defectivity. In this note, we combine numerical algebraic geometry with representation theory to accomplish this task. In particular, numerical algebraic geometric algorithms implemented in Bertini [BHSW06] are used to determine the degrees of several hypersurfaces with representation theory using this data as input to understand the hypersurface. This approach allows us to answer [AW13, Problem 6.5] and show 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.