Computations and equations for Segre-Grassmann hypersurfaces

Abstract

In 2013, Abo and Wan studied Waring’s problem for systems of skewsymmetric 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. In these cases, one aims to obtain both a defining polynomial for these hypersurfaces along with a representation theoretic description of the defectivity. We combine numerical algebraic geometry with representation theory to accomplish this task. 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 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. Mathematics Subject Classification (2010). Primary 14M12; Secondary 14M15, 14Q10, 14M17, 15A69 , 15A72.