Hankel Like Catalecticants

The meaning of catalecticant here does not refer to the classical (generic) catalecticant from invariant theory, a notion that only retrieves the present one in the special case of a Hankel matrix. The classical one is a symmetric matrix in the square case, representing a linear map attached to a given form in the dual space, induced by the Macaulay inverse action. The matrices in this chapter constitute a natural extension of the Hankel logistic, consisting in fixing an integer \(r\geq 1\) and starting the second row \(r+1\) steps off (instead of two steps). In the square case it is only symmetric for \(r=1\) , which retrieves the usual Hankel matrix. In the case these matrices have only two rows, up to column permutations they may be written as concatenations of Hankel matrices, hence, define rational normal scrolls. Here one focuses on generic such matrices, moving away from the nature of its linear sections, emphasizing instead a method of relating to such a matrix another matrix mimicking a famous construction of Gruson and Peskine. Such a method has been used before, but here one expands on the related algebraic invariants. In the square case a discussion of the dual variety to the determinant is spelled, with a handle to the problem of parabolism stated in previous chapters.