On catalecticant perfect ideals of codimension 2

Abstract

One deals with catalectic codimension two perfect ideals and certain degenerations thereof, with a view towards the nature of their symbolic powers. In the spirit of [10] one considers linearly presented such ideals, only now in the situation where the number of variables is sufficiently larger than the size of the matrix, yet still stays within reasonable bounds.

Homaloidal determinants

“…An important result proved in [11] is that the generic Hankel matrix of arbitrary size m × n is 1-generic. Using this result, it has been proved in [27] that all generic catalecticants of arbitrary size are 1-generic. The advantage of this notion is that it implies, in particular, that the determinant of a square such a matrix is irreducible.…”

“…where the rightmost matrix is the syzygy K in (26), viewed as a column vector, where l = n − 1. By reasoning as in the argument that ensues (27), one gets that the ith entry of g 4 is −r n−i , 1 ≤ i ≤ n − 1. As a result, the leftmost map in (30) is ψ := (x n−1 n−1 , −x n , g t 4 ) = (x n−1 n−1 , −x n , −r n−1 , .…”

Blow-up algebras of secant varieties of rational normal scrolls

“…Proof. The first inequality is from (31). With Observation 3.8, we notice that each strictly increasing subsequence has a length at most r. Therefore,…”

“…, x c+(r−1)d ]. This kind of matrix is also called d-leap catalecticant or d-catalecticant in [31]. And when d = 1, it is precisely the ordinary Hankel matrix, which is also known as a Toelitz matrix in other fields like functional analysis, orthogonal polynomial theory, moment problem, and probability.…”