Superfat points and associated tensors

We consider 0-dimensional schemes supported at a single point in n-space that are m-symmetric, i.e. that intersect any smooth curve passing through the point with length m, and the ones among them that are maximal with respect to inclusion (called m-superfat points). We study properties of such schemes, in particular for n=2. We give a first application of the simplest such schemes, namely 2-superfat points in the plane, by studying varieties defined by them on Veronese and Segre-Veronese varieties and the (symmetric or partially symmetric) tensors they parameterize.