Postulation for 2-superfat points in the plane

TL;DR

This paper investigates the postulation of unions of 2-superfat points in the plane, demonstrating they have the expected Hilbert function, and extends results to include a general 3-fat point, with applications to interpolation problems.

Contribution

It proves that unions of 2-superfat points in the plane have good postulation and extends this to include a general 3-fat point, addressing a specific interpolation problem.

Findings

Unions of 2-superfat points have the expected Hilbert function.

Adding a general 3-fat point preserves good postulation.

Results apply to a particular interpolation problem.

Abstract

We study the postulation of 0-dimensional schemes given by unions of 2-superfat points in general position in the plane, i.e., the union of local schemes defined by the intersection of two distinct double lines. We prove that such schemes have good postulation, i.e., they have the expected Hilbert function. We also show the good postulation of such schemes when we add a general 3-fat point. Finally, we use these results to answer a peculiar kind of interpolation problem.