Hilbert functions of schemes of double and reduced points

TL;DR

This paper demonstrates how to realize any valid Hilbert function of zero-dimensional schemes in the projective plane using a combination of double and reduced points, and explores conditions related to star configurations.

Contribution

It provides a construction method for matching Hilbert functions with schemes of double and reduced points, and characterizes when these functions correspond to star configurations.

Findings

Any valid Hilbert function can be realized with double and reduced points.

For certain functions, only double points are needed.

Conditions are established for star configuration support.

Abstract

It remains an open problem to classify the Hilbert functions of double points in $\mathbb{P}^2$. Given a valid Hilbert function $H$ of a zero-dimensional scheme in $\mathbb{P}^2$, we show how to construct a set of fat points $Z \subseteq \mathbb{P}^2$ of double and reduced points such that $H_Z$, the Hilbert function of $Z$, is the same as $H$. In other words, we show that any valid Hilbert function $H$ of a zero-dimensional scheme is the Hilbert function of a set of a positive number of double points and some reduced points. For some families of valid Hilbert functions, we are also able to show that $H$ is the Hilbert function of only double points. In addition, we give necessary and sufficient conditions for the Hilbert function of a scheme of a double points, or double points plus one additional reduced point, to be the Hilbert function of points with support on a star configuration of lines.