Hilbert functions of fat point subschemes of the plane: the two-fold way

TL;DR

This paper explores two methods for calculating Hilbert functions of fat point schemes in the plane, providing complete classifications for specific configurations and extending previous results.

Contribution

It introduces a dual approach framework for determining Hilbert functions, extending known classifications to more complex point configurations.

Findings

Complete classification of Hilbert functions for 9 double points.

Determination of Hilbert functions for n≥9 m-multiple points on an irreducible cubic.

Extended results for double points on cubic curves.

Abstract

Two approaches for determining Hilbert functions of fat point subschemes of $\mathbb P^2$ are demonstrated. A complete determination of the Hilbert functions which occur for 9 double points is given using the first approach, extending results obtained in a previous paper using the second approach. In addition the second approach is used to obtain a complete determination of the Hilbert functions for $n\geq 9$ $m$-multiple points for every $m$ if the points are smooth points of an irreducible plane cubic curve. Additional results are obtained using the first approach for $n\geq 9$ double points when the points lie on an irreducible cubic (but now are not assumed to be smooth points of the cubic).