Hilbert functions of double point schemes in P^2

TL;DR

This paper investigates the minimal Hilbert functions of double point schemes in the projective plane, providing new results for specific cases and extending previous findings for certain numbers of points.

Contribution

It offers new evidence and affirmative answers for the minimal Hilbert functions when the number of points is a binomial coefficient plus one, especially for s=11.

Findings

Confirmed minimal Hilbert functions for s=11 points

Extended known results to cases where s equals a binomial coefficient plus one

Provided evidence supporting the minimal Hilbert function conjecture

Abstract

We study the question of whether there is a minimum Hilbert functions for double point schemes whose support is $s$ points with the generic Hilbert functions. Previous work shows that the question has an affirmative answer for $s \le 9$ and for $s$ equal to a binomial coefficient number. In this paper, we give evidence in the case when $s$ equals to a binomial coefficient number plus 1, and give an affirmative answer to the question when $s = 11$.