# Distinguishing $\Bbbk$-configurations

## TL;DR

This paper introduces a method to distinguish $kk$-configurations in projective plane by analyzing the number of lines containing a specific number of points, using the Hilbert function of fat points.

## Contribution

It establishes a novel connection between the geometric configuration of points and the algebraic Hilbert function, enabling classification of $kk$-configurations.

## Key findings

- Number of lines with $d_s$ points equals $ riangle H_{mbX}(m d_s -1)$ for large $m$.
- Provides a new invariant for classifying $kk$-configurations.
- Links geometric properties with algebraic invariants via Hilbert functions.

## Abstract

A $\Bbbk$-configuration is a set of points $\mathbb{X}$ in $\mathbb{P}^2$ that satisfies a number of geometric conditions. Associated to a $\Bbbk$-configuration is a sequence $(d_1,\ldots,d_s)$ of positive integers, called its type, which encodes many of its homological invariants. We distinguish $\Bbbk$-configurations by counting the number of lines that contain $d_s$ points of $\mathbb{X}$. In particular, we show that for all integers $m \gg 0$, the number of such lines is precisely the value of $\Delta \mathbf{H}_{m\mathbb{X}}(m d_s -1)$. Here, $\Delta \mathbf{H}_{m\mathbb{X}}(-)$ is the first difference of the Hilbert function of the fat points of multiplicity $m$ supported on $\mathbb{X}$.