Distinguishing $\mathbb{k}$-configurations

Abstract

A $\mathbb{k}$-configuration is a set of points $\mathbb{X}$ in ${\mathbb{P}}^2$ that satisfies a number of geometric conditions. Associated to a $\mathbb{k}$-configuration is a sequence $(d_1,\dots ,d_s)$ of positive integers, called its type, which encodes many of its homological invariants. We distinguish $\mathbb{k}$-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 $\hat{I}”{\mathbf{H}}_{m\mathbb{X}}(m{d}_s-1)$. Here, $\hat{I}”{\mathbf{H}}_{m\mathbb{X}}(-)$ is the first difference of the Hilbert function of the fat points of multiplicity $m$ supported on $\mathbb{X}$.

Revised version of paper; most changes minor except the proof of Lemma 4.1 which has been rewritten; to appear in Illinois Journal of Mathematics