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arXiv · 1705.09195

Distinguishing $\Bbbk$-configurations

Federico Galetto·Yong-Su Shin·Adam Van Tuyl
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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 $\mathbb{X}$.

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Federico Galetto, Yong-Su Shin, Adam Van Tuyl. 2017-05-25. Distinguishing $\Bbbk$-configurations. https://arxiv.org/abs/1705.09195

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