## Illinois Journal of Mathematics

### Distinguishing $\Bbbk$-configurations

#### 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 $m\gg0$, the number of such lines is precisely the value of $\Delta\mathbf{H}_{m\mathbb{X}}(md_{s}-1)$. Here, $\Delta\mathbf{H}_{m\mathbb{X}}(-)$ is the first difference of the Hilbert function of the fat points of multiplicity $m$ supported on $\mathbb{X}$.

#### Article information

Source
Illinois J. Math., Volume 61, Number 3-4 (2017), 415-441.

Dates
Revised: 15 February 2018
First available in Project Euclid: 22 August 2018

https://projecteuclid.org/euclid.ijm/1534924834

Mathematical Reviews number (MathSciNet)
MR3845728

Zentralblatt MATH identifier
06932511

#### Citation

Galetto, Federico; Shin, Yong-Su; Van Tuyl, Adam. Distinguishing $\Bbbk$-configurations. Illinois J. Math. 61 (2017), no. 3-4, 415--441. https://projecteuclid.org/euclid.ijm/1534924834

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