Illinois Journal of Mathematics

Distinguishing $\Bbbk$-configurations

Federico Galetto, Yong-Su Shin, and Adam Van Tuyl

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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

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

Received: 25 May 2017
Revised: 15 February 2018
First available in Project Euclid: 22 August 2018

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Zentralblatt MATH identifier

Primary: 13D40: Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series 14M05: Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) [See also 13F15, 13F45, 13H10]


Galetto, Federico; Shin, Yong-Su; Van Tuyl, Adam. Distinguishing $\Bbbk$-configurations. Illinois J. Math. 61 (2017), no. 3-4, 415--441. doi:10.1215/ijm/1534924834.

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