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Fall and Winter 2017 Distinguishing $\Bbbk$-configurations
Federico Galetto, Yong-Su Shin, Adam Van Tuyl
Illinois J. Math. 61(3-4): 415-441 (Fall and Winter 2017). DOI: 10.1215/ijm/1534924834


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 $\mathbb{X}$.


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Federico Galetto. Yong-Su Shin. Adam Van Tuyl. "Distinguishing $\Bbbk$-configurations." Illinois J. Math. 61 (3-4) 415 - 441, Fall and Winter 2017.


Received: 25 May 2017; Revised: 15 February 2018; Published: Fall and Winter 2017
First available in Project Euclid: 22 August 2018

zbMATH: 06932511
MathSciNet: MR3845728
Digital Object Identifier: 10.1215/ijm/1534924834

Primary: 13D40 , 14M05

Rights: Copyright © 2017 University of Illinois at Urbana-Champaign


Vol.61 • No. 3-4 • Fall and Winter 2017
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