Illinois Journal of Mathematics

Distinguishing $\Bbbk$-configurations

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

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

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

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1534924834

Digital Object Identifier
doi:10.1215/ijm/1534924834

Mathematical Reviews number (MathSciNet)
MR3845728

Zentralblatt MATH identifier
06932511

Subjects
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]

Citation

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


Export citation

References

  • J. Abbott, A. Bigatti and G. Lagorio, CoCoA-5: A system for doing Computations in Commutative Algebra; available at http://cocoa.dima.unige.it.
  • A. Bigatti, A. V. Geramita and J. Migliore, Geometric consequences of extremal behavior in a theorem of Macaulay, Trans. Amer. Math. Soc. 346 (1994), 203–235.
  • M. V. Catalisano, N. V. Trung and G. Valla, A sharp bound for the regularity index of fat points in general position, Proc. Amer. Math. Soc. 118 (1993), 717–724.
  • L. Chiantini and J. Migliore, Almost maximal growth of the Hilbert function, J. Algebra 431 (2015), 38–77.
  • S. Cooper, B. Harbourne and Z. Teitler, Combinatorial bounds on Hilbert functions of fat points in projective space, J. Pure Appl. Algebra 215 (2011), 2165–2179.
  • A. V. Geramita, B. Harbourne and J. Migliore, Star configurations in $\mathbb{P}^n$, J. Algebra 376 (2013), 279–299.
  • A. V. Geramita, T. Harima and Y. S. Shin, Extremal point sets and Gorenstein ideals, Adv. Math. 152 (2000), 78–119.
  • A. V. Geramita, T. Harima and Y. S. Shin, An alternative to the Hilbert function for the ideal of a finite set of points in $\mathbb{P}^n$, Illinois J. Math. 45 (2001), 1–23.
  • A. V. Geramita, T. Harima and Y. S. Shin, Decompositions of the Hilbert function of a set of points in $\mathbb{P}^n$, Canad. J. Math. 53 (2001), 923–943.
  • A. V. Geramita, J. Migliore and L. Sabourin, On the first infinitesimal neighborhood of a linear configuration of points in $\mathbb{P}^2$, J. Algebra 298 (2006), 563–611.
  • A. V. Geramita and Y. S. Shin, $k$-configurations in $\mathbb{P}^3$ all have extremal resolutions, J. Algebra 213 (1999), 351–368.
  • T. Harima, Some examples of unimodal Gorenstein sequences, J. Pure Appl. Algebra 103 (1995), 313–324.
  • L. G. Roberts and M. Roitman, On Hilbert functions of reduced and of integral algebras, J. Pure Appl. Algebra 56 (1989), 85–104.
  • O. Zariski and P. Samuel, Commutative algebra, vol. II, Springer, New York, 1960.