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Combinatorica

, Volume 35, Issue 1, pp 63–74 | Cite as

k-nets embedded in a projective plane over a field

  • Gábor Korchmáros
  • Gábor P. Nagy
  • Nicola Pace
Original paper

Abstract

We investigate k-nets with k≥4 embedded in the projective plane PG(2,\(\mathbb{K}\)) defined over a field \(\mathbb{K}\); they are line configurations in PG(2,\(\mathbb{K}\)) consisting of k pairwise disjoint line-sets, called components, such that any two lines from distinct families are concurrent with exactly one line from each component. The size of each component of a k-net is the same, the order of the k-net. If \(\mathbb{K}\) has zero characteristic, no embedded k-net for k≥5 exists; see [11,14]. Here we prove that this holds true in positive characteristic p as long as p is sufficiently large compared with the order of the k-net. Our approach, different from that used in [11,14], also provides a new proof in characteristic zero.

Mathematics Subject Classification (2000)

52C30 05B25 

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References

  1. [1]
    A. Blokhuis, G. Korchmáros and F. Mazzocca:On the structure of 3-nets embedded in a projective plane, J. Combin. Theory Ser-A. 118 (2011), 1228–1238.CrossRefzbMATHMathSciNetGoogle Scholar
  2. [2]
    I. Dolgachev: Pencils of plane curves with completely reducible members, Oberwolfach Reports 26/2008, 1436–1438.Google Scholar
  3. [3]
    M. Falk and S. Yuzvinsky: Multinets, resonance varieties, and pencils of plane curves, Compos. Math. 143 (2007), 1069–1088.zbMATHMathSciNetGoogle Scholar
  4. [4]
    J. W. P. Hirschfeld, G. Korchmáros and F. Torres: Algebraic Curves Over a Finite Field, Princeton Univ. Press, Princeton and Oxford, 2008, xx+696.CrossRefzbMATHGoogle Scholar
  5. [5]
    D. R. Hughes and F. C. Piper: Projective Planes, Graduate Texts in Mathematics 6, Springer, New York, 1973, x+291 pp.Google Scholar
  6. [6]
    G. Korchmáros, G. P. Nagy and N. Pace: 3-nets realizing a group in a projective plane, to appear in J. Algebraic Combin. Google Scholar
  7. [7]
    G. Lunardon: Private communication, 2012.Google Scholar
  8. [8]
    Á. Miguel and M. Buzunáriz: A description of the resonance variety of a line combinatorics via combinatorial pencils, Graphs and Combinatorics 25 (2009), 469–488.CrossRefzbMATHMathSciNetGoogle Scholar
  9. [9]
    G. P. Nagy and N. Pace: On small 3-nets embedded in a projective plane over a field, J. Combin. Theory Ser-A 120 (2013), 1632–1641.CrossRefMathSciNetGoogle Scholar
  10. [10]
    J. V. Pereira and S. Yuzvinsky: Completely reducible hypersurfaces in a pencil, Adv. Math. 219 (2008), 672–688.CrossRefzbMATHMathSciNetGoogle Scholar
  11. [11]
    J. Stipins: Old and new examples of k-nets in ℙ2, math.AG/0701046.Google Scholar
  12. [12]
    G. Urzúa: On line arrangements with applications to 3-nets, Adv. Geom. 10 (2010), 287–310.CrossRefzbMATHMathSciNetGoogle Scholar
  13. [13]
    S. Yuzvinsky: Realization of finite abelian groups by nets in ℙ2, Compos. Math. 140 (2004), 1614–1624.CrossRefzbMATHMathSciNetGoogle Scholar
  14. [14]
    S. Yuzvinsky: A new bound on the number of special fibers in a pencil of curves, Proc. Amer. Math. Soc. 137 (2009), 1641–1648.CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Gábor Korchmáros
    • 1
  • Gábor P. Nagy
    • 2
  • Nicola Pace
    • 3
  1. 1.Dipartimento di Matematica e InformaticaUniversità della Basilicata Contrada Macchia RomanaPotenzaItaly
  2. 2.Bolyai InstituteUniversity of SzegedSzegedHungary
  3. 3.Inst. de Ciências Matemãticas e de ComputaçãoUniversidade de São PauloSão CarlosBrazil

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