, 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


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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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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