k-nets embedded in a projective plane over a field

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.

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Correspondence to Gábor Korchmáros.

Additional information

The research was performed while the first author was a visiting professor at the Bolyai Institute of University of Szeged during the second semester of the academic year 2011-12. The visit was financially supported by the TAMOP-4.2.1/B-09/1/KONV-2010-0005 project

The research was supported by the European Union and co-funded by the European Social Fund; project number TAMOP-4.2.2.A-11/1/KONV-2012-0073

The research was supported by FAPESP (Fundação de Amparo a Pesquisa do Estado de São Paulo), proc. no. 12/03526-0.

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Korchmáros, G., Nagy, G.P. & Pace, N. k-nets embedded in a projective plane over a field. Combinatorica 35, 63–74 (2015). https://doi.org/10.1007/s00493-011-3055-z

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Mathematics Subject Classification (2000)

  • 52C30
  • 05B25