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Line arrangements with the maximal number of triple points

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Abstract

The purpose of this note is to study configurations of lines in projective planes over arbitrary fields having the maximal number of intersection points where three lines meet. We give precise conditions on ground fields \(\mathbb F\) over which such extremal configurations exist. We show that there does not exist a field admitting a configuration of 11 lines with 17 triple points, even though such a configuration is allowed combinatorially. Finally, we present an infinite series of configurations which have a high number of triple intersection points.

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Acknowledgments

This notes originated in a workshop on Arrangements of Lines held in Lanckorona in April 2014. We thank the Jagiellonian University in Cracow for financial support. We thank also Brian Harbourne and Witold Jarnicki for helpful discussions. We are grateful to the referee for remarks which led to considerable improvements of the final version of this paper.

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Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, Jagiellonian University, Łojasiewicza 6, 30-348, Kraków, Poland

    M. Dumnicki, Ł. Farnik & H. Tutaj-Gasińska

  2. Department of Mathematics, Pedagogical University of Cracow, Podchora̧żych 2, 30-084, Kraków, Poland

    A. Główka, M. Lampa-Baczyńska, G. Malara, T. Szemberg & J. Szpond

  3. Mathematisches Institut, Albert-Ludwigs-Universität Freiburg, 79104, Freiburg, Germany

    M. Lampa-Baczyńska, G. Malara & T. Szemberg

Authors
  1. M. Dumnicki
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  2. Ł. Farnik
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  3. A. Główka
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  4. M. Lampa-Baczyńska
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  5. G. Malara
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  6. T. Szemberg
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  7. J. Szpond
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  8. H. Tutaj-Gasińska
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Corresponding author

Correspondence to T. Szemberg.

Additional information

M.D., T.S., J.S. and H.TG were partially supported by the National Science Centre, Poland, Grant 2014/15/B/ST1/02197.

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Dumnicki, M., Farnik, Ł., Główka, A. et al. Line arrangements with the maximal number of triple points. Geom Dedicata 180, 69–83 (2016). https://doi.org/10.1007/s10711-015-0091-7

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  • DOI: https://doi.org/10.1007/s10711-015-0091-7

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