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Embeddings of hermitian unitals into pappian projective planes

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Abstract

Every embedding of a hermitian unital with at least four points on a block into any pappian projective plane is standard, i.e. it originates from an inclusion of the pertinent fields. This result about embeddings also allows us to determine the full automorphism groups of (generalized) hermitian unitals.

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References

  1. Artin, E.: Geometric Algebra. Interscience Publ, New York (1957)

    MATH  Google Scholar 

  2. Barwick, S., Ebert, G.: Unitals in Projective Planes. Springer, New York (2008)

    MATH  Google Scholar 

  3. Bichara, A., Korchmáros, G.: \(n^{2}\)-sets in a projective plane which determine exactly \(n^{2}+n\) lines. J. Geom. 15, 175–181 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bruhat, F., Tits, J.: Groupes réductifs sur un corps local, I. Données radicielles valuées. Inst. Hautes Études Sci. Publ. Math. 41, 5–252 (1972)

    Article  MATH  Google Scholar 

  5. Cohn, P.M.: Further Algebra and Applications. Springer, London (2003)

    Book  MATH  Google Scholar 

  6. De Medts, T., Haot, F., Knop, R., Van Maldeghem, H.: On the uniqueness of the unipotent subgroups of some Moufang sets. In: A. Hulpke et al. (eds), Finite geometries, groups, and computation (Proceedings of the Conference held at Colorado State University, 2004), pp. 43–66. De Gruyter, Berlin (2006)

  7. Dickson, L.E.: The points of inflexion of a plane cubic curve. Ann. Math. (2) 16, 50–66 (1914/15)

  8. Dienst, K.J.: Verallgemeinerte Vierecke in Pappusschen projektiven Räumen. Geom. Dedic. 9, 199–206 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  9. Faina, G., Korchmáros, G.: A graphic characterization of Hermitian curves. Combinatorics ’81 (Rome, 1981). Ann. Discret. Math. 18(1983), 335–342 (1981)

    MATH  Google Scholar 

  10. Grove, L.C.: Classical Groups and Geometric Algebra. American Mathematical Society, Providence (2002)

    Google Scholar 

  11. Grundhöfer, T., Krinn, B., Stroppel, M.: Non-existence of isomorphisms between certain unitals. Des. Codes Cryptogr. 60, 197–201 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  12. Grundhöfer, T., Stroppel, M.J., Van Maldeghem, H.: Unitals admitting all translations. J. Combin. Des. 21, 419–431 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hahn, A.J., O’Meara, O.T.: The Classical Groups and K-Theory. Springer, Berlin (1989)

    Book  MATH  Google Scholar 

  14. Hirschfeld, J.W.P.: Projective Geometries over Finite Fields, 2nd edn. Oxford University Press, Oxford (1998)

    MATH  Google Scholar 

  15. Hughes, D.R., Piper, F.C.: Projective Planes. Springer, Berlin (1973)

    MATH  Google Scholar 

  16. Jungnickel, D.: On a theorem of Rigby. J. Geom. 107, 257–265 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  17. Knarr, N., Stroppel, M.: Polarities and planar collineations of Moufang planes. Monatsh. Math. 169, 383–395 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  18. Knarr, N., Stroppel, M.: Baer involutions and polarities in Moufang planes of characteristic two. Adv. Geom. 13, 533–546 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  19. Korchmáros, G., Siciliano, A., Szőnyi, T.: Embedding of classical polar unitals in \({\rm PG}(2, q^2)\). J. Combin. Theory (A) 153, 67–75 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  20. Lefèvre-Percsy, C.: Characterization of Hermitian curves. Arch. Math. 39, 476–480 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  21. O’Nan, M.E.: Automorphisms of unitary block designs. J. Algebra 20, 495–511 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  22. Ostrom, T.G., Sherk, F.A.: Finite projective planes with affine subplanes. Can. Math. Bull. 7, 549–559 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  23. Pasini, A.: Embedded polar spaces revisited. Innov. Incid. Geom. 15, 31–72 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  24. Pickert, G.: Near-embeddings of the affine plane with 9 points into Desarguesian projective and affine planes. Note Mat. 27, 11–19 (2007)

    MathSciNet  MATH  Google Scholar 

  25. Pralle, H.: Affine generalized quadrangles–an axiomatization. Geom. Dedic. 84, 1–23 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  26. Rigby, J.F.: Affine subplanes of finite projective planes. Can. J. Math. 17, 977–1009 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  27. Salzmann, H., Grundhöfer, T., Hähl, H., Löwen, R.: The Classical Fields. Cambridge University Press, Cambridge (2007)

    Book  MATH  Google Scholar 

  28. Stroppel, M.: Orthogonal polar spaces and unitals. Innov. Incid. Geom. 12, 167–179 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  29. Stroppel, M., Van Maldeghem, H.: Automorphisms of unitals. Bull. Belg. Math. Soc. Simon Stevin 12, 895–908 (2005)

    MathSciNet  MATH  Google Scholar 

  30. Taylor, D.E.: The Geometry of the Classical Groups. Heldermann, Berlin (1992)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Institut für Mathematik, Universität Würzburg, Am Hubland, 97074, Würzburg, Germany

    Theo Grundhöfer

  2. LExMath, Universität Stuttgart, 70550, Stuttgart, Germany

    Markus J. Stroppel

  3. Department Mathematics: Algebra & Geometry, Universiteit Gent, Krijgslaan 281, S25, 9000, Gent, Belgium

    Hendrik Van Maldeghem

Authors
  1. Theo Grundhöfer
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  2. Markus J. Stroppel
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  3. Hendrik Van Maldeghem
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Correspondence to Hendrik Van Maldeghem.

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Grundhöfer, T., Stroppel, M.J. & Van Maldeghem, H. Embeddings of hermitian unitals into pappian projective planes. Aequat. Math. 93, 927–953 (2019). https://doi.org/10.1007/s00010-019-00652-x

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  • DOI: https://doi.org/10.1007/s00010-019-00652-x

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