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Notes on Cooperstein Ovoids in Finite Geometries of Type 𝖤 6,1

Author

Listed:
  • Hendrik Van Maldeghem

    (Department of Mathematics, Computer Science and Statistics, Ghent University, Krijgslaan 281, S9, 9000 Gent, Belgium)

Abstract

A Cooperstein ovoid is a set of q 8 + q 4 + 1 pairwise non-collinear points in the Lie incidence geometry E 6 , 1 ( q ) . They were introduced by Cooperstein twenty-six years ago, motivated by the fact that possible non-existence of them would imply non-existence of ovoids in hyperbolic quadrics of rank 5. Since then, no progress has been made on their existence question. We prove that Cooperstein ovoids do not exist under some natural additional conditions. In particular, Cooperstein ovoids intersecting every symplecton of E 6 , 1 ( q ) do not exist, Cooperstein ovoids which are the fixed points of a collineation do not exist, and Cooperstein ovoids which are the absolute points of a polarity of E 6 , 1 ( q ) do not exist.

Suggested Citation

  • Hendrik Van Maldeghem, 2024. "Notes on Cooperstein Ovoids in Finite Geometries of Type 𝖤 6,1," Mathematics, MDPI, vol. 12(23), pages 1-11, November.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:23:p:3720-:d:1530652
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