A characterisation of spreads ovally-derived from Desarguesian spreads

Abstract

We characterise all spreads that are obtainable from Desarguesian spreads by replacing a partial spread consisting of translation ovals; the corresponding “ovally-derived” planes are generalised André planes, of order 2N, although not all generalised André planes are ovallyderived from Desarguesian planes. Our analysis allows us to obtain a complete classification of all nearfield planes that are ovally-derived from Desarguesian planes. It turns out that whether or not a nearfield plane is ovally-derived from a Desarguesian plane depends solely on the parametersq andr, where GF (q) is the kern, andr is the dimension of the plane. Our results also imply that a Hall plane of even orderq 2 can be ovally-derived from a Desarguesian spread if and only ifq is a square.

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Jha, V., Johnson, N.L. A characterisation of spreads ovally-derived from Desarguesian spreads. Combinatorica 14, 51–61 (1994). https://doi.org/10.1007/BF01305950

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AMS subject classification codes (1991)

  • 51 A 40
  • 51 E 21