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Pseudo-ovals in even characteristic and Laguerre planes.

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Pseudo-arcs are the higher-dimensional generalisation of arcs in a projective plane: a pseudo-arc is a set A of (n-1)-spaces in PG(3n-1; q) such that any three span the whole space. Pseudo-arcs of size q^n + 1 are called pseudo-ovals, while pseudo-arcs of size q^n + 2 are called pseudo-hyperovals. A pseudo-arc is called elementary if it arises from applying field reduction to an arc in PG(2; q^n).

We explain the connection between dual pseudo-ovals and elation Laguerre planes and show that an elation Laguerre plane is ovoidal if and only if it arises from an elementary dual pseudo-oval. The main theorem of this paper shows that a pseudo-(hyper)oval in PG(3n - 1; q), where q is even and n is prime, such that every element induces a Desarguesian spread is elementary. As a corollary, this gives a characterisation of certain ovoidal Laguerre planes in terms of the derived affine planes.
Tijdschrift: J. Comb. Theory Ser. A
Issue: 129
Pagina's: 105-121
Jaar van publicatie:2014
Trefwoorden:pseudo-arcs, Laguerre plane, Desarguesian spread, pseudo-oval