Pseudo-ovals in even characteristic and ovoidal Laguerre planes

Abstract

Pseudo-arcs are the higher dimensional analogues 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 qn+1 are called pseudo-ovals, while pseudo-arcs of size qn+2 are called pseudo-hyperovals. A pseudo-arc is called elementary if it arises from applying field reduction to an arc in PG(2,qn).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, we give a characterisation of certain ovoidal Laguerre planes in terms of the derived affine planes.