On a new family of flag-transitive semibiplanes

Each of the d-dimensional dual hyperovals S(m)(h) discovered by Yoshiara [20] gives rise, via affine expansion, to a flag-transitive semibiplane Af (S(m)(h)). We prove that, if m + h = d + 1, then Af (S(m)(h)) is an elation semibiplane. In the other cases, if d > 2 then Af (S(m)(h)) is not isomorphic to any of the examples we are aware of, except possibly for certain semibiplanes obtained from D(n)-buildings defined over G F(2). However, many semibiplanes live hidden as quotients inside halved hypercubes. It is thus quite natural to ask whether any of our semibiplanes are like that. We prove that Af (S(m)(h)) is a quotient of a halved hypercube if and only if h = m. (C) 2001 Academic Press.