A note on tensor products of polar spaces over finite fields

Abstract

A symplectic or orthogonal space admitting a hyperbolic basis over a nite eld is tensored with its Galois conjugates to obtain a symplectic or orthogonal space over a smaller eld. A mapping between these spaces is dened which takes absolute points to absolute points. It is shown that caps go to caps. Combined with a result of Dye’s one obtains a simple proof of a result due to Blokhuis and Moorehouse that ovoids do not exist on hyperbolic quadrics in dimension ten over a eld of characteristic two. Let k = GF (q), q a prime power, and K = GF (q m ) for some positive integer