Notes on Cooperstein Ovoids in Finite Geometries of Type 𝖤6,1

A Cooperstein ovoid is a set of q8+q4+1 pairwise non-collinear points in the Lie incidence geometry E6,1(q). They were introduced by Cooperstein twenty-six years ago, motivated by the fact that possible non-existence of them would imply non-existence of ovoids in hyperbolic quadrics of rank 5. Since then, no progress has been made on their existence question. We prove that Cooperstein ovoids do not exist under some natural additional conditions. In particular, Cooperstein ovoids intersecting every symplecton of E6,1(q) do not exist, Cooperstein ovoids which are the fixed points of a collineation do not exist, and Cooperstein ovoids which are the absolute points of a polarity of E6,1(q) do not exist.