Solids in the space of the Veronese surface in even characteristic

Abstract

We classify the orbits of solids in the projective space PG(5, q), q even, under the setwise stabiliser K ∼ = PGL(3, q) of the Veronese surface. For each orbit, we provide an explicit representative S and determine two combinatorial invariants: the point-orbit distribution and the hyperplane-orbit distribution. These invariants characterise the orbits except in two specific cases (in which the orbits are distinguished by their line-orbit distributions). In addition, we determine the stabiliser of S in K, thereby obtaining the size of each orbit. As a consequence, we obtain a proof of the classification of pencils of conics in PG(2, q), q even, which to the best of our knowledge has been heretofore missing in the literature.