Solids in the space of the Veronese surface in even characteristic

Official URL: https://dx.doi.org/10.1016/j.ffa.2022.102068

This paper forms part of a project aiming to determine the orbits of subspaces of PG(5,q) under the subgroup PGL(3,q) of the setwise stabiliser of the Veronese surface in PGL(6,q), where Fq is the finite field of order q. Specifically, we classify the orbits of solids of PG(5,q) in the case where q is even. We also determine two useful combinatorial invariants of each type of solid, namely their point-orbit and hyperplane-orbit distributions. Additionally, we calculate the stabiliser in PGL(3,q) of each (type of) solid S, and thereby determine the size of each orbit. The classification of solids is equivalent to the classification of pencils of conics in PG(2,q), q even, under coordinate transformations. The latter classification is known, but it seems that no proof is recorded in the literature, which generally points to a 1927 article of Campbell containing only an incomplete classification. Our approach is different and independent of Campbell's work, which we correct and complete.