Orbits of tensors over finite fields

This thesis forms part of a project aiming to classify subspaces of PG(5, q) under the action of the subgroup K < PGL(6, q) stabilising the Veronese surface V(Fq), where Fq is the finite field of order q. Firstly, we determine the K-orbits of solids of PG(5, q) in the case where q is even. We compute as well two useful combinatorial invariants of each type of solids, namely their point-orbit and hyperplane-orbit distributions. Additionally, we calculate the stabiliser of each orbit representative, and thereby obtain the size of each orbit. The classification of solids in PG(5, q) corresponds to the classification of pencils of conics in PG(2, q), q even. The latter classification was incompletely obtained by Campbell in 1927. Our results complete Campbell’s work and correct two of his claims. Moreover, we give a partial classification of planes in PG(5, q), q even. Specifically, we determine the K-orbits of planes intersecting the Veronese surface in at least one point. Our proof is geometric based on studying the different types of points that are incident with a plane PG(5, q). In some cases, point orbit-distributions are not sufficient to characterise each orbit, and we tend to determine stronger geometric-combinatorial invariants such as iv line-orbit distributions and inflexion points. Finally, we introduce the GAP package, T233, which uses some functionality from the FinInG package to determine G-orbits and ranks of points in PG(F2q F3q F3q ) = PG(17, q), where G is the group stabilising the Segre variety S1,2,2(Fq). Note that, the algorithms defined in T233 and the combinatorial tools introduced earlier can be generalised to higher-ordered tensor product spaces, and thus one may extend these implementation tools and classifications to higher-ordered tensor product spaces