Solids in the space of the Veronese surface in even characteristic

TL;DR

This paper classifies the orbits of solids in a projective space over a finite field of even characteristic under a group action, providing explicit representatives, invariants, and stabilizers, and applies this to classify pencils of conics.

Contribution

It offers a complete classification of solids in PG(5,q) under the stabilizer group for even q, including explicit representatives and invariants, and completes the classification of pencils of conics.

Findings

Classified all orbits of solids under the stabilizer group in PG(5,q) for even q.

Provided explicit representatives and invariants for each orbit.

Determined the stabilizers and orbit sizes, and proved the classification of pencils of conics.

Abstract

We classify the orbits of solids in the projective space $\text{PG}(5,q)$, $q$ even, under the setwise stabiliser $K \cong \text{PGL}(3,q)$ 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 $\text{PG}(2,q)$, $q$ even, which to the best of our knowledge has been heretofore missing in the literature.