Nets of conics of rank one in PG(2,q), q odd

TL;DR

This paper classifies nets of conics of rank one in projective planes over finite fields of odd order, using geometric methods to analyze orbit structures under group actions, completing a nearly century-old classification effort.

Contribution

It provides a complete geometric classification of nets of conics of rank one in PG(2,q) for odd q, extending Wilson's earlier partial results.

Findings

Classified the orbits of planes in PG(5,q) meeting the Veronesean in at least one point.

Connected the classification to group actions of PGL(3,q) on these planes.

Completed a nearly century-old classification of nets of conics of rank one.

Abstract

We classify nets of conics in Desarguesian projective planes over finite fields of odd order, namely, two-dimensional linear systems of conics containing a repeated line. Our proof is geometric in the sense that we solve the equivalent problem of classifying the orbits of planes in $\text{PG}(5,q)$ which meet the quadric Veronesean in at least one point, under the action of $\text{PGL}(3,q) \leqslant \text{PGL}(6,q)$ (for $q$ odd). Our results complete a partial classification of nets of conics of rank one obtained by A. H. Wilson in the article "The canonical types of nets of modular conics", American Journal of Mathematics 36 (1914) 187-210.