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

Official URL: http://dx.doi.org/10.1007/s00022-020-00548-1

We classify nets of conics of rank one 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 PG(5, q) which meet the quadric Veronesean in at least one point, under the action of PGL(3, q) <= PGL(6, q) (for q odd). Our results complete a partial classification of nets of conics of rank one obtained by Wilson (Am J Math 36:187-210, 1914).