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

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).