Cyclic 2-spreads in \(V(6, q)\) and flag-transitive linear spaces (Q6597206)

This is a contribution to the classification of all finite linear spaces \(S\) admitting a flag-transitive group \(G\) of collineations; see [\textit{F. Buekenhout} et al., Geom. Dedicata 36, No. 1, 89--94 (1990; Zbl 0707.51017)] for a survey. Many cases of this classification are covered by \textit{M. W. Liebeck} [J. Comb. Theory, Ser. A 84, No. 2, 196--235 (1998; Zbl 0918.51009)] and \textit{J. Saxl} [J. Comb. Theory, Ser. A 100, No. 2, 322--348 (2002; Zbl 1031.51009)]; in the remaining open case, \(S\) has \(q\) points and \(G\le \mathrm {A\Gamma L}(1,q)\) for some prime power \(q\).\N\NThe authors of the paper under review use the polynomial approach of \textit{M. Pauley} and \textit{J. Bamberg} [Finite Fields Appl. 14, No. 2, 537--548 (2008; Zbl 1137.51007)] in the special situation where \(S\) has \(q^6\) points and \(G\) contains a cyclic subgroup of \(\mathrm{GL}(1,q^6)\) that acts transitively on a 2-spread in the vector space \(\mathbb F_q^6\) (i.e.\ on a set of 2-dimensional subspaces that yield a partition of \(\mathbb F_q^6 \setminus \{0\})\). For prime powers \(q\) coprime to \(6\), they describe all possible examples, determine the number of equivalence classes for \(S\), and show that these 2-spreads arise from certain cubic polynomials over \(\mathbb F_{q^2}\). Some new examples appear in this fashion.