On nets of order \(q^ 2\) and degree \(q+1\) admitting GL(2,\(q\)) (Q1311937)

This paper classifies all nets of order \(q^ 2\) and degree \(q+1\) which admit a collineation group \(G\) with a regular (w.r.t. the action on points) normal subgroup \(T\) such that \(G/T\) is isomorphic to \(\text{GL} (2,q)\) \((q>3)\). The nets that satisfy these conditions are all obtained as follows. Let \({\mathcal S}\) be a set of \(q+1\) mutually skew lines (partial spread) of \(\text{PG} (3,q)\) and view the latter as the space at infinity of the affine space \(\text{AG} (4,q)\). The points of the net are the points of \(\text{AG} (4,q)\); the lines are the planes of \(\text{AG} (4,q)\) that have a member of \({\mathcal S}\) as line at infinity and incidence is the natural one. For the set \({\mathcal S}\), there are 4 possibilities. Either one takes a regulus of a hyperbolic quadric, or one takes \(q+1\) mutually skew lines on the surface with equation \(x_ 1x^ \sigma_ 2+x_ 3x^ \sigma_ 4=0\) (with \(\sigma\) a nontrivial field automorphism), or \({\mathcal S}\) defines a subnet of a Hering plane of order \(q^ 2\), or \({\mathcal S}\) is a sporadic example in \(\text{PG} (3,5)\). The proof of this result is rather long and technical, but in my opinion clearly written and well structured.