Classification of kk-nets

A finite kk-net   of order nn is an incidence structure consisting of k≥3k≥3 pairwise disjoint classes of lines, each of size nn, such that every point incident with two lines from distinct classes is incident with exactly one line from each of the kk classes. Deleting a line class from a kk-net, with k≥4k≥4, gives a derived   (k−1k−1)-net of the same order. Finite kk-nets embedded in a projective plane PG(2,K) coordinatized by a field KK of characteristic 0 only exist for k=3,4k=3,4, see Korchmáros et al. (2014). In this paper, we investigate 3-nets embedded in PG(2,K) whose line classes are in perspective position with an axis rr, that is, every point on the line rr incident with a line of the net is incident with exactly one line from each class. The problem of determining all such 3-nets remains open whereas we obtain a complete classification for those coordinatizable by a group. As a corollary, the (unique) 4-net of order 3 embedded in PG(2,K) turns out to be the only 4-net embedded in PG(2,K) with a derived 3-net which can be coordinatized by a group. Our results hold true in positive characteristic under the hypothesis that the order of the kk-net considered is smaller than the characteristic of KK.