Classification of k-nets

A finite k-net of order n is an incidence structure consisting of k ≥ 3 pairwise disjoint classes of lines, each of size n, such that every point incident with two lines from distinct classes is incident with exactly one line from each of the k classes. Deleting a line class from a k-net, with k ≥ 4, gives a derived (k−1)-net of the same order. Finite k-nets embedded in a projective plane PG(2,K) coordinatized by a field K of characteristic 0 only exist for k = 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 r, that is, every point on the line r 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 k-net considered is smaller than the characteristic of K.