Classification of k-nets

TL;DR

This paper classifies certain 3-nets in projective planes, especially those with group-coordinatizable structures, and explores their properties and uniqueness, including implications for 4-nets of order 3.

Contribution

It provides a complete classification of group-coordinatizable 3-nets in projective planes and characterizes the unique 4-net of order 3 with a derived 3-net.

Findings

Classification of 3-nets with perspective line classes

Identification of the unique 4-net of order 3 with a group-coordinatizable derived 3-net

Results applicable in positive characteristic under certain order constraints

Abstract

A finite \emph{$k$-net} of order $n$ is an incidence structure consisting of $k\ge 3$ 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 $k\ge 4$, gives a \emph{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 \cite{knp_k}. 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 $K$.