On the structure of 3-nets embedded in a projective plane

TL;DR

This paper studies the geometric structure of irregular 3-nets embedded in projective planes over various fields, revealing conditions under which lines are tangent to conics or cubics and classifying small cases.

Contribution

It provides new results on the structure of irregular 3-nets, including conditions for concurrency and tangency, and offers a complete classification for nets of order 4.

Findings

If lines from two classes are tangent to the same irreducible conic, the third class lines are concurrent.

In the complex plane, irregular 3-nets are tangent to a cubic curve, except for a known sporadic case.

Complete classification of irregular 3-nets of order 4 in positive characteristic.

Abstract

We investigate finite 3-nets embedded in a projective plane over a (finite or infinite) field of any characteristic p. Such an embedding is regular when each of the three classes of the 3-net comprises concurrent lines, and irregular otherwise. It is completely irregular when no class of the 3-net consists of concurrent lines. We are interested in embeddings of 3-nets which are irregular but the lines of one class are concurrent. For an irregular embedding of a 3-net of order n greater than 4 we prove that, if all lines from two classes are tangent to the same irreducible conic, then all lines from the third class are concurrent. We also prove the converse provided that the order n of the 3-net is smaller than p. In the complex plane, apart from a sporadic example of order n=5 due to Stipins, each known irregularly embedded 3-net has the property that all its lines are tangent to a plane cubic curve. Actually, the procedure of constructing irregular 3-nets with this property works over any field. In positive characteristic, we present some more examples for n greater than 4 and give a complete classification for n=4.