Le rang des tissus de Nakai

TL;DR

This paper investigates Nakai's webs, a special class of planar 4-webs characterized by constant cross-ratio and no hexagonal 3-subweb, proving their rank is limited to 0 or 1 and providing methods to construct examples.

Contribution

It establishes the rank classification of Nakai's webs and introduces a universal construction method for rank 1 examples.

Findings

Nakai's webs have rank 0 or 1.

Provided explicit examples of rank 1 Nakai's webs.

Presented a universal construction method for these webs.

Abstract

According to Alain H\'enaut, a planar 4-web is called Nakai's web if the cross-ratio of the tangents to the four foliations at each point is constant and if it has no hexagonal 3-subweb. We prove that Nakai's webs have rank 0 or 1. We give examples with rank 1 and present a universal way to build such examples.