On the linearizability of 3-webs: end of controversy

TL;DR

This paper resolves a controversy by proving that a specific 3-web is linearizable, confirming the earlier theory and clarifying the correct criteria for linearizability in 3-webs.

Contribution

It provides a short proof confirming the linearizability of a 3-web, settling a debate between two conflicting theories.

Findings

The 3-web W is linearizable.

Confirms the theory from the 2001 article.

Clarifies the correct linearizability condition for 3-webs.

Abstract

There are two theories describing the linearizability of 3-webs: one is developed in the article "On the linearizability of 3-webs" (Nonlinear analysis 47, (2001) pp.2643-2654) and another in the article "On the Blaschke conjecture for 3-webs" (J. Geom. Anal. 16, 1 (2006), 69-115). Unfortunately, they cannot be both correct because on an explicit 3-web W they contradict: the first predicts that W is linearizable while the second states that W is not linearizable. The essential question beyond this particular 3-web is: which theory describes correctly the linearizability condition? In this paper we present a very short proof, due to J.-P.~Dufour, that W is linearizable, confirming the result of the first article.