The Classification of Exceptional CDQL Webs on Compact Complex Surfaces

TL;DR

This paper classifies exceptional CDQL webs on compact complex surfaces, identifying specific families and sporadic examples on the projective plane, advancing understanding of webs with maximal rank.

Contribution

It provides a global classification of exceptional CDQL webs on compact complex surfaces, including explicit families and sporadic examples on the projective plane.

Findings

Exactly four countable families of exceptional CDQL webs on the projective plane.

Thirteen sporadic exceptional CDQL webs identified.

Classification results extend understanding of webs of maximal rank.

Abstract

Codimension one webs are configurations of finitely many codimension one foliations in general position. Much of the classical theory evolved around the concept of abelian relation: a functional relation among the first integrals of the foliations defining the web reminiscent of Abel's addition theorem in classical algebraic geometry. The abelian relations of a given web form a finite dimensional vector space with dimension (the rank of the web) bounded by Castelnuovo number p(n,k) where n is the dimension of the ambient space and k is the number of foliations defining the web. A fundamental problem in web geometry is the classification of exceptional webs, that is, webs of maximal rank not equivalent to the dual of a projective curve. Recently, J.-M. Trepreau proved that there are no exceptional k-webs for n>2 and k > 2n-1. In dimension two there are examples of exceptional k-webs for arbitrary k and the classification problem is wide open. In this paper, we classify the exceptional Completely Decomposable Quasi-Linear (CDQL) webs globally defined on compact complex surfaces. By definition, the CDQL (k+1)-webs are formed by the superposition of k linear foliations and one non-linear foliation. For instance, we show that up to projective transformations there are exactly four countable families and thirteen sporadic exceptional CDQL webs on the projective plane.