The classification of exceptional CDQL webs on compact complex surfaces

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 equation among the first integrals of the foliations defining the web reminiscent of Abel's Addition Theorem. The abelian relations of a given web form a finite-dimensional vector space with dimension (the rank of the web) bounded by Castelnuovo number π(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, Trépreau proved that there are no exceptional k-webs for n ≥ 3 and k ≥ 2n. In dimension two, there are examples for arbitrary k ≥ 5 and the classification of exceptional webs is wide open. In this paper, we classify the exceptional completely decomposable quasi-linear (CDQL) webs globally defined on compact complex surfaces. By definition, these are the exceptional (k + 1)-webs on compact complex surfaces that are formed by the superposition of k "linear" and one non-linear foliations. For instance, we show that up to projective transformations there are exactly four countable families and thirteen sporadic examples of exceptional CDQL webs on the projective plane.