The Classification of Exceptional CDQL Webs on Compact Complex Surfaces
Jorge Vitório Pereira, Luc Pirio · International Mathematics Research Notices · 2009
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.