  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.
