  The paper provides a combinatorial method to decide when the space of local
systems with non vanishing first cohomology on the complement to an arrangement
of lines in a complex projective plane has as an irreducible component a
subgroup of positive dimension. Partial classification of arrangements having
such a component of positive dimension and a comparison theorem for cohomology
of Orlik-Solomon algebra and cohomology of local systems are given. The methods
are based on Vinberg-Kac classification of generalized Cartan matrices and
study of pencils of algebraic curves defined by mentioned positive dimensional
components.
