  We explore the relation between the positive dimensional irreducible
components of the characteristic varieties of rank one local systems on a
smooth surface and the associated (rational or irrational) pencils. Our study,
which may viewed as a continuation of D. Arapura's work, yields new geometric
insight into the translated components relating them to the multiplicities of
curves in the associated pencil, in a close analogy to the compact situation
treated by A. Beauville. The new point of view is the key role played by the
constructible sheaves naturally arising from local systems.
