  In this paper we provide a purely algebraic characterization of the exponents
of one-dimensional direct images of a structure sheaf by a rational function,
related to the vanishing of the cohomologies of a certain Koszul complex
associated with such a morphism. This can be extended to a more general family
of Gauss-Manin systems. As an application, we calculate a set of possible
exponents of the Gauss-Manin cohomology of some arrangements of hyperplanes
with multiplicities, relevant to Dwork families and mirror symmetry.
