  Smooth complex surfaces polarized with an ample and globally generated line
bundle of degree three and four, such that the adjoint bundle is not globally
generated, are considered. Scrolls of a vector bundle over a smooth curve are
shown to be the only examples in degree three. Two classes of examples in
degree four are presented, one of which is shown to characterize regular such
pairs. A Reider-type theorem is obtained in which the assumption on the degree
of the polarization is removed.
