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

On polarized surfaces of low degree whose adjoint bundles are not spanned