Complex projective surfaces of nonnegative Kodaira dimension polarized by an ample and spanned line bundle of genus four

Let X be a complex projective smooth surface endowed with an ample line bundle L, which is spanned by its global sections. Pairs (X,L) as above are classified under the assumption that X is not birationally ruled and that g(L)=4. The result is obtained by combining the geometry of the finite morphism associated to |L| with the abstract structure of X.