Numerically positive divisors on algebraic surfaces

Numerically positive line bundles on a complex projective smooth algebraic surface S are studied. In particular, for any such line bundle L \in Pic(S) we prove the following facts: (i) g(L) \geq 0 and (ii) L is a mple if g(L) \leq 1, g standing for the arithmetic genus. Some applications are discussed. We also investigate numerically positive non-ample line bundles L with g(L)=2.