Pluriadjoint bundles of polarized surfaces

Let X be a complex connected projective algebraic surface and let L be an ample line bundle on X. The maps associated with the pluriadjoint bundles t(K_X+L), t \geq 2, are studied by combining an ampleness result for K_X+L with a very recent result by Reider. It turns out that apart from some exceptions and up to reductions: 1) 3(K_X+L) is very ample; 2) 2(K_X+L) is ample and spanned by global sections, and is very ample unless either g(L)=2 (arithmetic genus of L) or X contains an elliptic curve E with E^2 = 0, EL=1; 3) when 2(K_X+L) is not very ample, the associated map has degree \leq 4, equality implying that g(L)=2 and \chi(O_X)=0.