Polarized surfaces of positive Kodaira dimension with canonical bundle of small degree

Let X be a smooth complex projective surface and letL be an ample line bundle on it. Assumme that X has positive Kodaira dimension, so that the intersection index LKX —- k of L with the canonical bundle of X is always positive. Here we classify pairs (X, L) as above when k = \,2. We also specify our results in the case whenZ, is spanned by its global sections.