Topological Formulae for the Zeroth Cohomology of Line Bundles on Surfaces

We identify a set of transforms on the Picard lattice of non-singular complex projective surfaces that map effective line bundles to nef line bundles, while preserving the dimension of the zeroth cohomology. These transforms can often be used in conjunction with vanishing theorems to compute the dimension of the zeroth cohomology in terms of a topological index. The method is illustrated on del Pezzo and Hirzebruch surfaces.