Divisors class groups of singular surfaces

Abstract: We compute divisors class groups of singular surfaces. Most notably we produce an exact sequence that relates the Cartier divisors and almost Cartier divisors of a surface to the those of its normalization. This generalizes Hartshorne's theorem for the cubic ruled surface in P^3. We apply these results to limit the possible curves that can be set-theoretic complete intersection in P^3 in characteristic zero.