  We study the fixed singularities imposed on members of a linear system of
surfaces in P^3_C by its base locus Z. For a 1-dimensional subscheme Z \subset
P^3 with finitely many points p_i of embedding dimension three and d >> 0, we
determine the nature of the singularities p_i \in S for general S in |H^0 (P^3,
I_Z (d))| and give a method to compute the kernel of the restriction map Cl S
\to Cl O_{S,p_i}. One tool developed is an algorithm to identify the type of an
A_n singularity via its local equation. We illustrate the method for
representative Z and use Noether-Lefschetz theory to compute Pic S.
