Grothendieck-Lefschetz Theorem with Base Locus
  We compute the divisor class group of the general hypersurface Y of a complex projective normal variety X of dimension at least four containing a fixed base locus Z. We deduce that completions of normal local complete intersection domains of finite type over the complex numbers of dimension $\ge 4$ are completions of UFDs of finite type over the complex numbers. 
John Brevik and Scott Nollet
math.AG