In an algebraic function field $F/k$ we have the degree zero divisor class group $\text{Cl}^0(F/k)$. Now since any such function field corresponds to the function field of a normal projective curve over $k$ (in particular regular, Noetherian), it seems to me quite reasonable to ask about the corresponding object in this geometric case. How does it look like, is it simply the same, only dealing with cartier divisors?
And further: Does anything change if we don't start with the function field but with a $\underline{\text{non-regular}}$ integral and complete (projective) curve over $k$?