Let $C$ be an effective Cartier divisor on a smooth projective surface $S$ over an algebraically closed field $k$ of characteristic $0$.

Assume that $C$ has a **reduced** irreducible component $C_1$ (even though $C$ may not be reduced) and assume that $h^1(C, \mathcal{O}_C) =1$.**(Edited)**

Let $x_1 \in C_1$ be a closed point on the smooth part $C_1^0$ of $C_1$.

**Question**
(1)
Is it possible to define $a \colon C_1^0 \rightarrow {\rm Pic}^0 C ; x \mapsto \mathcal{O}_C(x-x_1)$?

(2) If so, is $a$ a non-constant morphism?