Divisors of degree zero on curves of positive genus

Question

Let $C$ be a projective curve of genus $g\geq 1$, and $D$ a divisor of degree $d > 0$ on $C$. Let $P\in C$ be a point and $D_0 = D-dP$.

Then $D_0$ is a divisor of degree zero on $C$. If the point $p\in C$ is general do we have that $D_0$ is not a torsion divisor ($kD_0 \nsim \mathcal{O}_C$ for all $k\geq 1$) ?

Answer 1

Taking a point $P \in C$ to the divisor class $D - dP$ defines a regular morphism $$ f \colon C \to \mathrm{Pic}^0(C) $$ which is generically finite onto its image. In particular, $f(C)$ is a curve in $\mathrm{Pic}^0(C)$. Therefore, its general point does not belong to the countable set of torsion points of $\mathrm{Pic}^0(C)$.