Let $C$ be a smooth projective curve of genus $g$ over a field $k$ and let $J$ denote its Jacobian. Let $P$ be a $k$-rational point on $C$, and let $r$ be a natural number. Then there is a morphism from the symmetric power $C^{(r)}$ of $C$ to $J$ which is informally defined as follows:
$\sum P_i \mapsto [\sum P_i - rP]$
Let $D$ be a divisor on $C$ of degree $0$ and let $\mathcal{L}$ be the corresponding invertible sheaf. Then the fiber of this map above $\mathcal{L}$ can be identified with the collection of all divisors on $C$ of degree $r$ which are linearly equivalent to $D+rP$. These in turn can be identified with the one dimensional subspaces of the vector space of global sections $\Gamma(C, \mathcal{L})$. If $r$ is sufficiently large depending on the genus then by Riemann Roch the above is always a vector space of dimension $r - g + 1$ so morally the morphism should locally look like a projection $J \times \mathbb{P}^{r - g} \mapsto J$. However, I am not sure if this is true or how to prove this...