Let $C$ be a smooth curve and $J$ its Jacobian. Let $p$ be a point on $C$ and $j: C \to J$ be the map $x \mapsto x-p$. Let $\theta$ be the Theta divisor on $J$, i.e. the locus $\{ x_1 + \cdots + x_{g-1} - (g-1)p \mid x_i \in C\}$. It follows from Poincare formula that $j^* \mathcal{O}(\theta)$ is a degree $g$ line bundle on $C$. The question is: what is this line bundle? This seems to be something well-known, but I couldn't find any references.

Let $C$ be a smooth curve and $J$ its Jacobian. Let $p$ be a point on $C$ and $j: C \to J$ be the map $x \mapsto x-p$. Let $\theta$ be the Theta divisor on $J$, i.e. the locus $\{ x_1 + \cdots + x_{g-1} - (g-1)p \mid x_i \in C\}$. It follows from Poincaré formula that $j^* \mathcal{O}(\theta)$ is a degree $g$ line bundle on $C$. The question is: what is this line bundle? This seems to be something well-known, but I couldn't find any references.