[Math] pullback of theta divisor

Question

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.

Answer 1

There is no canonical way to distinguish a particular theta divisor within its algebraic equivalence class. So, in order to get a meaningful answer to any question of this kind, one must specify which theta divisor is actually considering.

For some particular choice of $\Theta$, the answer can be found in [Birkhenake-Lange, Complex Abelian Varieties, Exercise 10 p. 361]. For the reader's convenience I will restate it here.

Proposition. Let $C$ be a smooth curve of genus $g$ and let $\alpha_c \colon C \to J$ be the embedding with respect to the point $c \in C$.

Let $\Theta \subset J$ be the theta divisor defined by $$\alpha_L^* \Theta = W_{g-1},$$
where $L=\omega_g \otimes \mathscr{O}((1-g) \cdot c)$. Then one has $$\alpha_c^* \mathscr{O}_J(\Theta) = \mathscr{O}_C (g \cdot c).$$