Is there a unique line bundle in the Kummer surface which pulls back to a totally symmetric line bundle?

Question

Let $X=Jac(C)$ be an abelian surface over $\mathbb{C}$, the Jacobian of a genus 2 curve. Let $L$ be a symmetric line bundle. Let $Y$ be the Kummer surface, quotient of $X$ by the action of involution. Then $L^2$ is totally symmetric, hence there is a line bundle $L'$ on $Y$ which pulls back to $L^2$. Further since $L^2$ embeds $Y$ in $\mathbb{P}^3$, $L'$ is very ample.

1) Is $L'$ the unique line bundle which pulls back to $L^2$?

2) Consider a curve $C\in |L^2|$, which is smooth, preserved under involution and avoiding the 16 double points of $X$. Then the image of $C'$ is smooth and avoids the 16 singular points of $Y$. Can we say that $C'\in |L'|$?

Thanks!

Answer 1

1) Yes. If there is another one, it differs from $L'$ by a line bundle $M$ with $M^{2}\cong \mathcal{O}_Y$. Consider the resolution $\pi :\hat{Y}\rightarrow Y$ obtained by blowing up the double points $p_1,\ldots ,p_{16}$. Since $\hat{Y}$ is simply connected, we have $\pi^* M\cong \mathcal{O}_{\hat{Y}}\ $.

Thus $M_{|Y\smallsetminus \{p_i\}}$ is trivial, and this implies that $M$ is trivial.

2) Yes by 1), since the pull-back of $\mathcal{O}_Y(C')$ to $X$ is $\mathcal{O}_X(C)=L^2$.