Let $Y$ be a smooth projective complex curve of genus 2 and $f : X \to Y$ a finite etale cover. Choose two distinct points A and B on $Y$ and let $\Sigma\subset X$ be the set of complex points $P$ such that $f(P)$ is A or B. For any $x\in \Sigma$ choose an element $L(x) \in Pic(Y)[2]$. Does there exist an element $L\in Pic(X\times Y)[2]$ such that, for every $x \in \Sigma$, the restriction of $L$ to $x\times Y = Y$ is equal to $L(x)$?

Angelo says that the restriction of $L$ to $x\times Y = Y$ is constant. Is it so and why?

OK. Thank you for the answer.