line bundles and jacobians

Question

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.

Answer 1

If $Y$ is a complex projective algebraic variety, the Picard group $\operatorname{Pic}Y$ has the structure of an algebraic variety; if $X$ is another algebraic variety, any line bundle $L$ gives a regular map $X \to \operatorname{Pic}Y$ by sending $x \in X$ to the class of $L \mid x\times Y$. If $L$ is 2-torsion, then the map $X \to \operatorname{Pic}Y$ has image contained in $\operatorname{Pic}(Y)[2]$; but $\operatorname{Pic}(Y)[2]$ is finite, so if $X$ is connected, the map has to be constant.