# line bundles and jacobians

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 $$L(x) \in Pic(Y)$$. Does there exist an element $$L\in Pic(X\times Y)$$ 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.

• What is the relationship between $x \in \Sigma$ and $L(x)$? – Simon Rose Dec 8 '19 at 8:29
• None. Another way to phrase the question: is the restriction map $Pic(X\timesY) \ra Pic(\Sigma\times Y)$ surjective? – user95246 Dec 8 '19 at 8:50
• Given an element $L\in Pic(X\times Y)$, the associated map $X to Pic(Y)$ is constant. – Angelo Dec 8 '19 at 8:53

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 $$\operatorname{Pic}(Y)$$; but $$\operatorname{Pic}(Y)$$ is finite, so if $$X$$ is connected, the map has to be constant.