Practice problem

Sheaves and Algebraic Geometry

In this problem, a curve over a field $k$ will refer to a reduced separated scheme of finite type over $k$ , of dimension one. In particular, we allow reducible and/or singular curves.

Let $k = \overline{k}$ be an algebraically closed field. Let $\pi\colon X\to Y$ be a birational morphism of curves over $k$ . How are the Picard groups of $X,Y$ related via $\pi$ ?