Let $A \to B$ be a Galois cover of curves. In particular, $B \cong G \backslash A$, where $G$ is the Galois group. Why is it the case that the image of the injective map of Jacobians $Pic^0(B) \to Pic^0(A)$ is the connected component of the identity of the $G-$invariant sub-group variety of $Pic^0(A)$?
More in detail:
- Why is the map of Jacobians injective?
- The characterization of the image seems reasonable, because $Pic^0$ is a contravariant functor. But is this a special property of this specific functor? And shouldn’t this require exactness?
- Why do we need to take the connected component of the identity?