The Hilbert scheme of 2 points on an elliptic curve $C$, $Hilb^2(C)$, has a natural structure of ruled surface, given by the map $f:Hilb^2(C) \to C$ such that $f(P,Q)=P+Q$.


What can we say about the corresponding locally free sheaf of rank 2 on $C$?



Hilbert scheme of 2 points on an elliptic curve
In this case the corresponding locally free sheaf or rank $2$ on $C$ is the unique indecomposable one.


