X and Y are irreducible curves, and f:X-->Y a morphism. Let E be a vector bundle over X.


When does there exist a vector bundle F over Y such that f\*F=E and when will it be unique?



When does a vector bundle descend?
If $f$ does not factor through a point then $f$ is flat. So, you can use the usual descent condition --- let $p\_1,p\_2:X\times\_Y X \to X$ be the projections. Then $E$ descends if there is an isomorphism $p\_1^\*E \cong p\_2^\*E$ satisfying the cocycle condition on the triple fiber product. Each such isomorphism gives a descent.


