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?
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?
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.