I recently finished reading this paper, and was wondering about a couple of things relating to theorem 1, which says that for any curve X there is a curve Y and f:Y->X such that pushforward is a dominant rational map from the Jacobian of Y to the moduli of semistable vector bundles on X (with numerical invariants fixed to make things more definite.) So I had two questions:

1. Given a morphism of curves f:Y->X, is there a good characterization of the line bundles L on Y with f_*(L) semistable (or not semistable, equivalently)?

2. Given a morphism of curves f:Y->X, is there a good characterization of which semistable bundles are in/not in the image of f_*?

(those with more rep than me, can we add a vector-bundles tag?)

I recently finished reading this paper, and was wondering about a couple of things relating to theorem 1, which says that for any curve X there is a curve Y and f:Y->X such that pushforward is a dominant rational map from the Jacobian of Y to the moduli of semistable vector bundles on X (with numerical invariants fixed to make things more definite.) So I had two questions:

1. Given a morphism of curves f:Y->X, is there a good characterization of the line bundles L on Y with f_*(L) semistable (or not semistable, equivalently)?

2. Given a morphism of curves f:Y->X, is there a good characterization of which semistable bundles are in/not in the image of f_*?