I recently finished reading [this](http://math.unice.fr/~beauvill/pubs/bnr.pdf) 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\_\*?



Pushforwards of Line Bundles and Stability
Re: first question.


For semistability, we need a homomorphism from a line bundle U -> X of a certain degree to the pushdown of a line bundle L, which is the same thing as having a section on Y of f ^ \* U ^ \* \otimes L.


This can be expressed in terms of special spaces of divisors, and you can find details worked out explicitly for an example in rank 2 (i.e. a holomorphic double covering f: Y -> X) in pages 103-105 of [[NJH,87]](http://plms.oxfordjournals.org/cgi/pdf_extract/s3-55/1/59).


