MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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_*?

share|cite|improve this question
up vote 2 down vote accepted

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].

share|cite|improve this answer
Just got a chance to look at it, and this is pretty much what I was after. Thanks – Charles Siegel Nov 6 '09 at 0:36

A later paper of Beauville (here, preprint from 2000) seems to address your first question, but more generally by taking direct images of vector bundles. He makes the following conjecture:

If f: X'-> X is a finite morphism of smooth projective curves and E is a generic vector bundle on X', then f_*E is stable if g(X) \geq 2 and is semistable if g(x) = 1.

The problem is actually equivalent to that of a pushing down a line bundle L (from a different cover).

In the paper, he shows that the conjecture holds with some restrictions on L (e.g. when \chi(X) is small), although obviously one wants it to hold more generally. He shows, as well, that the conjecture holds as worded above whenever f is an etale covering.

Does this help?

share|cite|improve this answer
Eh, not really. I'm looking for more "what does generic mean in this case?" type of answer, and I don't really see anything in there addressing this. – Charles Siegel Oct 28 '09 at 5:03
Yeah, I realise now that you were looking for more, and that I should have read the questions better. Please see my next attempt below... (Revision note for those following this thread: originally I had directly replaced my old answer with the new one, but I think this made things confusing, since Charles' comment above was directed at my original attempt. So now my new answer is a separate post below. Sorry for the confusion.) – 1-- Nov 1 '09 at 23:32

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.