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Re: first question.

For semistability, you 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 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].

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?

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?

Re: first question.

For semistability, you 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 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].

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 (from a different cover).

In the paper, he proves the conjecture with certain restrictions; in particular, it holds whenever f is an etale covering.

Does this help?

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?