Number of fibrations over a curve.

Question

Fix a non-singular complex projective curve $C$. I would like to know how many non-singular complex projective surfaces $S$ have the following properties (up to isomorphism):

1. There is a fibration $S\longrightarrow C$.
2. All the fibers have genus $g=0$.

Finitely many surfaces? Countable many or ucontable many surfaces?

Edit: Suppose also that the fibration is relatively minimal.

Many Thanks

Answer 1

Your question refers to the class of ruled surfaces over a smooth connected curve $C$.

1. Each ruled surface is isomorphic to the projective bundle $\mathbb P(V) \longrightarrow C$ of a vector bundle on $C$ of rank = 2.

2. Hence, the classification of ruled surfaces reduces to the classification of vector bundles on $C$ of rank = 2. Apparently $\mathbb P(V) \cong \mathbb P(V \otimes \mathscr O_C)$.

3. A survey of the theory of moduli spaces for semi-stable vector bundles on $C$ you find in

http://www.math.harvard.edu/~chaoli/doc/StableVectorBundles.html