Curves on elliptic ruled surfaces? - MathOverflow

Curves on elliptic ruled surfaces?

2009-12-18T18:10:25Z

Let $S\overset{\pi}{\to} E$ be a ruled surface over an elliptic curve over complex field. Clearly, there are rational curves and elliptic curves on $S$. Is there any higher genus curves on $S$. Are all the elliptic curves isomorphic. The reason for the second question is that, all the smooth sections are isomorphic elliptic curves, except this sections is there any other elliptic curves?If so, how do I find them?

Answer by Felipe Voloch for Curves on elliptic ruled surfaces?

2009-12-18T18:59:45Z

Any surface has lots of curves of high genus. Just take a generic hypersurface section of high degree.

Any other elliptic curve will be isogenous to $E$ with $\pi$ inducing the isogeny. I think you can embed any isogenous curve in $S$.

Answer by Joel Fine for Curves on elliptic ruled surfaces?

2009-12-19T00:35:31Z

To find higher genus curves without using a specific embedding $S \subset \mathbb{P}^n$, it could help to think first about the case when your surface is actually a product $S=\mathbb{P}^1 \times E$. Let $C$ be a curve which admits two branched covers, $f\colon C \to E$ and $g \colon C \to \mathbb{P}^1$. Then the product $f \times g \colon C \to S$ maps into the surface $S$. If the branch points of $f$ and $g$ are different then $f \times g$ will even be an embedding.

In general, let $V \to E$ be your rank-two vector bundle, so $S=\mathbb{P}(V)$. Given a banched cover $f \colon C \to E$, you pull back $V$ to a bundle $V' \to C$. Now every time you have a line sub-bundle $L$ of $V'\to C$ you get a section of $\mathbb{P}(V')$ which plays the role of $g$ in the first paragraph. It can be combined with $f \colon C \to E$ to give a map $C \to S$. Depending on how much you know about $E$ and $V$, hopefully this should help you find plenty of explicit curves in $S$.