ag.algebraic geometry - Curves on elliptic ruled surfaces?

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, $fcolon C to E$ 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 $S$.

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