Let $\mathbb{P}(X)$ be a non-trivial $\mathbb{P}^1$-bundle over a curve $C$ (here $X$ is a vector bundle of rank 2 over $C$). A minimal section of $\mathbb{P}(X)$ is a section of minimal self-intersection.
My questions are the following:
- Can $\mathbb{P}(X)$ admit a infinite number of minimal section? If not, why?
- Is there a known example of non trivial $\mathbb{P}^1$-bundle with $3$ distinct minimal sections (or infinite number of minimal section, and it would also answer my first question)?
The related examples that I know are the following:
- If $\mathbb{P}(X)\simeq \mathbb{F}_n$ is a Hirzebruch surface (i.e $C\simeq \mathbb{P}^1$) with $n\neq 0$. Then there exists an unique minimal section of self-intersection $-n$.
- If $C$ is an elliptic curve. It follows from the article (plus a little extra work): Maruyama, M., On automorphism groups of ruled surfaces, J. Math. Kyoto Univ. 11, 89-112 (1971). ZBL0213.47803., that all $\mathbb{P}^1$-bundles over $C$ have at most $2$ minimal sections.
Actually Maruyama shows for all curve $C$, independently of the genus, that if the minimal self-intersection is negative then the minimal section is unique. In particular, if the example I am looking for exists, then it should be over a curve of genus $\geq2$ and the minimal self-intersection is non-negative. Unfortunately, I don't know anything on vector bundles of rank $2$ over a curve of genus $\geq 2$ and I have not found any concrete examples ...
Thank you in advance for your time!