I am interested in finding families $X\to C$ of elliptic curves over a projective curve $C$. The fibers are not all isomorphic, not all smooth, and the singular fibers should be nodal stable curves. Moreover, one of the following is true:

  • there are two sections $C\to X$, that are conjugated under some elliptic involution of the surface $X$, or
  • there are two sections $C\to X$, that are both fixed under some elliptic involution (i.e. if we use one to give a group structure on fibers, the other one will be identically $2$-torsion).

In any example, I would like to understand how many singular fibers there are, and their topological type.

I basically have one example of this sort in mind, which is the blowup at the $9$ base points of an elliptic pencil in $\mathbb{P}^2$. There are $9$ sections, and they don't intersect, but I don't know if I can choose them to satisfy these additional requirements.

Even if this turns out to work, I would still be interested in more examples, because I suspect that there are many, although I could not find any. In that case, references are very welcome.

  • 2
    $\begingroup$ One way to construct a lot of examples is to take hypersurfaces of type $(D,3)$ in $C \times \mathbb P^2$ for $D$ some base-point free divisor on $C$. This gives an explicit Weierstraß equation, so most things you want to compute should be fairly explicit. $\endgroup$ Feb 24 '18 at 3:59

This could be just a comment, but I make it an answer because it will be more useful to people looking for a reference.

These lectures by Rick Miranda http://www.math.colostate.edu/~miranda/BTES-Miranda.pdf contain the answers to all the above questions, and much more.

Fibrations of genus 1 are an old and well studied subject, starting with Kodaira who classified all the possible singular fibers. An account of the teory can be found also in Barth-Hulek-Peters Van de Ven book on complex surfaces.


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