The moduli space of curves of genus two covering elliptic curves
Let \(S_ n\) be the set of isomorphism classes of complex curves of genus \(2\) having a ``maximal'' map of degree \(n\) onto an elliptic curve (maximal means that the map does not factorize through an unramified covering of the elliptic curve). The purpose of this article is to prove that the class of a curve belongs to \(S_ n\) if and only if the class of its Jacobian belongs to the Humbert surface \(H_{n^ 2}\) in the moduli space of principally polarized abelian surfaces. This amounts to proving that it has a period matrix of the form \((\begin{smallmatrix} z_ 1 & 1/n & 1 & 0 \\ 1/n & z_ 2 & 0 & 1 \end{smallmatrix} )\) with \(\text{Im} z_ 1\), \(\text{Im} z_ 2 > 0\).
moduli space of principally polarized abelian surfaces
scientific article; zbMATH DE number 701158
scientific article