Covers of elliptic curves with unique, totally ramified branch points (Q2862564)

This paper gives, for each \(g \geq 1\), an explicit formula for a family of degree \(2g-1\) maps from genus \(g\) curves (in characteristic zero) to elliptic curves. Every elliptic curve appears as a target curve in the family of maps. The maps are all branched at one point. The formula is first given and proven, but the real content of the paper is the derivation of the formula (which is not necessary for the proof).NEWLINENEWLINEThe derivation is given for the example of a degree \(3\) family of maps from genus \(2\) curves. The key observation is that one starts by writing down a map from a \textit{degenerate} genus \(2\) curve to a nodal cubic. This is relatively easy to do explicitly, by first writing down the map on the normalizations (which are just \(\mathbb{P}^1\)'s). Then, one makes an (honest, not formal) one-parameter deformation of this map, such that the other fibers are maps of smooth curves.