Moduli of abelian covers of elliptic curves

TL;DR

This paper investigates the structure of moduli spaces of abelian covers of elliptic curves, revealing their components, Picard groups, and birational relations, with specific results for bielliptic curves.

Contribution

It identifies irreducible components of these moduli spaces and characterizes their Picard groups, especially in the totally ramified and bielliptic cases.

Findings

Moduli space has trivial rational Picard group in the totally ramified case.

The moduli space is birational to M_{1,n} for n branch points.

Boundary divisors form a basis of the Picard group in the bielliptic case.

Abstract

For any finite abelian group G, we study the moduli space of abelian $G$-covers of elliptic curves, in particular identifying the irreducible components of the moduli space. We prove that, in the totally ramified case, the moduli space has trivial rational Picard group, and it is birational to the moduli space M_{1,n}, where n is the number of branch points. In the particular case of moduli of bielliptic curves, we also prove that the boundary divisors are a basis of the rational Picard group of the admissible covers compactification of the moduli space. Our methods are entirely algebro-geometric.