Simply branched covers of an elliptic curve and the moduli space of curves

TL;DR

This paper studies the geometry of a family of genus g curves covering elliptic curves simply branched at 2g-2 points, analyzing its slope, genus, and components to understand the moduli space of curves.

Contribution

It introduces a new approach using admissible covers to analyze the geometry of these special covers and their implications for the moduli space.

Findings

Determines the slope of the family W of covers.

Analyzes the genus and component structure of W.

Provides applications to the slopes of effective divisors on the moduli space.

Abstract

Consider genus g curves that admit degree d covers to an elliptic curve simply branched at 2g-2 points. Vary a branch point and the locus of such covers forms a one-parameter family W. We investigate the geometry of W by using admissible covers to study its slope, genus and components. The results can also be applied to study slopes of effective divisors on the moduli space of genus g curves.