The Moduli of Klein Covers of Curves

TL;DR

This paper investigates the structure and compactification of the moduli space of Klein four covers of algebraic curves, analyzing boundary components, intersections, and the canonical divisor to deepen understanding of these geometric objects.

Contribution

It constructs a new space with a basis choice for Klein four groups, analyzes boundary components, and computes the canonical divisor of the compactified moduli space.

Findings

Identifies intersection of components along boundary

Determines degrees of boundary components over moduli space

Computes the canonical divisor of the compactified space

Abstract

We study the moduli space ${V}_4\mathcal{M}_{g}$ of Klein four covers of genus $g$ curves and its natural compactification. This requires the construction of a related space which has a choice of basis for the Klein four group. This space has the interesting property that the two components intersect along a component of the boundary. Further, we carry out a detailed analysis of the boundary, determining components, degrees of the components over their images in $\overline{\mathcal{M}_g}$, and computing the canonical divisor of $\overline{{V}_4\mathcal{M}_{g}}$.