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 $\overline{ { V}_4\mathcal{ M}_{ g}}$.
Categories: math.AG