The class of a Hurwitz divisor on the moduli of curves of even genus

Abstract

We study the geometry of the natural map from the Hurwitz space $\overline{H}_{2k,k+1}$ to the moduli space $\overline{\mathcal{M}}_{2k}$. We calculate the cycle class of the Hurwitz divisor $D_2$ on $\overline{\mathcal{M}}_g$ for $g = 2k$ given by the degree $k + 1$ covers of $\mathbb{P}^1$ with simple ramification points, two of which lie in the same fibre. This has applications to bounds on the slope of the moving cone of $\overline{\mathcal{M}}_g$, the calculation of other divisor classes and motivated an algebraic proof for the formula of the Hodge bundle of the Hurwitz space.