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

TL;DR

This paper computes the cycle class of a specific Hurwitz divisor on the moduli space of even genus curves, revealing geometric properties of admissible covers and their relation to stable curves.

Contribution

It provides an explicit calculation of the cycle class of a Hurwitz divisor on the moduli space of even genus curves and explores the geometry of the associated Hurwitz space map.

Findings

Cycle class of the Hurwitz divisor $D_2$ explicitly computed.

Insights into the geometry of the Hurwitz space of admissible covers.

Relationship between Hurwitz space and moduli space of stable curves analyzed.

Abstract

We calculate the cycle class of the Hurwitz divisor $D_2$ on the moduli space of stable curves of genus $g=2k$ given by the degree $k+1$ covers of the projective line with simple ramification points, two of which lie in the same fibre. We also study some aspects of the geometry of the natural map of the Hurwitz space of admissible covers of degree $k+1$ and genus $2k$ to the moduli space of stable curves of genus $2k$.