Rational correspondences between moduli spaces of curves defined by Hurwitz spaces

TL;DR

This paper constructs a rational map from Hurwitz spaces to moduli spaces of stable curves using trace curves, analyzing its impact on divisor class groups, thus linking algebraic curves, covers, and moduli spaces.

Contribution

It introduces a new rational map between Hurwitz spaces and moduli spaces of stable curves via trace curves, exploring its divisor class group effects.

Findings

Established a rational map from Hurwitz spaces to moduli spaces of stable curves.

Analyzed the induced map's effect on divisor class groups.

Connected the geometry of admissible covers with moduli space structures.

Abstract

By associating to a curve C of genus g=2k and a pencil of degree d=k+1 the so-called trace curve (resp. the reduced trace curve) we define a rational map from the Hurwitz space of admissible covers of genus g=2k and degree d=k+1 to a moduli space of stable curves. We study the induced map between the divisor class groups of these moduli spaces of curves.