Stable maps and Hurwitz schemes in mixed characteristic

TL;DR

This paper introduces a new definition for complete Hurwitz schemes in mixed characteristic, extending classical concepts to a broader setting and analyzing their properties, including reduction modulo 2 for genus 1 covers.

Contribution

It proposes a novel approach to defining complete Hurwitz stacks in mixed characteristic using stable maps and explores their properties and reductions, connecting to existing work on stable maps into stacks.

Findings

The complete Hurwitz stack is the closure of Hurwitz covers in characteristic 0.

Reduction modulo 2 of genus 1 double covers exhibits complex behavior.

Speculations on relationships with stable maps into Deligne-Mumford stacks.

Abstract

In this short note, we propose a definition of complete Hurwitz schemes (and stacks) in mixed characteristic. We follow an idea of R. Pandharipande, and define the complete Hurwitz stack as a substack of stable maps of degree d of nodal pointed curves of genus g to the universal curve over the moduli stack of stable pointed curves of genus h. This substack is simply the closure of the locus of Hurwitz covers in characteristic 0. We describe in detail the reduction modulo 2 of the complete Hurwitz stack of genus 1 double covers of the projective line. The result is wilder than you would imagine in your wildest dreams. We end by speculating about a possible relationship with the work of Abramovich-Vistoli about stable maps into Deligne-Mumford stacks.