Stable maps and Hurwitz schemes in mixed characteristic

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.

preliminatry version, 8 pages, 4 figures

References in corpus (3)

• Quotients by Groupoids
• A high fibered power of a family of varieties of general type dominates a variety of general type
• A geometric construction of Getzler's relation

Cited by in corpus (8)

• Stable maps and Quot schemes
• Compactifying the space of stable maps
• Parabolic reductions of principal bundles
• Twisted bundles and admissible covers
• The elliptic Grothendieck-Springer resolution as a simultaneous log resolution of algebraic stacks
• Stable maps and singular curves on K3 surfaces
• p-divisibility for coherent cohomology
• Reduction of the Hurwitz space of metacyclic covers