Stable Maps and Hurwitz Schemes in Mixed Characteristic

The Hurwitz scheme was originally conceived as a parameter space for simply branched covers of the projective line. A variant of this is a parameter spaec for simply branched covers of the projective line, up to automorphisms of P the so called unparametrized Hurwitz scheme. Other variants involve fixing branching types which are not necessarily simple (see [D-D-H]). A rigorous algebraic definition of the Hurwitz scheme in characteristic 0 was given by Fulton in [F] the unparametrized version is its quotient by the PGL(2) action on P. A natural compactification of the unparametrized Hurwitz scheme in characteristic 0 was given by Harris and Mumford [H-M] the space of admissible covers. A rigorous treatment using logarithmic structures can be found in [Mo], who puts some order in the zoo of variants one can think of (parametrized vs. unparametrized, ordered branch points vs. unordered branch points, stack vs. coarse moduli scheme etc.). A new treatment using twisted principal bundles and stable maps into stacks can be found in [א-V, א-C-V]. In [P] the space of admissible covers is identified as a closed subscheme in the space of stable maps into M0,n+1. We know of no existing treatment of Hurwitz schemes in positive or mixed characteristic when the degree exceeds the characteristic of the residue fields. In this note we follow Pandharipande’s idea in [P] and define the compactified Hurwitz scheme as a subscheme of the space of stable maps into M0,n+1. Variants with target curves of higher genus are defined as well. In order to follow this idea we have to treat moduli of stable maps in mixed characteristic. This is implicit in [B-M] but has not been available in the litterature in this generality. We show (Theorem 2.8) that for any base scheme S, and any integers g, n, r and d there exists an Artin algebraic stack with finite stabilizers, denoted Mg,n(PS , d), which is proper over S, parametrizing stable maps of n-pointed curves of genus g into P over S. This stack admits a projective coarse moduli scheme Mg,n(P r S , d). One immediately derives the existence of a similar stack Mg,n(X, d) parametrizing stable maps of degree d into a projective S-scheme of finite presentation X ⊂ PS . Denote by C0,n → M0,n the universal family of stable n-pointed curves of genus 0. We propose to define the (unparametrized) Hurwitz stack (with ordered simple branch points) to be the closure in Mg,n(C0,n/M0,n, d) of the locus of Hurwitz covers over Q: one identifies an admissible cover C → D with ordered simple branchings P1, . . . , Pn as a stable map from the n-pointed curve (C,Q1, . . . Qn) to the stable n-pointed rational curve (D,P1, . . . Pn), where Qi are the ramification points. Here the pointed curve (D,P1, . . . Pn) is identified uniquly as a fiber of C0,n → M0,n. There are some interesting things which happen in this construction. For the sake of comparison, recall that in characteristic 0