Stable maps and Hurwitz schemes in mixed characteristics (Q2752190)

After the classical work of Hurwitz, the Hurwitz scheme in characteristic 0, parametrizing simply branched covers of the projective line, was first rigorously introduced by \textit{W. Fulton} [Ann. Math. (2) 90, 542--575 (1969; Zbl 0194.21901)]. Successively several variants of this scheme have been introduced and studied, e.g. the unparametrized Hurwitz scheme, which is the quotient of the Hurwitz scheme by the PGL(2) action on \(\mathbb P^1\). A natural compactification of this scheme, the space of admissible covers, was given by \textit{J. Harris} and \textit{D. Mumford} [Invent. Math. 67, 23--86 (1982; Zbl 0506.14016)]. Recently \textit{R. Pandharipande} [Math. Ann. 313, 715--729 (1999; Zbl 0933.14035)] identified it as a closed subscheme in the space of stable maps into the stack \(\overline{\mathcal M}_{0,n+1}\).NEWLINENEWLINEIn the paper under review, following the idea of Pandharipande, the authors introduce compactified Hurwitz stacks in mixed characteristic. They study then in detail the example of the compactified stack of double covers of \(\mathbb P^1\) branched in 4 points, in characteristic 2, and compare it with the characteristic 0 scheme.NEWLINENEWLINEFor the entire collection see [Zbl 0966.00024].