(Log) twisted curves.

*(English)*Zbl 1138.14017The present paper gives a different point of view on (\(n\)-pointed) twisted curves, a type of (\(n\)-pointed) nodal curve with a stack structure at finitely many points (including the nodes) introduced by D. Abramovich and A. Vistoli [in: Recent progress in intersection theory. Based on the international conference on intersection theory, Bologna, Italy, December 1997. Boston, MA: Birkhäuser. Trends in Mathematics. 1–31 (2000; Zbl 0979.14018)] in their search for a compactification of Kontsevich’s moduli stack of stable maps. The author introduces the notion of a (\(n\)-pointed) log twisted curve, which is an ordinary nodal curve endowed with a logarithmic structure in the sense of Fontaine and Illusie. Given a base scheme \(S\), he then proves that there is an equivalence of groupoids between the groupoid of \(n\)-pointed twisted curves over \(S\) and the groupoid of \(n\)-pointed log twisted curves over \(S\). This equivalence is compatible with base change \(S'/S\). As a consequence, the author proves that, étale locally on the base, any twisted curve has a finite flat cover by a scheme. Here, it is worth recalling that the interest in such a result comes from applications to intersection theory of stacks, or to the study of the Brauer map of schemes, that is to say the morphism from the usual Brauer group to the cohomological Brauer group. Another noteworthy consequence is another proof of the fact that the Abramovich-Vistoli stack of twisted stable maps into a tame Deligne-Mumford stack is a Deligne-Mumford stack of finite presentation.

Reviewer: Matthieu Romagny (Paris)