\'Etale cohomology of a DM curve-stack with coefficients in G_m

TL;DR

This paper computes the étale cohomology groups H^i(X,G_m) for certain one-dimensional algebraic stacks, providing complete results for orbicurves and cyclic stabilizers, and revealing dependence on stabilizer structure.

Contribution

It offers new calculations of étale cohomology for DM curve-stacks, especially for orbicurves with cyclic or abelian stabilizers, extending understanding of their cohomological properties.

Findings

H^2(X, G_m) depends only on the underlying orbicurve and generic stabilizer when stabilizers are abelian.

Complete results are obtained for orbicurves and twisted nodal curves with cyclic stabilizers.

Partial results and examples are provided for the general case.

Abstract

We compute the \'etale cohomology groups H^i(X,G_m) in several cases, where X is a smooth tame Deligne-Mumford stack of dimension 1 over an algebraically closed field. We have complete results for orbicurves (and, more generally, for twisted nodal curves) and in the case all stabilizers are cyclic; we give some partial results and examples in the general case. In particular we show that if the stabilizers are abelian then H^2(X, G_m) does not depend on X but only on the underlying orbicurve and on the generic stabilizer.