On Grothendieck's section conjecture for orbicurves

Abstract

As already noted by Niels Borne and Michel Emsalem, there is a natural generalization of the section conjecture for proper orbicurves. Combined with the reformulation by Niels Borne and Angelo Vistoli of the conjecture in terms of the étale fundamental gerbe, this suggests an even stronger conjecture for orbicurves, asking an equivalence of categories instead of a mere bijection. We prove that the three versions of the conjecture are in fact equivalent, and that "injectivity" (i.e. full faithfulness) holds in the case of orbicurves. As a byproduct, we obtain a new proof of the fact that the section conjecture for proper curves implies the section conjecture for open curves.

The results of the paper are now part of a more comprehensive treatment of anabelian geometry for DM stacks, see arXiv:1804.07176

References in corpus (3)

• Packets in Grothendieck's Section Conjecture
• The Nori fundamental gerbe of a fibered category
• Nori fundamental gerbe of essentially finite covers and Galois closure of towers of torsors