On Grothendieck's section conjecture for orbicurves

TL;DR

This paper explores a generalized form of Grothendieck's section conjecture for orbicurves, establishing the equivalence of multiple conjecture versions and proving injectivity for orbicurves, with implications for proper and open curves.

Contribution

It proves the equivalence of three versions of the conjecture for orbicurves and demonstrates injectivity, extending the conjecture's validity to broader classes of curves.

Findings

Three versions of the conjecture are equivalent.

Injectivity (full faithfulness) holds for orbicurves.

A new proof links the section conjecture for proper and open curves.

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 \'etale 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.