On the nilpotent section conjecture for finite group actions on curves

TL;DR

This paper provides a new geometric proof of the section conjecture for finite group actions on complex projective curves, including its nilpotent analogue and explicit descriptions for prime order groups.

Contribution

It introduces a novel geometric proof for the section conjecture and its nilpotent version, along with explicit abelianised maps for prime order groups.

Findings

Proof of the section conjecture for finite group actions on curves

Explicit description of abelianised section map for prime order groups

Verification of the 2-nilpotent section conjecture

Abstract

We give a new, geometric proof of the section conjecture for fixed points of finite group actions on projective curves of positive genus defined over the field of complex numbers, as well as its natural nilpotent analogue. As a part of our investigations we give an explicit description of the abelianised section map for groups of prime order in this setting. We also show a version of the 2-nilpotent section conjecture.