On geometrically $C_1$ fields

TL;DR

This paper investigates the property of geometrically $C_1$ fields, showing how it lifts from residue fields to valued fields under certain conditions and establishing that specific classes of valued fields are geometrically $C_1$.

Contribution

It demonstrates the lifting of the geometrically $C_1$ property from residue fields to henselian valued fields with divisible value groups and proves that algebraically maximal valued fields with finite residue fields are geometrically $C_1$.

Findings

Property lifts from residue field to valued field in certain cases

Algebraically maximal valued fields with finite residue are geometrically $C_1$

Maximal totally ramified extensions of local fields are geometrically $C_1$

Abstract

A field $k$ is called geometrically $C_1$ if every smooth projective separably rationally connected $k$-variety has a $k$-rational point. Given a henselian valued field of equal characteristic $0$ with divisible value group, we show that the property of being geometrically $C_1$ lifts from the residue field to the valued field. We also prove that algebraically maximal valued fields with divisible value group and finite residue field are geometrically $C_1$. In particular, any maximal totally ramified extension of a local field is geometrically $C_1$.