On geometrically $C_1$ fields

Konstantinos Kartas

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arXiv · 2407.19986

On geometrically $C_1$ fields

Konstantinos Kartas

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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 $C_1$.

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Konstantinos Kartas. 2024-07-29. On geometrically $C_1$ fields. https://arxiv.org/abs/2407.19986

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