Submission date: 29 Julyy 2024

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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Full text arXiv 2407.19986: pdf, ps.