Degenerations of rationally connected varieties and PAC fields

TL;DR

This paper proves that degenerations of separably rationally connected varieties over perfect PAC fields containing an algebraically closed field always have rational points, extending known results for Fano complete intersections.

Contribution

It establishes that all degenerations of separably rationally connected varieties over such fields possess rational points, broadening the class of varieties with this property.

Findings

Degenerations of separably rationally connected varieties have rational points over perfect PAC fields.

Extension of the $C_1$ property from Fano complete intersections to a wider class of varieties.

Supports the conjecture that rational points exist on degenerations of rationally connected varieties.

Abstract

A perfect PAC field containing an algebraically closed field is known to be $C_1$, i.e., every degeneration of a Fano complete intersection has a point. We prove that also every degeneration of a separably rationally connected variety has a point.