Smooth rationally connected threefolds contain all smooth curves

TL;DR

This paper proves that smooth rationally connected threefolds can embed any smooth projective curve, and characterizes rationally connected varieties of dimension three or more based on this property.

Contribution

It establishes that all smooth curves can be embedded into smooth rationally connected threefolds and provides a characterization of such varieties in higher dimensions.

Findings

Any smooth projective curve embeds into a smooth rationally connected threefold.

A property related to embedding curves characterizes rationally connected varieties of dimension ≥ 3.

Details are provided for the case of toric varieties.

Abstract

We show that if X is a smooth rationally connected threefold and C is a smooth projective curve then C can be embedded in X. Furthermore, a version of this property characterises rationally connected varieties of dimension at least 3. We give some details about the toric case.