Threefolds containing all curves are rationally connected

TL;DR

The paper proves that a threefold in which every curve embeds must be rationally connected, establishing that 'all curves embed' is a birational property for threefolds.

Contribution

It demonstrates that the property of containing all curves characterizes rationally connected threefolds, providing a new birational criterion.

Findings

Any curve embeds into a rationally connected threefold.

If every curve embeds into a threefold, then the threefold is rationally connected.

'All curves embed' is a birational property for threefolds.

Abstract

Any smooth projective curve embeds into $\mathbb{P}^3$. More generally, any curve embeds into a rationally connected variety of dimension at least three. We prove conversely that if every curve embeds in a threefold $X$, then $X$ is rationally connected. In particular "all curves embed" is a birational property for threefolds.