Threefolds containing all curves are rationally connected

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 variety $X$, then $X$ contains a rationally connected subvariety of dimension at least three. In particular, "all curves embed" is a birational property for threefolds.

7 pages. Substantially revised and expanded version; accepted for publication in Journal of Algebra