Threefolds containing all curves are rationally connected
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.
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