Unirationality is the same thing as Rational Connectedness in Characteristic Zero

Abstract

In this paper we prove that unirationality, rational connectedness and rational chain connectedness coincide for smooth projective varieties over a field $k$ of characteristic zero. Our approach uses the MRC fibration to show that if $X$ is a smooth projective variety, then there exists a variety $\mathrm{MU}(X)$, together with rational maps $\pi :X⇢\mathrm{MU}(X)$ and $\lambda :\mathrm{MU}(X)⇢\mathrm{MRC}(X)$, such that i) if $\nu :X⇢\mathrm{MRC}(X)$, then $\lambda \circ \pi =\nu$ on the appropriate domains; ii) the very general fibres of $\pi$ are unirational; iii) the very general fibres of $\lambda$ are rationally connected but not unirational. We then apply an induction argument to show that $\mathrm{MU}(X)$ is birationally equivalent to $\mathrm{MRC}(X)$.