Unirationality is the Same as Rational Connectedness in Characteristic Zero

Abstract

In this paper we describe a fibration for a smooth, projective variety $ X $ over a field of characteristic zero. This fibration is similar to the MRC fibration, and we call it the MU fibration of $ X $. The MU fibration $ \pi: X \dashrightarrow MU(X) $ is characterized by the following properties: i) The very general fibres of $ \pi $ are unirational, ii) if $ Z $ is a unirational sub-variety of $ X $, $ z $ is a very general point of $ MU(X) $ (i.e., a point in the complement of a countable union of Zariski closed sub-sets of $ MU(X) $), and $ Z $ intersects $ \pi^{-1}(z) $ non-trivially, then $ Z $ is contained in $ \pi^{-1}(z) $, iii) The variety $ MU(X) $ is unique up to birational equivalence. If we call $ MU(X) $ a maximal unirational quotient, then $ X $ is unirational if and only if the dimension of any maximal unirational quotient is equal to zero. We use this work to show that unirationality, rational connectedness, and rational chain connectedness are equivalent for smooth varieties over a field of characteristic zero, and that the MRC quotient of a smooth, projective variety over a field of characteristic zero is not uniruled.