Unirationality is the Same as Rational Connectedness in Characteristic Zero

TL;DR

This paper introduces the MU fibration for smooth projective varieties over characteristic zero fields, proving that unirationality, rational connectedness, and rational chain connectedness are equivalent in this setting.

Contribution

It defines the MU fibration, characterizes its properties, and establishes the equivalence of key rationality notions for smooth varieties in characteristic zero.

Findings

Unirationality, rational connectedness, and rational chain connectedness are equivalent.

The MU fibration has unique properties related to unirational subvarieties.

The MRC quotient of a smooth projective variety is not uniruled.

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.