Unirationality is the same thing as Rational Connectedness in Characteristic Zero

Technical Summary: Unirationality and Rational Connectedness in Characteristic Zero

Problem Statement
The paper addresses a fundamental question in algebraic geometry regarding the relationship between unirationality and rational connectedness for smooth projective varieties over a field $k$ of characteristic zero. While it is well-established that rational varieties are unirational, and unirational varieties are rationally connected (and rationally chain connected), the converse implications have historically been subtle.

• Unirationality is defined as the existence of a generically finite, dominant rational map $\mathbb{P}^n_k \dashrightarrow Z$.
• Rational Connectedness is defined by the existence of a dominant rational map $\mathbb{P}^1_k \times M \dashrightarrow Z$ such that the induced map on pairs is dominant.
• Rational Chain Connectedness requires any two points to be connected by a chain of rational curves.

In characteristic zero, smooth rationally chain connected varieties are rationally connected. However, the question remains whether every rationally connected variety is unirational. Previous counterexamples, such as the smooth cubic threefold (Clemens-Griffiths), demonstrated that unirationality does not imply rationality (being birational to $\mathbb{P}^n$). The paper seeks to determine if the weaker condition of rational connectedness is equivalent to unirationality in the smooth projective setting over characteristic zero, distinct from the stronger condition of rationality.

Methodology
The author employs the theory of the Maximal Rationally Connected (MRC) fibration, originally established by Campana, Kollár, Miyaoka, and Mori. The core strategy involves constructing a specific intermediate variety, denoted $MU(X)$, to bridge the gap between the variety $X$ and its MRC quotient.

1. Construction of $MU(X)$: The paper proves the existence of a variety $MU(X)$ equipped with rational maps $\pi: X \dashrightarrow MU(X)$ and $\lambda: MU(X) \dashrightarrow MRC(X)$ satisfying three conditions:

   • The composition $\lambda \circ \pi$ recovers the MRC fibration $\nu: X \dashrightarrow MRC(X)$.
   • The very general fibers of $\pi$ are unirational.
   • The very general fibers of $\lambda$ are rationally connected but not unirational.

2. Inductive Argument: The proof utilizes an induction on the dimension of the variety. The author establishes that if $X$ is rationally connected, then $MRC(X)$ is a point (specifically $\text{Spec}(k)$). Consequently, $MU(X)$ becomes the very general fiber of $\lambda$.

   • If $MU(X)$ has positive dimension, the induction hypothesis (applied to lower-dimensional rationally connected varieties) would imply $MU(X)$ is unirational.
   • However, by construction, the fibers of $\lambda$ are not unirational.
   • This contradiction forces $MU(X)$ to be a point, implying that the fibers of $\pi$ (which are $X$ itself in this case) must be unirational.

3. Zorn's Lemma Application: To ensure the existence of a "maximal" unirational fibration, the paper orders the set of rational maps with unirational fibers by the inclusion of their function fields. Using Zorn's Lemma, the author demonstrates the existence of a unique maximal element (up to birational equivalence), which serves as $MU(X)$.

Key Contributions and Results

• Equivalence Theorem: The primary result (Theorem 9) proves that for any smooth projective variety $X$ over a field $k$ of characteristic zero, the following properties are equivalent:
  1. $X$ is unirational.
  2. $X$ is rationally connected.
  3. $X$ is rationally chain connected.
     Note: This equivalence holds specifically for the relationship between unirationality and rational connectedness, distinct from the property of rationality. The paper acknowledges that varieties can be unirational and rationally connected without being rational (e.g., the smooth cubic threefold).
• Existence of Maximal Unirational Fibration: The paper establishes the existence and uniqueness (up to birational equivalence) of a maximal unirational fibration $\pi: X \dashrightarrow MU(X)$ for any smooth projective variety.
• Structural Decomposition: The paper provides a structural decomposition of any smooth projective variety $X$ into a sequence of fibrations where the "unirational part" is separated from the "non-unirational rationally connected part." Specifically, it shows that if the MRC quotient is trivial, the variety itself must be unirational.

Significance and Claims
The paper claims to resolve the equivalence of unirationality and rational connectedness for smooth projective varieties in characteristic zero. The author notes that while rational connectedness is often easier to verify than unirationality (e.g., via the existence of very free curves), this result establishes that for smooth projective varieties in characteristic zero, the two notions are identical.

The work relies on the machinery of the MRC fibration and generic smoothness in characteristic zero. It explicitly distinguishes its results from the positive characteristic case, where separability conditions are required, and from the question of rationality (where counterexamples like the cubic threefold exist). The paper does not claim to solve the rationality problem but rather clarifies the hierarchy between unirationality and rational connectedness, showing that the latter implies the former in the specified context.