Example of a 3-dimensional rationally connected variety that is not rational.

Question

We will call a proper variety $X$ rationally connected if any two general points of $X$ are in the image of some map $\boldsymbol{P} \to X.$ A variety is rational if it is birationally equivalent to $\boldsymbol{P}^n$ for some $n$.

Any rational variety must be rationally connected. These two notions coincide exactly for varieties of dimension two or fewer, but this is not the case in higher dimensions. What is an illustrative example of a three-dimensional variety that is rationally connected, but is not rational?

Answer 1

This question is very much related to the Lüroth Problem:

A variety $X$ of dimension $n$ is unirational if there exists a dominant rational map $\mathbf{P}^n \dashrightarrow X$. Does there exist a unirational variety that is not rational?

Since a unirational variety must be rationally connected, finding a unirational variety that is not rational will answer this question. In this paper on The Lüroth Problem, Arnaud Beauville summarizes the following three examples.

• In Clemens and Griffith's paper The Intermediate Jacobian of the Cubic Threefold, the authors prove that a smooth cubic threefold $V_3 \subset \mathbf{P}^4$ is not rational by showing that it's intermediate Jacobian is not a Jacobian. It had been shown prior to this paper that $V_3$ is unirational. A particular example of such a threefold is the vanishing set of $\sum_{i \in \mathbf{Z}/5} X_i^2 X_{i+1}$ in $\mathbf{P}^4$.
• In the paper Three-dimensional quartics and counterexamples to the Lüroth problem Iskovskikh and Manin prove that any smooth quartic threefold $V_4 \subset \mathbf{P}^4$ is not rational. Since examples of unirational quartic threefolds already existed, this also answers the Lüroth problem. They proved that such a variety is not rational by proving that the group of birational automorphisms of $V_4$ is finite, whereas the group of birational automorphisms of $\mathbf{P}^3$ is very large.
• Artin and Mumford in Some Elementary Examples of Unirational Varieties Which are Not Rational provide a specific example of a variety $X$ that is a double covering of $\mathbf{P}^3$ branched along a quartic surface in $\mathbf{P}^3$ with ten nodes that is unirational but not rational. Their approach involves considering the torsion part of $\operatorname{H}^3(X,\mathbf{Z})$.