# Example of a projective variety over a field of characteristic zero which is uniruled but not ruled

A variety $$Z$$ over a field $$k$$ of characteristic zero is ruled if there is a variety $$M$$ and a dominant, birational map $$\phi: M \times \mathbb{P}^{1}_{k} \dashrightarrow Z$$. A variety $$Z$$ over a field $$k$$ of characteristic zero is uniruled if there is a variety $$M$$ and a dominant, rational map $$\phi: M \times \mathbb{P}^{1}_{k} \dashrightarrow Z$$. Does anyone know of a projective variety $$Z$$ over a field $$k$$ of characteristic zero which is uniruled, but not ruled?

• I don't know much about this, but I would guess a cubic threefold. – roy smith Jul 6 '20 at 21:17

I think you just need to know that there exists a threefold $$X$$ which is unirational but not rational (e.g. the cubic threefold). If $$X$$ is birational to $$S\times \mathbb{P}^1$$, there is a dominant rational map $$X -\!-\!\!\!> S$$, thus $$S$$ is unirational, hence rational by Castelnuovo's theorem. Therefore $$X$$ is rational.

Some conic bundles that are not birationally trivial do the job. For explicit examples, see pages 143-148 of

K. Matsuki: Introduction to the Mori program, Universitext. New York, NY: Springer (ISBN 0-387-98465-8/hbk). xxiii, 478 p. (2002). ZBL0988.14007,

The case of a cubic threefold $$W_3 \subset \mathbb{P}^4$$, cited by Roy Smith in his comment, belongs to this family of counterexamples. In fact, the blow-up $$X=\mathrm{Bl}_L(W_3)$$ of $$W_3$$ along a line $$L \subset W_3$$ is a conic bundle over $$\mathbb{P}^2$$. By Clemens-Griffths we know that $$W_3$$ is not rational, so $$X$$ is not rational, and this implies that its conic bundle $$X \to \mathbb{P}^2$$ is not birationally trivial.