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?



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


