The question is stated in the title, but I would like to add some motivation.

I've been teaching a course on complex tori and abelian varieties this semester and I would like to end it by showing some significant application of abelian varieties in algebraic geometry. I've come across a very beautiful recent proof by Beauville that a certain specific sextic threefold as in the title is not rational and I have decided give an outline of it in my last lecture. Beauville refers to a paper of Enriques of 1912 for the proof of unirationality. I've got Enriques paper but I cannot make sense of it, so I'm looking either for another reference or for a sketch of proof.

Just in case it helps, here's what I've been able to understand from Enriques' proof. Let $V\_6=Q\_2\cap C\_3$ be the threefold, where $Q\_2$ is a smooth quadric and $C\_3$ a smooth cubic. Let $P\in V\_6=Q\_2\cap C\_3$ be a point. Then $Q\_2$ contains two families of planes through $P$, each parametrized by a $\mathbb P^1$. If $H$ is such a plane $H\cap C\_3$ is a plane cubic $C\_H$ and we can associate with $P$ the residual intersection $Q\_H$ with $C\_H$ of the tangent line to $C\_H$ at $P$. As $H$ varies in one of the families of planes through $P$, $Q\_H$ describes a rational curve $K$ in $V\_6$. Of course, as $P$ varies, the curves $K$ fill up $V\_6$. At this point, Enriques just claims that the curves $K$ thus defined ``correspond to the lines through a point in $\mathbb P^3$''.