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 $\mathbb P^3$''.