# How does one prove that the complete intersection of a quadric and a cubic of $\mathbb P^5$ is unirational?

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 $\mathbb P^3$''.

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You can look at the short paper by Conte, Marchisio end Murre On the k-unirationality of the cubic complex (2007).

It contains a proof of the unirationality of $V_6$ over a field $k$ of any characteristic $\neq 2,3$, under the assumption that $V_6$ has a $k$-rational point $p$ and that one of the two planes through $p$ on the quadric is also rational over $k$.

In the introduction, the authors write "we follow closely Enriques construction, our only contribution being to fully explain and justify his statements".

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Thank you! it's exaclty what I needed. –  rita Dec 13 '12 at 11:15
You are welcome. –  Francesco Polizzi Dec 13 '12 at 14:11

EDIT: here is a short, but non-elementary argument to prove that the threefold in question is uniruled:

The canonical class of your complete intersection is $(-6+2+3)H=-H$ where $H$ is the hyperplane section. Therefore this is a Fano and hence uniruled by the Miyaoka-Mori criterion, which is essentially an application of Bend and Break.

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Thank you, Sándor. You are completely right: 1) I did not know the Miyaoka-Mori criterion, and 2) actually I am looking for an elementary proof, specific to the situation. The MM criterion says Fano varieties are uniruled. How does one show that they are actually unirational? –  rita Dec 13 '12 at 7:42
Oh, sorry. I was sloppy. I was thinking uniruled all the way. So, this is not an answer after all. Sorry! –  Sándor Kovács Dec 13 '12 at 8:18
I now feel obligated to think about this, but only tomorrow. That is, after I sleep a little. (I've just realized it is already tomorrow).... –  Sándor Kovács Dec 13 '12 at 8:20
As you can see, Francesco found a reference. Thanks again! –  rita Dec 13 '12 at 11:16