Discussion of Luroth's problem in an article of Beauville

2012-04-26T21:27:45Z

I am reading a wonderful article of Arnaud Beauville, called La théorie de Hodge et quelques applications http://math.unice.fr/~beauvill/conf/Bordeaux2.pdf

There is one place on page 12 that I can not understand. Beauville seem to claim the following:

Claim. Denote by $K$ the field of meromorphic functions of the (complex) cubic $x_1^3+x_2^3+x_3^3+x_4^3+x_5^3=0$. Then one can show that there exists an embedding $K\subset \mathbb C(y_1,y_2,y_3)$.

Question. How to show this?

I have to say, I doubt this statement (not anymore). Since the cubic is unirational, we have $K\subset \mathbb C(y_1,...,y_n)$ for some $n$. But it sounds strange that one can chose $n=3$. Is this a misprint or I miss something?

(for a connection to Luroth problem see http://en.wikipedia.org/wiki/Rational_variety)

PS. I guess Artie's comment explains that Beauville is 100% correct. So I would like to ask one more (non-trivial ?) question:

Question 2 Is it known what is the minimal number $d$ such that there is a degree $d$ (dominant) morphism from a rational complex projective three-fold to the cubic?

Answer by Artie Prendergast-Smith for Discussion of Luroth's problem in an article of Beauville

2012-04-27T15:10:00Z

(As requested, I'm turning my comment into an answer.)

The issue seems to be that there are differing definitions of unirational. One is that there exists a dominant rational map $\mathbf{P}^n \dashrightarrow X$ for some $n$. (That corresponds to the inclusion of function fields $K \subset \mathbb C(y_1,...,y_n)$.) But it's easy to show that it's equivalent to ask for a dominant rational map $\mathbf{P}^d \dashrightarrow X$, where $d$ is the dimension of X. (In your case, this gives the inclusion $K \subset \mathbb C(y_1,y_2,y_3)$.)

To see the equivalence of the two, suppose $\phi: \mathbf{P}^n \dashrightarrow X$ is a dominant rational map with $n$ strictly larger than dim X. Then a general hyperplane $H$ in $\mathbf{P}^n$ will intersect the general fibre of $\phi$, so the restriction of $\phi$ to $H$ is still dominant. By induction, you're done.

Discussion of Luroth's problem in an article of Beauville - MathOverflow

Discussion of Luroth's problem in an article of Beauville

Answer by Artie Prendergast-Smith for Discussion of Luroth's problem in an article of Beauville