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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 $x_1^2+x_2^2+x_3^2+x_4^3+x_5^3=0$. 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?

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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 $x_1^2+x_2^2+x_3^2+x_4^3+x_5^3=0$. 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?

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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 $x_1^2+x_2^2+x_3^2+x_4^3=0$. x_1^2+x_2^2+x_3^2+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. 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)
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