Discussion of Luroth's problem in an article of Beauville - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T02:20:28Z http://mathoverflow.net/feeds/question/95306 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/95306/discussion-of-luroths-problem-in-an-article-of-beauville Discussion of Luroth's problem in an article of Beauville aglearner 2012-04-26T21:27:45Z 2012-04-27T15:10:00Z <p>I am reading a wonderful article of Arnaud Beauville, called <em>La théorie de Hodge et quelques applications</em> <a href="http://math.unice.fr/~beauvill/conf/Bordeaux2.pdf" rel="nofollow">http://math.unice.fr/~beauvill/conf/Bordeaux2.pdf</a></p> <p>There is one place on page 12 that I can not understand. Beauville seem to claim the following: </p> <p><strong>Claim.</strong> 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 $K\subset \mathbb C(y_1,y_2,y_3)$. </p> <p><strong>Question.</strong> How to show this?</p> <p>I have to say, I doubt this statement (<em>not anymore</em>). Since the cubic is <em>unirational</em>, 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?</p> <p>(for a connection to Luroth problem see <a href="http://en.wikipedia.org/wiki/Rational_variety" rel="nofollow">http://en.wikipedia.org/wiki/Rational_variety</a>)</p> <p><strong>PS.</strong> I guess Artie's comment explains that Beauville is 100% correct. So I would like to ask one more (non-trivial ?) question:</p> <p><strong>Question 2</strong> 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? </p> http://mathoverflow.net/questions/95306/discussion-of-luroths-problem-in-an-article-of-beauville/95367#95367 Answer by Artie Prendergast-Smith for Discussion of Luroth's problem in an article of Beauville Artie Prendergast-Smith 2012-04-27T15:10:00Z 2012-04-27T15:10:00Z <p>(As requested, I'm turning my comment into an answer.)</p> <p>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 <em>some</em> $n$. (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 $K \subset \mathbb C(y_1,y_2,y_3)$.)</p> <p>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.</p>