Talk:Rationally connected variety

There is not the image of a rational curve passing through every two points on all algebraic varieties of course. All surfaces would be rational then, or for example in higher dimension there are no non-constant maps from P^1->A where A is an abelian variety. I was hoping to initiate this article and people can beef it up. I have edited it a bit more to be more precise. — Preceding unsigned comment added by Fgounelas  (talk • contribs)  19:59, 17 February 2012 (UTC)


 * You have to say that the image of the projective line is inside the variety. A less jargon definition would be A rationally connected variety is an algebraic projective variety such that every two points are connected by a rational curve. D.Lazard (talk) 20:31, 17 February 2012 (UTC)


 * The image of a morphisme P^1 -> A is inside A. I really don't see your point. On the other hand, I agree that your formulation is more intuitive. I hope someone can improve this article because the rationally connected varieties are very exciting objects since 20 or 30 years (works of Kollar-Miyaoka-Mori, Campana, Graber-Harris-Starr., de Jong..). Liu (talk) 23:49, 17 February 2012 (UTC)


 * My point is that a definition, which may be correctly understood only by people who already know, it is not acceptable in Wikipedia. As a correct and understandable definition may be extracted from the given explanations, I have removed the prod template and rephrased the article.
 * As this article is very short and strongly related with Rational variety, which is also short, I intend to merge Rationally connected variety as a new section in Rational variety.
 * D.Lazard (talk) 10:55, 18 February 2012 (UTC)