Rational smooth complex projectives three fold with non-rational deformation - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T21:39:12Z http://mathoverflow.net/feeds/question/115707 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/115707/rational-smooth-complex-projectives-three-fold-with-non-rational-deformation Rational smooth complex projectives three fold with non-rational deformation aglearner 2012-12-07T12:03:08Z 2012-12-07T17:22:07Z <p>This question is prompted by a great talk of Beauville:</p> <p><a href="http://www.mathnet.ru/php/presentation.phtml?presentid=5821&amp;option_lang=rus" rel="nofollow">http://www.mathnet.ru/php/presentation.phtml?presentid=5821&amp;option_lang=rus</a></p> <p>The talk is called "Luroth problem". In this talk Beauville considers in particular Fano three-folds and says how one can prove that some of them are not rational. </p> <p>Still I was not able to figure out the following: is there any example of a rational (smooth of course) complex projective three fold that admits a deformation that is not rational? If yes what is the simplest example?</p> http://mathoverflow.net/questions/115707/rational-smooth-complex-projectives-three-fold-with-non-rational-deformation/115731#115731 Answer by Sasha for Rational smooth complex projectives three fold with non-rational deformation Sasha 2012-12-07T17:22:07Z 2012-12-07T17:22:07Z <p>A conjecture of Iskovskikh says that this never happens. To be more precise, it says that if there is a family of smooth projective threefolds with general threefold nonrational then all these threefolds are nonrational. </p> <p>The conjecture is not proved. On one hand it is not clear how this can be proved, on the other hand no counterexample known.</p>