This question is prompted by a great talk of Beauville:


<http://www.mathnet.ru/php/presentation.phtml?presentid=5821&option_lang=rus>


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. 


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?



Rational smooth complex projectives three fold with non-rational deformation
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. 


The conjecture is not proved. On one hand it is not clear how this can be proved, on the other hand no counterexample known.


