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 proof 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?

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?