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

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?

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If, instead of threefolds, you consider fourfolds, then the general cubic foufold is expected to be irrational but there do exist cubic fourfolds which are rational. –  ulrich Dec 8 '12 at 6:09
Ulrich, I know about this situation in dimension $4$, this is why I asked specifically about dim $3$ which looks a bit different somehow... In fact I heard also once that low degree hypersufaces in $\mathbb CP^n$ of very large dimension are expected to be rational, but I wonder if this is indeed expected and if yes, is it proven at least in one case (of degree $\ge 3$)... –  aglearner Dec 8 '12 at 11:49
Low degree hypersurfaces of large dimension are known to be unirational, but as far as I know rationality has not been proved in any degree $>2$ (for all hypersurfaces of a given dimension). –  ulrich Dec 9 '12 at 12:52