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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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    $\begingroup$ 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. $\endgroup$
    – naf
    Commented Dec 8, 2012 at 6:09
  • $\begingroup$ 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 $\ge 3$)... $\endgroup$
    – aglearner
    Commented Dec 8, 2012 at 11:49
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    $\begingroup$ 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). $\endgroup$
    – naf
    Commented Dec 9, 2012 at 12:52

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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.

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    $\begingroup$ Thank you Sasha! Do you have some reference of this conjecture? $\endgroup$
    – aglearner
    Commented Dec 7, 2012 at 19:11
  • $\begingroup$ As far as I know it hasn't been written down. $\endgroup$
    – Sasha
    Commented Dec 9, 2012 at 4:17

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