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What is the birational classification of (smooth projective) rationally connected 3-folds (over algebraically closed fields of characteristic $0$ or even $\mathbf{C}$, if $\mathrm{char}(k) = p > 0$, one should perhaps better ask for separably rationally connected 3-folds).

There is perhaps an article of Iskovskikh, but I do not have access to MathSciNet at the moment.

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    $\begingroup$ This is completely unknown, and probably hopeless. Just one example : taking the desingularization of a quartic threefold with 1,2, ...,45 nodes gives RC threefolds, most of which are very likely of different birational type. $\endgroup$
    – abx
    Sep 22, 2014 at 14:19
  • $\begingroup$ As a comment, this (birational) class is strictly larger than the class of Fano type varieties for dimension greater or equal than 6. arxiv.org/abs/1406.3752 $\endgroup$ Sep 23, 2014 at 12:15

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