In the Wikipedia page for 'Kodaira dimension', there is 'the classification table of algebraic three-folds' in the section 'Any dimension'.

What are those 'others' in the very bottom of this table? Are there some classifications for these 'others'?

Are there any examples of these 'others' that are not rational or birational to Fano threefolds?

  • $\begingroup$ Strange "classification", I wouldn't take it too seriously. Anyway, you can take a non-rational Fano threefold (e.g. a cubic threefold) and blow up whatever points or curves you like. Classification is of course impossible. $\endgroup$ – abx May 10 at 6:58
  • $\begingroup$ Any examples that are not rational or birational to Fano threefolds? I edited the question. $\endgroup$ – user69559 May 10 at 7:27
  • 1
    $\begingroup$ See the paper "Rationally connected non-Fano type varieties" by Igor Krylov for such examples. $\endgroup$ – naf May 10 at 8:27

The Artin-Mumford threefold (Proc. London Math. Soc. (3) 25 (1972), 75–95) is an example. It is unirational, hence has $h^{i}(\mathscr{O}_X)=0$ for $i>0$, and it has $\operatorname{Tors}H^3(X,\mathbb{Z}) \neq 0$. It is not birational to a Fano threefold because $\operatorname{Tors}H^3(X,\mathbb{Z}) $ is a birational invariant, which is zero for Fano threefolds (unfortunately the only proof I know of the latter fact is by the Iskovskikh-Mori-Mukai classification of Fano threefolds).


Here is a different example. Let $S$ be an Enriques surface. Then $X = S \times \mathbb{CP}^1$ is uniruled, hence has Kodaira dimension $-\infty$. But $b_{1}(X) = 0$, hence the irregularity satisfies $q(X) = h^{0,1}(X)=0$. Then we can can conclude that $X$ is not rational or birational to any Fano since for example $\pi_{1}(X) = \mathbb{Z}_{2}$.

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    $\begingroup$ Likewise, you could start with a fake projective plane instead of an Enriques surface, or really any surface with vanishing outer Hodge numbers. $\endgroup$ – R. van Dobben de Bruyn May 10 at 18:18

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