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I am mainly interested in varieties over an algebraic closed field $k$ (or $\mathbb{C}$). The classification of complex surface is established in the last century and known as Enriques–Kodaira classification. In higher dimension, people realized that mild singularities must be taken into account to build a reasonable theory and Mori and others studied Minimal model program, which is the birational classification of algebraic varieties. However, I am more interested in a concrete description of classification, like the Enriques–Kodaira classification (maybe except for general ones). If I understand correctly, Fano threefolds are pretty well-studied in a concrete manner by Mukai and others. Here is my question:

Is there an Enriques–Kodaira-like classification of Fano threefolds? Maybe over $\mathbb{C}$?

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up vote 6 down vote accepted

Yes, Fano threefolds have been completely classified and one has explicit projective models of them. See V.A. Iskovskih, Yu. G. Prokhorov: Algebraic Geometry V: Fano varieties. Encyclopaedia of Math. Sciences 47, Springer-Verlag, Berlin 1999. See also Andreas Ott's thesis on Fano threefolds of picard number $\rho\ge 2$.

In some sense the most complicated Fanos are the ones with Picard number one, i.e., $Pic(X)\simeq \mathbb Z$. Here there are 18 families of such threefolds and they have been classified by Iskovskih. The most basic invariant here is the $index$ of $X$, i.e., the maximal integer $r$ such that $K_X$ is divisible by $r$ in $Pic(X)$. Fano threefolds of higher Picard number was classified by Mori and Mukai. This was one of the early triumphs of Mori theory.

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Thank you for the nice answer. I suppose that Fano threefolds are the only class of threefolds whose explicit classification is known. Calabi-Yau threefolds must be very difficult and general ones are even worse, I guess. –  K Kim Feb 20 '13 at 0:44
    
Yes, that's correct: For Calabi-Yaus it is an open problem even to determine whether the set of CY-families is finite. –  J.C. Ottem Feb 20 '13 at 0:56
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