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](http://en.wikipedia.org/wiki/Enriques%E2%80%93Kodaira_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](http://en.wikipedia.org/wiki/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:



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> Is there an Enriques–Kodaira-like classification of Fano threefolds? Maybe over $\mathbb{C}$?
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Is there an Enriques–Kodaira-like classification of Fano threefolds?
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](http://people.mpim-bonn.mpg.de/andreaso/files/fano.pdf) 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. 


