Classification in Small Dimensions
Fano varieties in dimensions 1 are isomorphic to the projective line.
In dimension 2 they are del Pezzo surfaces and are isomorphic to either or to the projective plane blown up in at most 8 general points, and in particular are again all rational.
In dimension 3 there are non-rational examples. Iskovskih classified the Fano 3-folds with second Betti number 1 into 17 classes, and Mori & Mukai (1981) classified the ones with second Betti number at least 2, finding 88 deformation classes.
Read more about this topic: Fano Variety
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