Fano Variety
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High Quality Content by WIKIPEDIA articles! In algebraic geometry, a Fano variety, introduced by (Fano 1934, 1942), is a non-singular complete variety whose anticanonical bundle is ample. In particular Fano varieties all have Kodaira dimension . Fano varieties in dimensions 1 are isomorphic to the projective line. In dimension 2 they are del Pezzo surfaces and are isomorphic to either P1 x P1 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 1977, 1978, 1979) classified the Fano…mehr

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High Quality Content by WIKIPEDIA articles! In algebraic geometry, a Fano variety, introduced by (Fano 1934, 1942), is a non-singular complete variety whose anticanonical bundle is ample. In particular Fano varieties all have Kodaira dimension . Fano varieties in dimensions 1 are isomorphic to the projective line. In dimension 2 they are del Pezzo surfaces and are isomorphic to either P1 x P1 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 1977, 1978, 1979) classified the Fano 3-folds with second Betti number 1 into 18 classes, and Mori & Mukai (1981) classified the ones with second Betti number at least 2, finding 88 deformation classes.