Fano variety

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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 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.

[edit] References

• Fano, G. (1934), "Sulle varietà algebriche a tre dimensioni aventi tutti i generi nulu", Proc. Internat. Congress Mathematicians (Bologna) , 4 , Zanichelli, pp. 115–119 
• Fano, Gino (1942), "Su alcune varietà algebriche a tre dimensioni razionali, e aventi curve-sezioni canoniche", Commentarii Mathematici Helvetici 14: 202–211, MR0006445, ISSN 0010-2571, http://gdz.sub.uni-goettingen.de/no_cache/dms/load/img/?IDDOC=209966 
• Iskovskih, V. A. (1977), "Fano threefolds. I", Math. USSR-Izv. 11 (3): 485–527, doi:10.1070/IM1977v011n03ABEH001733, MR463151, ISSN 0373-2436 
• Iskovskih, V. A. (1978), "Fano 3-folds II", MATH USSR IZV 12 (3): 469–506, doi:10.1070/IM1978v012n03ABEH001994Fano 3-folds II, MR0463151 
• Iskovskih, V. A. (1979), "Anticanonical models of three-dimensional algebraic varieties", Current problems in mathematics, Vol. 12 (Russian), VINITI, Moscow, pp. 59–157, MR537685 
• Kulikov, Vik.S. (2001), "Fano variety", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104 
• Mori, Shigefumi; Mukai, Shigeru (1981), "Classification of Fano 3-folds with B₂≥2", Manuscripta Mathematica 36 (2): 147–162, doi:10.1007/BF01170131, MR641971, ISSN 0025-2611 Mori, Shigefumi; Mukai, Shigeru (2003), "Erratum: "Classification of Fano 3-folds with B₂≥2"", Manuscripta Mathematica 110 (3): 407, doi:10.1007/s00229-002-0336-2, MR1969009, ISSN 0025-2611