A Fano variety of dimension 2 is called a del Pezzo surface and is a rational surface. The multi-dimensional analogues of del Pezzo surfaces — Fano varieties of dimension — are not all rational varieties, for example the general cubic in the projective space . It is not known (1984) whether all Fano varieties are unirational.
Three-dimensional Fano varieties have been thoroughly investigated (see , ). Only isolated particular results are known about Fano varieties of dimension greater than 3.
The Picard group of a three-dimensional Fano variety is finitely generated and torsion-free. In case the ground field is , the rank of , which is equal to the second Betti number , does not exceed 10 (see ). If , then the Fano variety is isomorphic to , where is the del Pezzo surface of order . A Fano variety is called primitive if there is no monoidal transformation to a smooth variety with centre at a non-singular irreducible curve. If is a primitive Fano variety, then . If , then is a conic fibre space over , in other words, then there is a morphism each fibre of which is isomorphic to a conic, that is, an algebraic scheme given by a homogeneous equation of degree 2 in . A Fano variety with is a conic fibre space over the projective plane (see ). In the case there are 18 types of Fano varieties, which have been classified (see ).
For three-dimensional Fano varieties the self-intersection index of the anti-canonical divisor . The largest integer such that is isomorphic to for some divisor is called the index of the Fano variety. The index of a three-dimensional Fano variety can take the values 1, 2, 3, or 4. A Fano variety of index 4 is isomorphic to the projective space , and a Fano variety of index 3 is isomorphic to a smooth quadric . If , then the self-intersection index can take the values , with each of them being realized for some Fano variety. For a Fano variety of index 1 the mapping defined by the linear system has degree or 2. The Fano varieties of index 1 for which have been classified. If , then can be realized as a subvariety of degree in the projective space . The number is called the genus of the Fano variety and is the same as the genus of the canonical curve — the section of under the anti-canonical imbedding into . The Fano varieties the class of a hyperplane section of which is the same as the anti-canonical class and generates have been classified (see , ).
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Fano variety. Vik.S. Kulikov (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Fano_variety&oldid=14961