Fano variety

A smooth complete irreducible algebraic variety over a field whose anti-canonical sheaf is ample (cf. Ample sheaf). The basic research into such varieties was done by G. Fano ([1], [2]).

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 [3], ). 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 [4]). 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 [3]). In the case there are 18 types of Fano varieties, which have been classified (see [6]).

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 [4], ).

References

[1]	G. Fano, "Sulle varietà algebriche a tre dimensioni aventi tutti i generi nulu" , Proc. Internat. Congress Mathematicians (Bologna) , 4 , Zanichelli (1934) pp. 115–119
[2]	G. Fano, "Si alcune varietà algebriche a tre dimensioni razionali, e aventi curve-sezioni canoniche" Comment. Math. Helv. , 14 (1942) pp. 202–211
[3]	S. Mori, S. Mukai, "Classification of Fano 3-folds with " Manuscripta Math. , 36 : 2 (1981) pp. 147–162
[4]	L. Roth, "Sulle algebriche su cui l'aggiunzione si estingue" Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. , 9 (1950) pp. 246–250
[5a]	V.A. Iskovskikh, "Fano 3-folds. I" Math. USSR. Izv. , 11 : 3 (1977) pp. 485–527 Izv. Akad. Nauk SSSR Ser. Mat. , 41 : 3 (1977) pp. 516–562
[5b]	V.A. Iskovskikh, "Fano 3-folds. II" Math. USSR. Izv. , 12 : 3 (1978) pp. 469–506 Izv. Akad. Nauk SSSR Ser. Mat. , 42 : 3 (1978) pp. 506–549
[6]	V.A. Iskovskikh, "Anticanonical models of three-dimensional algebraic varieties" J. Soviet Math. , 13 : 6 (1980) pp. 745–850 Itogi Nauk. i Tekhn. Sovr. Probl. Mat. , 12 (1979) pp. 59–157