Fano variety

A smooth complete irreducible algebraic variety $ X $ over a field $ k $ whose anti-canonical sheaf $ K _ {X} ^ {-1} $ 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 $ > 2 $— are not all rational varieties, for example the general cubic in the projective space $ P ^ {4} $. 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 $ \mathop{\rm Pic} X $ of a three-dimensional Fano variety is finitely generated and torsion-free. In case the ground field $ k $ is $ \mathbf C $, the rank of $ \mathop{\rm Pic} X $, which is equal to the second Betti number $ b _ {2} ( X) $, does not exceed 10 (see [4]). If $ b _ {2} ( X) \geq 6 $, then the Fano variety is isomorphic to $ P ^ {1} \times S _ {11 - b _ {2} ( X) } $, where $ S _ {d} $ is the del Pezzo surface of order $ d $. A Fano variety $ X $ is called primitive if there is no monoidal transformation $ \sigma : X \rightarrow X _ {1} $ to a smooth variety $ X _ {1} $ with centre at a non-singular irreducible curve. If $ X $ is a primitive Fano variety, then $ b _ {2} ( X) \leq 3 $. If $ b _ {2} ( X) = 3 $, then $ X $ is a conic fibre space over $ S = P ^ {1} \times P ^ {1} $, in other words, then there is a morphism $ \pi : X \rightarrow S $ each fibre of which is isomorphic to a conic, that is, an algebraic scheme given by a homogeneous equation of degree 2 in $ P ^ {2} $. A Fano variety $ X $ with $ b _ {2} ( X) = 2 $ is a conic fibre space over the projective plane $ P ^ {2} $( see [3]). In the case $ b _ {2} ( X) = 1 $ there are 18 types of Fano varieties, which have been classified (see [6]).

For three-dimensional Fano varieties $ X $ the self-intersection index of the anti-canonical divisor $ (- K _ {X} ^ {3} ) \leq 64 $. The largest integer $ r \geq 1 $ such that $ H ^ {\otimes r } $ is isomorphic to $ K _ {X} ^ {-1} $ for some divisor $ H \in \mathop{\rm Pic} X $ 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 $ P ^ {3} $, and a Fano variety of index 3 is isomorphic to a smooth quadric $ Q \subset P ^ {4} $. If $ r = 2 $, then the self-intersection index $ d = H ^ {3} $ can take the values $ 1 \leq d \leq 7 $, with each of them being realized for some Fano variety. For a Fano variety of index 1 the mapping $ \phi _ {K _ {X} ^ {-1} } : X \rightarrow P ^ { \mathop{\rm dim} | K _ {X} ^ {-1} | } $ defined by the linear system $ | K _ {X} ^ {-1} | $ has degree $ \mathop{\rm deg} \phi _ {K _ {X} ^ {-1} } = 1 $ or 2. The Fano varieties of index 1 for which $ \mathop{\rm deg} \phi _ {K _ {X} ^ {-1} } = 2 $ have been classified. If $ \mathop{\rm deg} \phi _ {K _ {X} ^ {-1} } = 1 $, then $ X $ can be realized as a subvariety $ V _ {2g - 2 } $ of degree $ 2g - 2 $ in the projective space $ P ^ {g + 1 } $. The number $ g $ is called the genus of the Fano variety $ V _ {2g - 2 } $ and is the same as the genus of the canonical curve — the section of $ X $ under the anti-canonical imbedding into $ P ^ {g + 1 } $. The Fano varieties $ V _ {2g - 2 } \subset P ^ {g + 1 } $ the class of a hyperplane section of which is the same as the anti-canonical class and generates $ \mathop{\rm Pic} V _ {2g - 2 } $ 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 $B_2 \geq 2$" Manuscripta Math. , 36 : 2 (1981) pp. 147–162
[4]	L. Roth, "Sulle $V_3$ 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