# 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 " 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
How to Cite This Entry:
Fano variety. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fano_variety&oldid=46904
This article was adapted from an original article by Vik.S. Kulikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article