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A smooth complete irreducible [[Algebraic variety|algebraic variety]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038220/f0382201.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038220/f0382202.png" /> whose anti-canonical sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038220/f0382203.png" /> is ample (cf. [[Ample sheaf|Ample sheaf]]). The basic research into such varieties was done by G. Fano ([[#References|[1]]], [[#References|[2]]]).
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A Fano variety of dimension 2 is called a del Pezzo surface and is a [[Rational surface|rational surface]]. The multi-dimensional analogues of del Pezzo surfaces — Fano varieties of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038220/f0382204.png" /> — are not all rational varieties, for example the general cubic in the projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038220/f0382205.png" />. It is not known (1984) whether all Fano varieties are unirational.
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A smooth complete irreducible [[Algebraic variety|algebraic variety]]  $  X $
 +
over a field  $  k $
 +
whose anti-canonical sheaf  $  K _ {X}  ^ {-} 1 $
 +
is ample (cf. [[Ample sheaf|Ample sheaf]]). The basic research into such varieties was done by G. Fano ([[#References|[1]]], [[#References|[2]]]).
 +
 
 +
A Fano variety of dimension 2 is called a del Pezzo surface and is a [[Rational surface|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 [[#References|[3]]], ). Only isolated particular results are known about Fano varieties of dimension greater than 3.
 
Three-dimensional Fano varieties have been thoroughly investigated (see [[#References|[3]]], ). Only isolated particular results are known about Fano varieties of dimension greater than 3.
  
The [[Picard group|Picard group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038220/f0382206.png" /> of a three-dimensional Fano variety is finitely generated and torsion-free. In case the ground field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038220/f0382207.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038220/f0382208.png" />, the rank of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038220/f0382209.png" />, which is equal to the second [[Betti number|Betti number]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038220/f03822010.png" />, does not exceed 10 (see [[#References|[4]]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038220/f03822011.png" />, then the Fano variety is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038220/f03822012.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038220/f03822013.png" /> is the del Pezzo surface of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038220/f03822014.png" />. A Fano variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038220/f03822015.png" /> is called primitive if there is no [[Monoidal transformation|monoidal transformation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038220/f03822016.png" /> to a smooth variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038220/f03822017.png" /> with centre at a non-singular irreducible curve. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038220/f03822018.png" /> is a primitive Fano variety, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038220/f03822019.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038220/f03822020.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038220/f03822021.png" /> is a conic fibre space over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038220/f03822022.png" />, in other words, then there is a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038220/f03822023.png" /> each fibre of which is isomorphic to a conic, that is, an algebraic scheme given by a homogeneous equation of degree 2 in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038220/f03822024.png" />. A Fano variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038220/f03822025.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038220/f03822026.png" /> is a conic fibre space over the projective plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038220/f03822027.png" /> (see [[#References|[3]]]). In the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038220/f03822028.png" /> there are 18 types of Fano varieties, which have been classified (see [[#References|[6]]]).
+
The [[Picard group|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|Betti number]] $  b _ {2} ( X) $,  
 +
does not exceed 10 (see [[#References|[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|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 [[#References|[3]]]). In the case $  b _ {2} ( X) = 1 $
 +
there are 18 types of Fano varieties, which have been classified (see [[#References|[6]]]).
  
For three-dimensional Fano varieties <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038220/f03822029.png" /> the self-intersection index of the anti-canonical divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038220/f03822030.png" />. The largest integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038220/f03822031.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038220/f03822032.png" /> is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038220/f03822033.png" /> for some divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038220/f03822034.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038220/f03822035.png" />, and a Fano variety of index 3 is isomorphic to a smooth quadric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038220/f03822036.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038220/f03822037.png" />, then the self-intersection index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038220/f03822038.png" /> can take the values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038220/f03822039.png" />, with each of them being realized for some Fano variety. For a Fano variety of index 1 the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038220/f03822040.png" /> defined by the linear system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038220/f03822041.png" /> has degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038220/f03822042.png" /> or 2. The Fano varieties of index 1 for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038220/f03822043.png" /> have been classified. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038220/f03822044.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038220/f03822045.png" /> can be realized as a subvariety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038220/f03822046.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038220/f03822047.png" /> in the projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038220/f03822048.png" />. The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038220/f03822049.png" /> is called the genus of the Fano variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038220/f03822050.png" /> and is the same as the genus of the canonical curve — the section of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038220/f03822051.png" /> under the anti-canonical imbedding into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038220/f03822052.png" />. The Fano varieties <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038220/f03822053.png" /> the class of a hyperplane section of which is the same as the anti-canonical class and generates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038220/f03822054.png" /> have been classified (see [[#References|[4]]], ).
+
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 [[#References|[4]]], ).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Fano,  "Sulle varietà algebriche a tre dimensioni aventi tutti i generi nulu" , ''Proc. Internat. Congress Mathematicians (Bologna)'' , '''4''' , Zanichelli  (1934)  pp. 115–119</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G. Fano,  "Si alcune varietà algebriche a tre dimensioni razionali, e aventi curve-sezioni canoniche"  ''Comment. Math. Helv.'' , '''14'''  (1942)  pp. 202–211</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S. Mori,  S. Mukai,  "Classification of Fano 3-folds with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038220/f03822055.png" />"  ''Manuscripta Math.'' , '''36''' :  2  (1981)  pp. 147–162</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  L. Roth,  "Sulle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038220/f03822056.png" /> algebriche su cui l'aggiunzione si estingue"  ''Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat.'' , '''9'''  (1950)  pp. 246–250</TD></TR><TR><TD valign="top">[5a]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[5b]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  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</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Fano,  "Sulle varietà algebriche a tre dimensioni aventi tutti i generi nulu" , ''Proc. Internat. Congress Mathematicians (Bologna)'' , '''4''' , Zanichelli  (1934)  pp. 115–119</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G. Fano,  "Si alcune varietà algebriche a tre dimensioni razionali, e aventi curve-sezioni canoniche"  ''Comment. Math. Helv.'' , '''14'''  (1942)  pp. 202–211</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S. Mori,  S. Mukai,  "Classification of Fano 3-folds with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038220/f03822055.png" />"  ''Manuscripta Math.'' , '''36''' :  2  (1981)  pp. 147–162</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  L. Roth,  "Sulle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038220/f03822056.png" /> algebriche su cui l'aggiunzione si estingue"  ''Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat.'' , '''9'''  (1950)  pp. 246–250</TD></TR><TR><TD valign="top">[5a]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[5b]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  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</TD></TR></table>

Revision as of 19:38, 5 June 2020


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=14961
This article was adapted from an original article by Vik.S. Kulikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article