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An algebraic or analytic complete non-singular surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e0355501.png" /> having a fibration of elliptic curves (cf. [[Elliptic curve|Elliptic curve]]), that is, a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e0355502.png" /> onto a non-singular curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e0355503.png" /> whose generic fibre is a non-singular elliptic curve. Every elliptic surface is birationally (bimeromorphically) equivalent over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e0355504.png" /> to a unique minimal model, which is characterized by the fact that the fibre of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e0355505.png" /> does not contain exceptional curves of arithmetic genus 1. In what follows an elliptic surface is assumed to be minimal. Minimal elliptic surfaces have a more complicated structure than ruled surfaces. They can have singular fibres <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e0355506.png" /> (that is, fibres that are not non-singular elliptic curves). There is a classification
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of the singular fibres of elliptic surfaces. A singular fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e0355507.png" /> is called multiple if the greatest common divisor of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e0355508.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e0355509.png" />, and then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555011.png" /> is called the multiplicity of the fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555012.png" />.
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On a minimal elliptic surface the canonical class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555013.png" /> contains a divisor that is a rational combination of fibres, in particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555014.png" />. Moreover, the following formula holds for the canonical class (see [[#References|[1]]], ):
+
An algebraic or analytic complete non-singular surface  $  X $
 +
having a fibration of elliptic curves (cf. [[Elliptic curve|Elliptic curve]]), that is, a morphism  $  \pi : X \rightarrow B $
 +
onto a non-singular curve  $  B $
 +
whose generic fibre is a non-singular elliptic curve. Every elliptic surface is birationally (bimeromorphically) equivalent over  $  B $
 +
to a unique minimal model, which is characterized by the fact that the fibre of  $  \pi $
 +
does not contain exceptional curves of arithmetic genus 1. In what follows an elliptic surface is assumed to be minimal. Minimal elliptic surfaces have a more complicated structure than ruled surfaces. They can have singular fibres  $  X _ {t} = \pi  ^ {- 1} ( t) $ (that is, fibres that are not non-singular elliptic curves). There is a classification
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555015.png" /></td> </tr></table>
+
of the singular fibres of elliptic surfaces. A singular fibre  $  X _ {t} = \sum n _ {i} E _ {i} $
 +
is called multiple if the greatest common divisor of the  $  n _ {i} $
 +
is  $  m \geq  2 $,
 +
and then  $  X _ {t} = m F $
 +
and  $  m $
 +
is called the multiplicity of the fibre  $  X _ {t} $.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555016.png" /> are all the multiple fibres of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555018.png" /> is a divisor on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555019.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555020.png" />. The topological [[Euler characteristic|Euler characteristic]] satisfies the formula
+
On a minimal elliptic surface the canonical class  $  K _ {X} $
 +
contains a divisor that is a rational combination of fibres, in particular,  $  ( K _ {X}  ^ {2} ) = 0 $.  
 +
Moreover, the following formula holds for the canonical class (see [[#References|[1]]], ):
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555021.png" /></td> </tr></table>
+
$$
 +
K _ {X}  = \pi  ^ {*} ( K _ {B} - d ) +
 +
\sum ( m _ {i} - 1 ) F _ {i} ,
 +
$$
 +
 
 +
where  $  X _ {t _ {i}  } = m _ {i} F _ {i} $
 +
are all the multiple fibres of  $  \pi $
 +
and  $  d $
 +
is a divisor on  $  B $
 +
of degree  $  - \chi ( {\mathcal O} _ {X} ) $.
 +
The topological [[Euler characteristic|Euler characteristic]] satisfies the formula
 +
 
 +
$$
 +
e ( X)  = \sum e ( X _ {t _ {i}  } ) .
 +
$$
  
 
==The classification of elliptic fibrations.==
 
==The classification of elliptic fibrations.==
A fibration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555022.png" /> can be regarded as an elliptic curve over the function field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555023.png" />. This curve, generally speaking, does not have the structure of an Abelian variety over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555024.png" />. For this to happen it is necessary that it has a rational point over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555025.png" /> (and then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555026.png" /> is birationally isomorphic to the surface defined in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555027.png" /> by the Weierstrass equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555028.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555029.png" />). The specification of a rational point is equivalent to that of a section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555030.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555031.png" />; a necessary condition for the existence of a section is the absence of multiple fibres. Fibrations without multiple fibres are called reduced. Every fibration, after a suitable ramified covering of the base, has a section (that is, is reduced) . Every fibration can also be made reduced by a sequence of transformations inverse to logarithmic ones
+
A fibration $  \pi : X \rightarrow B $
 +
can be regarded as an elliptic curve over the function field $  k ( B) $.  
 +
This curve, generally speaking, does not have the structure of an Abelian variety over $  k ( B) $.  
 +
For this to happen it is necessary that it has a rational point over $  k ( B) $ (and then $  X $
 +
is birationally isomorphic to the surface defined in $  B \times A  ^ {2} $
 +
by the Weierstrass equation $  y  ^ {2} = x  ^ {3} - g _ {2} x - g _ {3} $,  
 +
where $  g _ {2} , g _ {3} \in k ( B) $).  
 +
The specification of a rational point is equivalent to that of a section $  e : B \rightarrow X $
 +
such that $  \pi e = \mathop{\rm id} $;  
 +
a necessary condition for the existence of a section is the absence of multiple fibres. Fibrations without multiple fibres are called reduced. Every fibration, after a suitable ramified covering of the base, has a section (that is, is reduced) . Every fibration can also be made reduced by a sequence of transformations inverse to logarithmic ones
  
 
— local surgery of the fibration in neighbourhoods of fibres.
 
— local surgery of the fibration in neighbourhoods of fibres.
  
Reduced elliptic fibrations may be described as follows. To every such fibration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555032.png" /> corresponds a unique fibration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555033.png" /> that is a [[Group object|group object]] and is such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555034.png" /> is a [[Principal homogeneous space|principal homogeneous space]] over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555035.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555036.png" /> is the Jacobi fibration for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555037.png" />; it characterizes the existence of a section. For a given Jacobi fibration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555038.png" />, the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555039.png" /> of isomorphism classes of fibrations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555040.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555041.png" /> has a cohomology description analogous to that of invertible sheaves (cf. [[Invertible sheaf|Invertible sheaf]]). Here the role of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555042.png" /> is played by the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555043.png" /> of local sections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555044.png" />. There is a natural one-to-one correspondence
+
Reduced elliptic fibrations may be described as follows. To every such fibration $  \pi : X \rightarrow B $
 +
corresponds a unique fibration $  {\mathcal J} _ {B} ( X) \rightarrow B $
 +
that is a [[Group object|group object]] and is such that $  X / B $
 +
is a [[Principal homogeneous space|principal homogeneous space]] over $  {\mathcal J} _ {B} ( X) / B $;  
 +
$  {\mathcal J} _ {B} ( X) / B $
 +
is the Jacobi fibration for $  X / B $;  
 +
it characterizes the existence of a section. For a given Jacobi fibration $  {\mathcal J} / B $,  
 +
the set $  I ( {\mathcal J} / B ) $
 +
of isomorphism classes of fibrations $  X / B $
 +
for which $  {\mathcal J} _ {B} ( X) \cong J $
 +
has a cohomology description analogous to that of invertible sheaves (cf. [[Invertible sheaf|Invertible sheaf]]). Here the role of $  {\mathcal O} _ {B}  ^ {*} $
 +
is played by the sheaf $  {\mathcal H}  ^ {0} ( {\mathcal J} / B ) $
 +
of local sections $  \tau : {\mathcal J} \rightarrow B $.  
 +
There is a natural one-to-one correspondence
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555045.png" /></td> </tr></table>
+
$$
 +
\theta : I ( {\mathcal J} / B )  \rightarrow  H  ^ {1}
 +
( B , {\mathcal H}  ^ {0} ( {\mathcal J} / B ) ) ,
 +
$$
  
under which the Jacobi fibration corresponds to the zero element. By means of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555046.png" /> one can distinguish between algebraic and non-algebraic fibrations: For a reduced fibration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555047.png" /> the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555048.png" /> is algebraic if and only if the element corresponding to it in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555049.png" /> is of finite order. The analogy with invertible sheaves can be pursued further. The analogue of the exact sequence
+
under which the Jacobi fibration corresponds to the zero element. By means of $  \theta $
 +
one can distinguish between algebraic and non-algebraic fibrations: For a reduced fibration $  \pi : X \rightarrow B $
 +
the surface $  X $
 +
is algebraic if and only if the element corresponding to it in $  H  ^ {1} ( B , {\mathcal H}  ^ {0} ( {\mathcal J} / B )) $
 +
is of finite order. The analogy with invertible sheaves can be pursued further. The analogue of the exact sequence
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555050.png" /></td> </tr></table>
+
$$
 +
0 \rightarrow  \mathbf Z  \rightarrow  {\mathcal O} _ {B}  \mathop \rightarrow \limits ^  {\mathop  {\rm exp}} {\mathcal O} _ {B}  ^ {*}  \rightarrow  1
 +
$$
  
 
is the exact sequence
 
is the exact sequence
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555051.png" /></td> </tr></table>
+
$$
 +
0 \rightarrow  R  ^ {1} \tau _ {0} \mathbf Z  \rightarrow  {\mathcal H}  ^ {0}
 +
( T ( {\mathcal J} ) / B )  \rightarrow  {\mathcal H}  ^ {0} ( {\mathcal J} / B )  \rightarrow  0 ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555052.png" /> is the sheaf of local sections of the bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555053.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555054.png" /> is the tangent space to the fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555055.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555056.png" />. The boundary homomorphism
+
where $  {\mathcal H}  ^ {0} ( T ( {\mathcal J} ) / B ) $
 +
is the sheaf of local sections of the bundle $  T ( {\mathcal J} )/ B $
 +
and $  T ( {\mathcal J} ) $
 +
is the tangent space to the fibre $  \tau  ^ {- 1 }( b) $
 +
at e ( b) $.  
 +
The boundary homomorphism
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555057.png" /></td> </tr></table>
+
$$
 +
\delta : H  ^ {1} ( B , {\mathcal H}  ^ {0}
 +
( {\mathcal J} / B ) )  \rightarrow  H  ^ {2} ( B , R  ^ {1} \tau _ {*} \mathbf Z )
 +
$$
  
allows one to recognize when one fibration is a deformation of another. For that it is necessary and sufficient that the elements corresponding to these fibrations have one and the same image under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555058.png" /> (see ).
+
allows one to recognize when one fibration is a deformation of another. For that it is necessary and sufficient that the elements corresponding to these fibrations have one and the same image under $  \delta $ (see ).
  
 
==The classification of algebraic elliptic surfaces.==
 
==The classification of algebraic elliptic surfaces.==
Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555059.png" />. For an elliptic surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555060.png" /> the canonical dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555061.png" />, that is, it is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555062.png" />, 0 or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555063.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555064.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555065.png" /> is said to be an elliptic surface of general type. These are characterized by the conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555066.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555067.png" />. Elliptic surfaces with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555068.png" /> or, more generally, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555069.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555070.png" />, are of general type.
+
Suppose that $  \mathop{\rm char}  k = 0 $.  
 +
For an elliptic surface $  X $
 +
the canonical dimension $  k ( X) \leq  1 $,  
 +
that is, it is equal to $  - 1 $,  
 +
0 or $  1 $.  
 +
If $  k ( X) = 1 $,  
 +
$  X $
 +
is said to be an elliptic surface of general type. These are characterized by the conditions $  12 K _ {X} \neq 0 $
 +
and $  | 12 K _ {X} | \neq \emptyset $.  
 +
Elliptic surfaces with $  p _ {g} \geq  2 $
 +
or, more generally, with $  P _ {m} \geq  2 $
 +
for some $  m $,  
 +
are of general type.
  
Elliptic surfaces with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555071.png" /> are characterized by the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555072.png" />. In this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555073.png" /> can take the three values 2, 1 or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555074.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555075.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555076.png" /> is an elliptic [[K3-surface|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555077.png" />-surface]] (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555078.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555079.png" />). In this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555080.png" /> is isomorphic to the projective line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555081.png" />, the fibration has no multiple fibres and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555082.png" /> has the invariants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555083.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555084.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555085.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555086.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555087.png" /> is an Enriques surface, that is, a surface with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555088.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555089.png" />. (Every Enriques surface is elliptic.) In this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555090.png" />, the fibration has two fibres of multiplicity 2, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555091.png" /> has the invariants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555092.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555093.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555094.png" />, then two cases are possible. Either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555095.png" /> is an Abelian variety (and then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555096.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555097.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555098.png" />); or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555099.png" /> is a hyper-elliptic surface, that is, a surface that has a finite unramified covering — the product of two elliptic curves. In that case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550100.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550101.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550102.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550103.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550104.png" /> has 3 or 4 multiple fibres with four possibilities for their multiplicity: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550105.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550106.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550107.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550108.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550109.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550110.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550111.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550112.png" />, respectively.
+
Elliptic surfaces with $  k ( X) = 0 $
 +
are characterized by the condition $  12 K _ {X} = 0 $.  
 +
In this case $  \chi ( {\mathcal O} _ {X} ) $
 +
can take the three values 2, 1 or 0 $.  
 +
If $  \chi ( {\mathcal O} _ {X} )= 2 $,  
 +
then $  X $
 +
is an elliptic [[K3-surface| $  K 3 $-surface]] ( $  q = 0 $,  
 +
$  K _ {X} = 0 $).  
 +
In this case $  B $
 +
is isomorphic to the projective line $  P  ^ {1} $,  
 +
the fibration has no multiple fibres and $  X $
 +
has the invariants $  p _ {g} = 1 $,  
 +
$  q = 0 $,  
 +
$  b _ {2} = 22 $.  
 +
If $  \chi ( {\mathcal O} _ {X} ) = 1 $,  
 +
then $  X $
 +
is an Enriques surface, that is, a surface with $  p _ {g} = q = 0 $,  
 +
$  2 K _ {X} = 0 $.  
 +
(Every Enriques surface is elliptic.) In this case $  B \simeq P  ^ {1} $,  
 +
the fibration has two fibres of multiplicity 2, and $  X $
 +
has the invariants $  p _ {g} = q = 0 $,  
 +
$  b _ {2} = 10 $.  
 +
If $  \chi ( {\mathcal O} _ {X} ) = 0 $,  
 +
then two cases are possible. Either $  X $
 +
is an Abelian variety (and then $  p _ {g} = 1 $,  
 +
$  q = 2 $,  
 +
$  b _ {2} = 6 $);  
 +
or $  X $
 +
is a hyper-elliptic surface, that is, a surface that has a finite unramified covering — the product of two elliptic curves. In that case $  p _ {g} = 0 $,  
 +
$  b _ {1} = 2 $,  
 +
$  b _ {2} = 2 $,  
 +
$  B = P  ^ {1} $,  
 +
and $  \pi $
 +
has 3 or 4 multiple fibres with four possibilities for their multiplicity: $  ( 3 , 3 , 3 ) $,
 +
$  ( 2 , 4 , 4 ) $,
 +
$  ( 2 , 3 , 6 ) $,  
 +
and $  ( 2 , 2 , 2 , 2 ) $,  
 +
and $  3 K _ {X} = 0 $,  
 +
$  4 K _ {X} = 0 $,  
 +
$  6 K _ {X} = 0 $,  
 +
and $  2 K _ {X} = 0 $,  
 +
respectively.
  
An elliptic surface with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550113.png" /> is ruled (cf. [[Ruled surface|Ruled surface]]). It is characterized by the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550114.png" />. Here two cases are possible: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550115.png" /> is a surface with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550116.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550117.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550118.png" /> has no multiple fibres or one; moreover, a surface without multiple fibres can be obtained as follows: one has to take a rational mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550119.png" /> determined by two cubics <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550120.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550121.png" /> and blow up their 9 points of intersection; or 2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550122.png" /> is a surface with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550123.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550124.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550125.png" />, and the multiplicities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550126.png" /> are subject to the inequality
+
An elliptic surface with $  k ( X) = - 1 $
 +
is ruled (cf. [[Ruled surface|Ruled surface]]). It is characterized by the condition $  | 12 K _ {X} | = \emptyset $.  
 +
Here two cases are possible: 1) $  X $
 +
is a surface with $  p _ {g} = q = 0 $,  
 +
$  b _ {2} = 10 $,  
 +
and $  \pi $
 +
has no multiple fibres or one; moreover, a surface without multiple fibres can be obtained as follows: one has to take a rational mapping $  P  ^ {2} \rightarrow P  ^ {1} $
 +
determined by two cubics $  F _ {0} $
 +
and $  F _ {1} $
 +
and blow up their 9 points of intersection; or 2) $  X $
 +
is a surface with $  p _ {g} = 0 $,  
 +
$  q = 1 $,  
 +
$  b _ {2} = 2 $,  
 +
and the multiplicities $  m _ {i} $
 +
are subject to the inequality
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550127.png" /></td> </tr></table>
+
$$
 +
\sum \left ( 1 -  
 +
\frac{1}{m _ {i} }
 +
\right )  < 2 .
 +
$$
  
 
The formula for the canonical class and the classification of elliptic surfaces can also be generalized to the case of a field of finite characteristic [[#References|[5]]], [[#References|[6]]].
 
The formula for the canonical class and the classification of elliptic surfaces can also be generalized to the case of a field of finite characteristic [[#References|[5]]], [[#References|[6]]].
  
 
==The classification of non-algebraic elliptic surfaces.==
 
==The classification of non-algebraic elliptic surfaces.==
The classification of non-algebraic elliptic surfaces. For non-algebraic surfaces the algebraic dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550128.png" /> is 1 or 0. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550129.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550130.png" /> is non-elliptic. All surfaces with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550131.png" /> are elliptic. Here the structure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550132.png" /> is determined almost canonically: Every such fibration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550133.png" /> is necessarily elliptic. The classification by the canonical dimension is precisely the same as for algebraic elliptic surfaces: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550134.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550135.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550136.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550137.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550138.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550139.png" />; and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550140.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550141.png" /> is of basic type) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550142.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550143.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550144.png" />.
+
The classification of non-algebraic elliptic surfaces. For non-algebraic surfaces the algebraic dimension $  a ( X) = \mathop{\rm tr}  \mathop{\rm deg}  M ( X) $
 +
is 1 or 0. If $  a ( X) = 0 $,  
 +
then $  X $
 +
is non-elliptic. All surfaces with $  a ( X) = 1 $
 +
are elliptic. Here the structure of $  \pi : X \rightarrow B $
 +
is determined almost canonically: Every such fibration $  \pi $
 +
is necessarily elliptic. The classification by the canonical dimension is precisely the same as for algebraic elliptic surfaces: $  k ( X) = - 1 $
 +
$  \iff $
 +
$  | 12 K _ {X} | = \emptyset $;  
 +
$  k ( X) = 0 $
 +
$  \iff $
 +
$  12K _ {X}  = 0 $;  
 +
and $  k ( X) = 1 $ ($  X $
 +
is of basic type) $  \iff $
 +
$  | 12 K _ {X} | = \emptyset $,  
 +
$  12 K _ {X} \neq 0 $.
  
Non-algebraic elliptic surfaces with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550145.png" /> belong to one of the following classes: 1) the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550146.png" />-surfaces (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550147.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550148.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550149.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550150.png" /> is simply connected); 2) the complex tori (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550151.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550152.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550153.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550154.png" />); 3) the Kodaira surfaces (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550155.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550156.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550157.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550158.png" />). Primary Kodaira surfaces are holomorphically locally trivial fibrations over an elliptic curve with an elliptic curve as typical fibre, and from the point of view of differentiability, bundles over a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550159.png" />-dimensional torus with a circle as fibre; or 4) the surfaces with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550160.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550161.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550162.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550163.png" />. For them <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550164.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550165.png" />, (analogous to hyper-elliptic surfaces). They have Kodaira surfaces as finite unramified coverings. In the cases 2), 3) and 4) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550166.png" /> is the universal covering of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550167.png" />.
+
Non-algebraic elliptic surfaces with $  k ( X) = 0 $
 +
belong to one of the following classes: 1) the $  K 3 $-surfaces ( $  \chi ( {\mathcal O} _ {X} )= 2 $,  
 +
$  b _ {1} = 0 $,  
 +
$  b _ {2} = 22 $,  
 +
$  X $
 +
is simply connected); 2) the complex tori ( $  K _ {X} = 0 $,  
 +
$  \chi ( {\mathcal O} _ {X} ) = 0 $,  
 +
$  b _ {1} = 4 $,  
 +
$  b _ {2} = 6 $);  
 +
3) the Kodaira surfaces ( $  K _ {X} = 0 $,  
 +
$  \chi ( {\mathcal O} _ {X} ) = 0 $,  
 +
$  b _ {1} = 3 $,  
 +
$  b _ {2} = 4 $).  
 +
Primary Kodaira surfaces are holomorphically locally trivial fibrations over an elliptic curve with an elliptic curve as typical fibre, and from the point of view of differentiability, bundles over a $  3 $-dimensional torus with a circle as fibre; or 4) the surfaces with $  \chi ( {\mathcal O} _ {X} ) = 0 $,  
 +
$  p _ {g} = 0 $,  
 +
$  b _ {1} = 1 $,  
 +
$  b _ {2} = 0 $.  
 +
For them $  m K _ {X} = 0 $
 +
with $  m = 2 , 3 , 4 $,  
 +
(analogous to hyper-elliptic surfaces). They have Kodaira surfaces as finite unramified coverings. In the cases 2), 3) and 4) $  \mathbf C  ^ {2} $
 +
is the universal covering of $  X $.
  
Non-algebraic elliptic surfaces with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550168.png" /> are Hopf surfaces, that is, their universal covering is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550169.png" />. For them <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550170.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550171.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550172.png" />. The proper Hopf surfaces are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550173.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550174.png" /> are real generators of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550175.png" />. They are homeomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550176.png" /> and are characterized by this property. Arbitrary elliptic Hopf surfaces are quotients of proper Hopf surfaces .
+
Non-algebraic elliptic surfaces with $  k ( X) = - 1 $
 +
are Hopf surfaces, that is, their universal covering is $  \mathbf C  ^ {2} \setminus  0 $.  
 +
For them $  \chi ( {\mathcal O} _ {X} ) = 0 $,  
 +
$  b _ {1} = 1 $,  
 +
$  b _ {2} = 0 $.  
 +
The proper Hopf surfaces are $  ( \mathbf C  ^ {2} \setminus  0 ) / T $,  
 +
where $  T ( z _ {1} , z _ {2} ) = ( \alpha _ {1} z _ {1} , \alpha _ {2} z _ {2} ) $
 +
are real generators of $  T $.  
 +
They are homeomorphic to $  S  ^ {1} \times S  ^ {2} $
 +
and are characterized by this property. Arbitrary elliptic Hopf surfaces are quotients of proper Hopf surfaces .
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> "Algebraic surfaces" ''Proc. Steklov Inst. Math.'' , '''75''' (1967) ''Trudy Mat. Inst. Steklov.'' , '''75''' (1965)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> D. Husemoller,   "Classification and embeddings of surfaces" R. Hartshorne (ed.) , ''Algebraic geometry (Arcata, 1974)'' , ''Proc. Symp. Pure Math.'' , '''29''' , Amer. Math. Soc. (1982) pp. 329–420</TD></TR><TR><TD valign="top">[3a]</TD> <TD valign="top"> K. Kodaira,   "On compact complex analytic surfaces I" ''Ann. of Math. (2)'' , '''71''' (10) pp. 111–152</TD></TR><TR><TD valign="top">[3b]</TD> <TD valign="top"> K. Kodaira,   "On compact complex analytic surfaces II" ''Ann. of Math. (2)'' , '''77''' (1963) pp. 563–626</TD></TR><TR><TD valign="top">[3c]</TD> <TD valign="top"> K. Kodaira,   "On compact complex analytic surfaces III" ''Ann. of Math. (2)'' , '''78''' (1963) pp. 1–40</TD></TR><TR><TD valign="top">[4a]</TD> <TD valign="top"> K. Kodaira,   "On the structure of compact complex analytic surfaces I" ''Amer. J. Math.'' , '''86''' (1964) pp. 751–798</TD></TR><TR><TD valign="top">[4b]</TD> <TD valign="top"> K. Kodaira,   "On the structure of compact complex analytic surfaces II" ''Amer. J. Math.'' , '''88''' (1966) pp. 682–721</TD></TR><TR><TD valign="top">[4c]</TD> <TD valign="top"> K. Kodaira,   "On the structure of compact complex analytic surfaces III" ''Amer. J. Math.'' , '''90''' (1968) pp. 55–83</TD></TR><TR><TD valign="top">[4d]</TD> <TD valign="top"> K. Kodaira,   "On the structure of compact complex analytic surfaces IV" ''Amer. J. Math.'' , '''90''' (1968) pp. 1048–1066</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> D. Mumford,   "Enriques' classification of surfaces in char <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550177.png" />. I" D.C. Spencer (ed.) S. Iyanaga (ed.) , ''Global analysis (papers in honor of K. Kodaira)'' , Princeton Univ. Press (1969) pp. 325–339</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> E. Bombieri,   D. Mumford,   "Enriques' classification of surfaces in char <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550178.png" />. II" W.L. Baily jr. (ed.) T. Shioda (ed.) , ''Complex Analysis and Algebraic geometry'' , Cambridge Univ. Press &amp; Iwanami Shoten (1977) pp. 23–42</TD></TR></table>
+
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> "Algebraic surfaces" ''Proc. Steklov Inst. Math.'' , '''75''' (1967) ''Trudy Mat. Inst. Steklov.'' , '''75''' (1965) {{MR|}} {{ZBL|0172.37901}} {{ZBL|0153.22401}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> D. Husemoller, "Classification and embeddings of surfaces" R. Hartshorne (ed.) , ''Algebraic geometry (Arcata, 1974)'' , ''Proc. Symp. Pure Math.'' , '''29''' , Amer. Math. Soc. (1982) pp. 329–420 {{MR|0506292}} {{ZBL|0326.14009}} </TD></TR><TR><TD valign="top">[3a]</TD> <TD valign="top"> K. Kodaira, "On compact complex analytic surfaces I" ''Ann. of Math. (2)'' , '''71''' (10) pp. 111–152 {{MR|0132556}} {{ZBL|0137.17501}} {{ZBL|0098.13004}} </TD></TR><TR><TD valign="top">[3b]</TD> <TD valign="top"> K. Kodaira, "On compact complex analytic surfaces II" ''Ann. of Math. (2)'' , '''77''' (1963) pp. 563–626 {{MR|165541}} {{ZBL|0193.37701}} {{ZBL|0133.16505}} </TD></TR><TR><TD valign="top">[3c]</TD> <TD valign="top"> K. Kodaira, "On compact complex analytic surfaces III" ''Ann. of Math. (2)'' , '''78''' (1963) pp. 1–40 {{MR|165541}} {{ZBL|0193.37701}} </TD></TR><TR><TD valign="top">[4a]</TD> <TD valign="top"> K. Kodaira, "On the structure of compact complex analytic surfaces I" ''Amer. J. Math.'' , '''86''' (1964) pp. 751–798 {{MR|0187255}} {{ZBL|0137.17501}} </TD></TR><TR><TD valign="top">[4b]</TD> <TD valign="top"> K. Kodaira, "On the structure of compact complex analytic surfaces II" ''Amer. J. Math.'' , '''88''' (1966) pp. 682–721 {{MR|0205280}} {{ZBL|0193.37701}} </TD></TR><TR><TD valign="top">[4c]</TD> <TD valign="top"> K. Kodaira, "On the structure of compact complex analytic surfaces III" ''Amer. J. Math.'' , '''90''' (1968) pp. 55–83 {{MR|0228019}} {{ZBL|0193.37701}} </TD></TR><TR><TD valign="top">[4d]</TD> <TD valign="top"> K. Kodaira, "On the structure of compact complex analytic surfaces IV" ''Amer. J. Math.'' , '''90''' (1968) pp. 1048–1066 {{MR|239114}} {{ZBL|}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> D. Mumford, "Enriques' classification of surfaces in char <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550177.png" />. I" D.C. Spencer (ed.) S. Iyanaga (ed.) , ''Global analysis (papers in honor of K. Kodaira)'' , Princeton Univ. Press (1969) pp. 325–339 {{MR|0491719}} {{MR|0491720}} {{ZBL|}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> E. Bombieri, D. Mumford, "Enriques' classification of surfaces in char <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550178.png" />. II" W.L. Baily jr. (ed.) T. Shioda (ed.) , ''Complex Analysis and Algebraic geometry'' , Cambridge Univ. Press &amp; Iwanami Shoten (1977) pp. 23–42 {{MR|0491719}} {{MR|0491720}} {{ZBL|}} </TD></TR></table>
 
 
 
 
  
 
====Comments====
 
====Comments====
 
A secondary Kodaira surface is a surface other than a primary one, admitting a primary Kodaira surface as an unramified covering. They are elliptic fibrations over a rational curve. Their first Betti number is 1.
 
A secondary Kodaira surface is a surface other than a primary one, admitting a primary Kodaira surface as an unramified covering. They are elliptic fibrations over a rational curve. Their first Betti number is 1.
  
The canonical dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550179.png" /> mentioned at the start of the section on classification of algebraic elliptic surfaces is the [[Kodaira dimension|Kodaira dimension]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550180.png" /> (with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550181.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550182.png" />).
+
The canonical dimension $  k ( X) $
 +
mentioned at the start of the section on classification of algebraic elliptic surfaces is the [[Kodaira dimension|Kodaira dimension]] $  \mathop{\rm Kod} ( X) $ (with $  k ( X) = - 1 $
 +
if $  \mathop{\rm Kod} ( X) = - \infty $).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> W. Barth,   C. Peters,   A. van der Ven,   "Compact complex surfaces" , Springer (1984)</TD></TR></table>
+
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> W. Barth, C. Peters, A. van der Ven, "Compact complex surfaces" , Springer (1984) {{MR|0749574}} {{ZBL|0718.14023}} </TD></TR></table>

Latest revision as of 05:46, 13 June 2022


An algebraic or analytic complete non-singular surface $ X $ having a fibration of elliptic curves (cf. Elliptic curve), that is, a morphism $ \pi : X \rightarrow B $ onto a non-singular curve $ B $ whose generic fibre is a non-singular elliptic curve. Every elliptic surface is birationally (bimeromorphically) equivalent over $ B $ to a unique minimal model, which is characterized by the fact that the fibre of $ \pi $ does not contain exceptional curves of arithmetic genus 1. In what follows an elliptic surface is assumed to be minimal. Minimal elliptic surfaces have a more complicated structure than ruled surfaces. They can have singular fibres $ X _ {t} = \pi ^ {- 1} ( t) $ (that is, fibres that are not non-singular elliptic curves). There is a classification

of the singular fibres of elliptic surfaces. A singular fibre $ X _ {t} = \sum n _ {i} E _ {i} $ is called multiple if the greatest common divisor of the $ n _ {i} $ is $ m \geq 2 $, and then $ X _ {t} = m F $ and $ m $ is called the multiplicity of the fibre $ X _ {t} $.

On a minimal elliptic surface the canonical class $ K _ {X} $ contains a divisor that is a rational combination of fibres, in particular, $ ( K _ {X} ^ {2} ) = 0 $. Moreover, the following formula holds for the canonical class (see [1], ):

$$ K _ {X} = \pi ^ {*} ( K _ {B} - d ) + \sum ( m _ {i} - 1 ) F _ {i} , $$

where $ X _ {t _ {i} } = m _ {i} F _ {i} $ are all the multiple fibres of $ \pi $ and $ d $ is a divisor on $ B $ of degree $ - \chi ( {\mathcal O} _ {X} ) $. The topological Euler characteristic satisfies the formula

$$ e ( X) = \sum e ( X _ {t _ {i} } ) . $$

The classification of elliptic fibrations.

A fibration $ \pi : X \rightarrow B $ can be regarded as an elliptic curve over the function field $ k ( B) $. This curve, generally speaking, does not have the structure of an Abelian variety over $ k ( B) $. For this to happen it is necessary that it has a rational point over $ k ( B) $ (and then $ X $ is birationally isomorphic to the surface defined in $ B \times A ^ {2} $ by the Weierstrass equation $ y ^ {2} = x ^ {3} - g _ {2} x - g _ {3} $, where $ g _ {2} , g _ {3} \in k ( B) $). The specification of a rational point is equivalent to that of a section $ e : B \rightarrow X $ such that $ \pi e = \mathop{\rm id} $; a necessary condition for the existence of a section is the absence of multiple fibres. Fibrations without multiple fibres are called reduced. Every fibration, after a suitable ramified covering of the base, has a section (that is, is reduced) . Every fibration can also be made reduced by a sequence of transformations inverse to logarithmic ones

— local surgery of the fibration in neighbourhoods of fibres.

Reduced elliptic fibrations may be described as follows. To every such fibration $ \pi : X \rightarrow B $ corresponds a unique fibration $ {\mathcal J} _ {B} ( X) \rightarrow B $ that is a group object and is such that $ X / B $ is a principal homogeneous space over $ {\mathcal J} _ {B} ( X) / B $; $ {\mathcal J} _ {B} ( X) / B $ is the Jacobi fibration for $ X / B $; it characterizes the existence of a section. For a given Jacobi fibration $ {\mathcal J} / B $, the set $ I ( {\mathcal J} / B ) $ of isomorphism classes of fibrations $ X / B $ for which $ {\mathcal J} _ {B} ( X) \cong J $ has a cohomology description analogous to that of invertible sheaves (cf. Invertible sheaf). Here the role of $ {\mathcal O} _ {B} ^ {*} $ is played by the sheaf $ {\mathcal H} ^ {0} ( {\mathcal J} / B ) $ of local sections $ \tau : {\mathcal J} \rightarrow B $. There is a natural one-to-one correspondence

$$ \theta : I ( {\mathcal J} / B ) \rightarrow H ^ {1} ( B , {\mathcal H} ^ {0} ( {\mathcal J} / B ) ) , $$

under which the Jacobi fibration corresponds to the zero element. By means of $ \theta $ one can distinguish between algebraic and non-algebraic fibrations: For a reduced fibration $ \pi : X \rightarrow B $ the surface $ X $ is algebraic if and only if the element corresponding to it in $ H ^ {1} ( B , {\mathcal H} ^ {0} ( {\mathcal J} / B )) $ is of finite order. The analogy with invertible sheaves can be pursued further. The analogue of the exact sequence

$$ 0 \rightarrow \mathbf Z \rightarrow {\mathcal O} _ {B} \mathop \rightarrow \limits ^ {\mathop {\rm exp}} {\mathcal O} _ {B} ^ {*} \rightarrow 1 $$

is the exact sequence

$$ 0 \rightarrow R ^ {1} \tau _ {0} \mathbf Z \rightarrow {\mathcal H} ^ {0} ( T ( {\mathcal J} ) / B ) \rightarrow {\mathcal H} ^ {0} ( {\mathcal J} / B ) \rightarrow 0 , $$

where $ {\mathcal H} ^ {0} ( T ( {\mathcal J} ) / B ) $ is the sheaf of local sections of the bundle $ T ( {\mathcal J} )/ B $ and $ T ( {\mathcal J} ) $ is the tangent space to the fibre $ \tau ^ {- 1 }( b) $ at $ e ( b) $. The boundary homomorphism

$$ \delta : H ^ {1} ( B , {\mathcal H} ^ {0} ( {\mathcal J} / B ) ) \rightarrow H ^ {2} ( B , R ^ {1} \tau _ {*} \mathbf Z ) $$

allows one to recognize when one fibration is a deformation of another. For that it is necessary and sufficient that the elements corresponding to these fibrations have one and the same image under $ \delta $ (see ).

The classification of algebraic elliptic surfaces.

Suppose that $ \mathop{\rm char} k = 0 $. For an elliptic surface $ X $ the canonical dimension $ k ( X) \leq 1 $, that is, it is equal to $ - 1 $, 0 or $ 1 $. If $ k ( X) = 1 $, $ X $ is said to be an elliptic surface of general type. These are characterized by the conditions $ 12 K _ {X} \neq 0 $ and $ | 12 K _ {X} | \neq \emptyset $. Elliptic surfaces with $ p _ {g} \geq 2 $ or, more generally, with $ P _ {m} \geq 2 $ for some $ m $, are of general type.

Elliptic surfaces with $ k ( X) = 0 $ are characterized by the condition $ 12 K _ {X} = 0 $. In this case $ \chi ( {\mathcal O} _ {X} ) $ can take the three values 2, 1 or $ 0 $. If $ \chi ( {\mathcal O} _ {X} )= 2 $, then $ X $ is an elliptic $ K 3 $-surface ( $ q = 0 $, $ K _ {X} = 0 $). In this case $ B $ is isomorphic to the projective line $ P ^ {1} $, the fibration has no multiple fibres and $ X $ has the invariants $ p _ {g} = 1 $, $ q = 0 $, $ b _ {2} = 22 $. If $ \chi ( {\mathcal O} _ {X} ) = 1 $, then $ X $ is an Enriques surface, that is, a surface with $ p _ {g} = q = 0 $, $ 2 K _ {X} = 0 $. (Every Enriques surface is elliptic.) In this case $ B \simeq P ^ {1} $, the fibration has two fibres of multiplicity 2, and $ X $ has the invariants $ p _ {g} = q = 0 $, $ b _ {2} = 10 $. If $ \chi ( {\mathcal O} _ {X} ) = 0 $, then two cases are possible. Either $ X $ is an Abelian variety (and then $ p _ {g} = 1 $, $ q = 2 $, $ b _ {2} = 6 $); or $ X $ is a hyper-elliptic surface, that is, a surface that has a finite unramified covering — the product of two elliptic curves. In that case $ p _ {g} = 0 $, $ b _ {1} = 2 $, $ b _ {2} = 2 $, $ B = P ^ {1} $, and $ \pi $ has 3 or 4 multiple fibres with four possibilities for their multiplicity: $ ( 3 , 3 , 3 ) $, $ ( 2 , 4 , 4 ) $, $ ( 2 , 3 , 6 ) $, and $ ( 2 , 2 , 2 , 2 ) $, and $ 3 K _ {X} = 0 $, $ 4 K _ {X} = 0 $, $ 6 K _ {X} = 0 $, and $ 2 K _ {X} = 0 $, respectively.

An elliptic surface with $ k ( X) = - 1 $ is ruled (cf. Ruled surface). It is characterized by the condition $ | 12 K _ {X} | = \emptyset $. Here two cases are possible: 1) $ X $ is a surface with $ p _ {g} = q = 0 $, $ b _ {2} = 10 $, and $ \pi $ has no multiple fibres or one; moreover, a surface without multiple fibres can be obtained as follows: one has to take a rational mapping $ P ^ {2} \rightarrow P ^ {1} $ determined by two cubics $ F _ {0} $ and $ F _ {1} $ and blow up their 9 points of intersection; or 2) $ X $ is a surface with $ p _ {g} = 0 $, $ q = 1 $, $ b _ {2} = 2 $, and the multiplicities $ m _ {i} $ are subject to the inequality

$$ \sum \left ( 1 - \frac{1}{m _ {i} } \right ) < 2 . $$

The formula for the canonical class and the classification of elliptic surfaces can also be generalized to the case of a field of finite characteristic [5], [6].

The classification of non-algebraic elliptic surfaces.

The classification of non-algebraic elliptic surfaces. For non-algebraic surfaces the algebraic dimension $ a ( X) = \mathop{\rm tr} \mathop{\rm deg} M ( X) $ is 1 or 0. If $ a ( X) = 0 $, then $ X $ is non-elliptic. All surfaces with $ a ( X) = 1 $ are elliptic. Here the structure of $ \pi : X \rightarrow B $ is determined almost canonically: Every such fibration $ \pi $ is necessarily elliptic. The classification by the canonical dimension is precisely the same as for algebraic elliptic surfaces: $ k ( X) = - 1 $ $ \iff $ $ | 12 K _ {X} | = \emptyset $; $ k ( X) = 0 $ $ \iff $ $ 12K _ {X} = 0 $; and $ k ( X) = 1 $ ($ X $ is of basic type) $ \iff $ $ | 12 K _ {X} | = \emptyset $, $ 12 K _ {X} \neq 0 $.

Non-algebraic elliptic surfaces with $ k ( X) = 0 $ belong to one of the following classes: 1) the $ K 3 $-surfaces ( $ \chi ( {\mathcal O} _ {X} )= 2 $, $ b _ {1} = 0 $, $ b _ {2} = 22 $, $ X $ is simply connected); 2) the complex tori ( $ K _ {X} = 0 $, $ \chi ( {\mathcal O} _ {X} ) = 0 $, $ b _ {1} = 4 $, $ b _ {2} = 6 $); 3) the Kodaira surfaces ( $ K _ {X} = 0 $, $ \chi ( {\mathcal O} _ {X} ) = 0 $, $ b _ {1} = 3 $, $ b _ {2} = 4 $). Primary Kodaira surfaces are holomorphically locally trivial fibrations over an elliptic curve with an elliptic curve as typical fibre, and from the point of view of differentiability, bundles over a $ 3 $-dimensional torus with a circle as fibre; or 4) the surfaces with $ \chi ( {\mathcal O} _ {X} ) = 0 $, $ p _ {g} = 0 $, $ b _ {1} = 1 $, $ b _ {2} = 0 $. For them $ m K _ {X} = 0 $ with $ m = 2 , 3 , 4 $, (analogous to hyper-elliptic surfaces). They have Kodaira surfaces as finite unramified coverings. In the cases 2), 3) and 4) $ \mathbf C ^ {2} $ is the universal covering of $ X $.

Non-algebraic elliptic surfaces with $ k ( X) = - 1 $ are Hopf surfaces, that is, their universal covering is $ \mathbf C ^ {2} \setminus 0 $. For them $ \chi ( {\mathcal O} _ {X} ) = 0 $, $ b _ {1} = 1 $, $ b _ {2} = 0 $. The proper Hopf surfaces are $ ( \mathbf C ^ {2} \setminus 0 ) / T $, where $ T ( z _ {1} , z _ {2} ) = ( \alpha _ {1} z _ {1} , \alpha _ {2} z _ {2} ) $ are real generators of $ T $. They are homeomorphic to $ S ^ {1} \times S ^ {2} $ and are characterized by this property. Arbitrary elliptic Hopf surfaces are quotients of proper Hopf surfaces .

References

[1] "Algebraic surfaces" Proc. Steklov Inst. Math. , 75 (1967) Trudy Mat. Inst. Steklov. , 75 (1965) Zbl 0172.37901 Zbl 0153.22401
[2] D. Husemoller, "Classification and embeddings of surfaces" R. Hartshorne (ed.) , Algebraic geometry (Arcata, 1974) , Proc. Symp. Pure Math. , 29 , Amer. Math. Soc. (1982) pp. 329–420 MR0506292 Zbl 0326.14009
[3a] K. Kodaira, "On compact complex analytic surfaces I" Ann. of Math. (2) , 71 (10) pp. 111–152 MR0132556 Zbl 0137.17501 Zbl 0098.13004
[3b] K. Kodaira, "On compact complex analytic surfaces II" Ann. of Math. (2) , 77 (1963) pp. 563–626 MR165541 Zbl 0193.37701 Zbl 0133.16505
[3c] K. Kodaira, "On compact complex analytic surfaces III" Ann. of Math. (2) , 78 (1963) pp. 1–40 MR165541 Zbl 0193.37701
[4a] K. Kodaira, "On the structure of compact complex analytic surfaces I" Amer. J. Math. , 86 (1964) pp. 751–798 MR0187255 Zbl 0137.17501
[4b] K. Kodaira, "On the structure of compact complex analytic surfaces II" Amer. J. Math. , 88 (1966) pp. 682–721 MR0205280 Zbl 0193.37701
[4c] K. Kodaira, "On the structure of compact complex analytic surfaces III" Amer. J. Math. , 90 (1968) pp. 55–83 MR0228019 Zbl 0193.37701
[4d] K. Kodaira, "On the structure of compact complex analytic surfaces IV" Amer. J. Math. , 90 (1968) pp. 1048–1066 MR239114
[5] D. Mumford, "Enriques' classification of surfaces in char . I" D.C. Spencer (ed.) S. Iyanaga (ed.) , Global analysis (papers in honor of K. Kodaira) , Princeton Univ. Press (1969) pp. 325–339 MR0491719 MR0491720
[6] E. Bombieri, D. Mumford, "Enriques' classification of surfaces in char . II" W.L. Baily jr. (ed.) T. Shioda (ed.) , Complex Analysis and Algebraic geometry , Cambridge Univ. Press & Iwanami Shoten (1977) pp. 23–42 MR0491719 MR0491720

Comments

A secondary Kodaira surface is a surface other than a primary one, admitting a primary Kodaira surface as an unramified covering. They are elliptic fibrations over a rational curve. Their first Betti number is 1.

The canonical dimension $ k ( X) $ mentioned at the start of the section on classification of algebraic elliptic surfaces is the Kodaira dimension $ \mathop{\rm Kod} ( X) $ (with $ k ( X) = - 1 $ if $ \mathop{\rm Kod} ( X) = - \infty $).

References

[a1] W. Barth, C. Peters, A. van der Ven, "Compact complex surfaces" , Springer (1984) MR0749574 Zbl 0718.14023
How to Cite This Entry:
Elliptic surface. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Elliptic_surface&oldid=17304
This article was adapted from an original article by Val.S. Kulikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article