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An [[Abelian variety|Abelian variety]] of dimension two, i.e. a complete connected group variety of dimension two over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a1100401.png" />. The group law is commutative. In the sequel, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a1100402.png" /> is assumed to be algebraically closed (cf. [[Algebraically closed field|Algebraically closed field]]).
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In the classification of algebraic surfaces, Abelian surfaces are exactly the smooth complete surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a1100403.png" /> with [[Kodaira dimension|Kodaira dimension]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a1100404.png" />, [[Geometric genus|geometric genus]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a1100405.png" /> and [[Irregularity|irregularity]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a1100406.png" />.
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For an Abelian surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a1100407.png" />, the dual Abelian variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a1100408.png" /> is again an Abelian surface. An invertible [[Sheaf|sheaf]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a1100409.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004010.png" /> defines the homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004012.png" />. The homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004013.png" /> depends only on the algebraic equivalence class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004014.png" />. The invertible sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004015.png" /> is ample (cf. [[Ample sheaf|Ample sheaf]]) if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004016.png" /> is an [[Isogeny|isogeny]] (i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004017.png" /> is surjective and has finite kernel) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004018.png" />. In this case, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004019.png" /> with a positive integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004020.png" /> and the [[Riemann–Roch theorem|Riemann–Roch theorem]] says that
+
An [[Abelian variety|Abelian variety]] of dimension two, i.e. a complete connected group variety of dimension two over a field  $  k $.  
 +
The group law is commutative. In the sequel,  $  k $
 +
is assumed to be algebraically closed (cf. [[Algebraically closed field|Algebraically closed field]]).
  
<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/a/a110/a110040/a11004021.png" /></td> </tr></table>
+
In the classification of algebraic surfaces, Abelian surfaces are exactly the smooth complete surfaces  $  A $
 +
with [[Kodaira dimension|Kodaira dimension]]  $  \kappa = 0 $,
 +
[[Geometric genus|geometric genus]]  $  p _ {g} = h  ^ {2} ( A, {\mathcal O} _ {A} ) =1 $
 +
and [[Irregularity|irregularity]]  $  q = h  ^ {1} ( A, {\mathcal O} _ {A} ) = 2 $.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004022.png" /> denotes the self-intersection number. Every Abelian surface admits an ample invertible sheaf and hence is projective (cf. [[Projective scheme|Projective scheme]]).
+
For an Abelian surface  $  A $,
 +
the dual Abelian variety  $  {\widehat{A}  } = { \mathop{\rm Pic} }  ^ {0} ( A ) $
 +
is again an Abelian surface. An invertible [[Sheaf|sheaf]]  $  L $
 +
on  $  A $
 +
defines the homomorphism  $  {\phi _ {L} } : A \rightarrow { {\widehat{A}  } } $,
 +
$  a \mapsto t _ {a}  ^ {*} L \otimes L ^ {- 1 } $.  
 +
The homomorphism  $  \phi _ {L} $
 +
depends only on the algebraic equivalence class of  $  L $.  
 +
The invertible sheaf $  L $
 +
is ample (cf. [[Ample sheaf|Ample sheaf]]) if and only if  $  \phi _ {L} $
 +
is an [[Isogeny|isogeny]] (i.e.,  $  \phi _ {L} $
 +
is surjective and has finite kernel) and  $  h  ^ {0} ( A,L ) \neq0 $.
 +
In this case,  $  { \mathop{\rm deg} } \phi _ {L} = d  ^ {2} $
 +
with a positive integer  $  d $
 +
and the [[Riemann–Roch theorem|Riemann–Roch theorem]] says that
  
A polarization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004023.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004024.png" /> is the algebraic equivalence class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004025.png" /> of an ample invertible sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004026.png" />. The degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004027.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004028.png" /> is by definition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004029.png" />. An Abelian surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004030.png" /> together with a polarization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004031.png" /> is a polarized Abelian surface. A principal polarization is a polarization of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004032.png" />. A principally polarized Abelian surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004033.png" /> is either the [[Jacobi variety|Jacobi variety]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004034.png" /> of a smooth projective curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004035.png" /> of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004036.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004037.png" /> is the class of the theta divisor, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004038.png" /> is the product of two elliptic curves (Abelian varieties of dimension one, cf. also [[Elliptic curve|Elliptic curve]]) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004039.png" /> the product polarization.
+
$$
 +
h  ^ {0} ( A,L ) = {
 +
\frac{1}{2}
 +
} ( L  ^ {2} ) = d,
 +
$$
  
If the degree of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004040.png" /> is prime to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004041.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004042.png" /> is said to be a separable polarization and the kernel of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004043.png" /> is isomorphic to the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004044.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004045.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004046.png" /> are positive integers such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004047.png" /> divides <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004048.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004049.png" />. The pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004050.png" /> is the type of the polarized Abelian surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004051.png" />.
+
where  $  ( L  ^ {2} ) $
 +
denotes the self-intersection number. Every Abelian surface admits an ample invertible sheaf and hence is projective (cf. [[Projective scheme|Projective scheme]]).
  
A polarization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004052.png" /> of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004053.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004054.png" /> defines a polarization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004055.png" /> on the dual Abelian surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004056.png" />. The polarization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004057.png" /> is again of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004058.png" /> and it is characterized by each of the following two equivalent properties:
+
A polarization $  \lambda $
 +
on  $  A $
 +
is the algebraic equivalence class  $  [ L ] $
 +
of an ample invertible sheaf  $  L $.  
 +
The degree  $  { \mathop{\rm deg} } \lambda $
 +
of $  \lambda $
 +
is by definition  $  d = \sqrt { { \mathop{\rm deg} } \phi _ {L} } $.  
 +
An Abelian surface  $  A $
 +
together with a polarization $  \lambda $
 +
is a polarized Abelian surface. A principal polarization is a polarization of degree  $  1 $.  
 +
A principally polarized Abelian surface  $  ( A, \lambda ) $
 +
is either the [[Jacobi variety|Jacobi variety]]  $  J ( H ) $
 +
of a smooth projective curve  $  H $
 +
of genus  $  2 $,
 +
and $  \lambda = \theta $
 +
is the class of the theta divisor, or  $  A $
 +
is the product of two elliptic curves (Abelian varieties of dimension one, cf. also [[Elliptic curve|Elliptic curve]]) with  $  \lambda $
 +
the product polarization.
  
<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/a/a110/a110040/a11004059.png" /></td> </tr></table>
+
If the degree of  $  \lambda = [ L ] $
 +
is prime to  $  { \mathop{\rm char} } ( k ) $,
 +
then  $  \lambda $
 +
is said to be a separable polarization and the kernel of  $  \phi _ {L} $
 +
is isomorphic to the group  $  ( \mathbf Z/d _ {1} \mathbf Z )  ^ {2} \times ( \mathbf Z/d _ {2} \mathbf Z )  ^ {2} $,
 +
where  $  d _ {1} $
 +
and  $  d _ {2} $
 +
are positive integers such that  $  d _ {1} $
 +
divides  $  d _ {2} $
 +
and  $  d _ {1} d _ {2} = { \mathop{\rm deg} } \lambda $.  
 +
The pair  $  ( d _ {1} ,d _ {2} ) $
 +
is the type of the polarized Abelian surface  $  ( A, \lambda ) $.
  
For a polarized Abelian surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004060.png" /> of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004061.png" />, the assignment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004062.png" /> defines a rational mapping from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004063.png" /> into the projective space of hyperplanes in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004064.png" />:
+
A polarization  $  \lambda = [ L ] $
 +
of type $  ( d _ {1} ,d _ {2} ) $
 +
on  $  A $
 +
defines a polarization  $  {\widehat \lambda  } = [ {\widehat{L}  } ] $
 +
on the dual Abelian surface  $  {\widehat{A}  } $.  
 +
The polarization  $  {\widehat \lambda  } $
 +
is again of type  $  ( d _ {1} ,d _ {2} ) $
 +
and it is characterized by each of the following two equivalent properties:
  
<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/a/a110/a110040/a11004065.png" /></td> </tr></table>
+
$$
 +
\phi _ {L}  ^ {*} {\widehat \lambda  } = d _ {1} d _ {2} \lambda  \iff  \phi _ { {\widehat{L}  }  } \phi _ {L} = d _ {1} d _ {2} { \mathop{\rm id} } _ {A} .
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004066.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004067.png" /> is everywhere defined. The Lefschetz theorem says that for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004068.png" /> the morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004069.png" /> is an embedding. Suppose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004070.png" />; then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004071.png" /> with a polarization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004072.png" /> of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004073.png" />. If the linear system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004074.png" /> has no fixed components, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004075.png" /> is an embedding.
+
For a polarized Abelian surface  $  ( A, \lambda = [ L ] ) $
 +
of type  $  ( d _ {1} ,d _ {2} ) $,
 +
the assignment  $  A \ni a \mapsto \{ {\sigma \in H  ^ {0} ( A,L ) } : {\sigma ( a ) = 0 } \} \subset  H  ^ {0} ( A,L ) $
 +
defines a rational mapping from  $  A $
 +
into the projective space of hyperplanes in  $  H  ^ {0} ( A,L ) $:
 +
 
 +
$$
 +
{\varphi _ {L} } : A \rightarrow {\mathbf P ( H  ^ {0} ( A,L )  ^ {*} ) } \simeq \mathbf P _ {k} ^ {d _ {1} d _ {2} - 1 } .
 +
$$
 +
 
 +
If  $  d _ {1} \geq  2 $,
 +
then  $  \varphi _ {L} $
 +
is everywhere defined. The Lefschetz theorem says that for $  d _ {1} \geq  3 $
 +
the morphism $  \varphi _ {L} $
 +
is an embedding. Suppose $  d _ {1} = 2 $;  
 +
then $  \lambda = 2 \mu $
 +
with a polarization $  \mu = [ M ] $
 +
of type $  ( 1, { {d _ {2} } / 2 } ) $.  
 +
If the linear system $  | M | $
 +
has no fixed components, then $  \varphi _ {L} $
 +
is an embedding.
  
 
==Complex Abelian surfaces.==
 
==Complex Abelian surfaces.==
An Abelian surface over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004076.png" /> of complex numbers is a [[Complex torus|complex torus]]
+
An Abelian surface over the field $  \mathbf C $
 +
of complex numbers is a [[Complex torus|complex torus]]
  
<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/a/a110/a110040/a11004077.png" /></td> </tr></table>
+
$$
 +
A = { {\mathbf C  ^ {2} } / \Lambda }
 +
$$
  
(with a lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004078.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004079.png" />) admitting a polarization. A polarization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004080.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004081.png" /> can be considered as a non-degenerate alternating form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004082.png" />, the elementary divisors of which are given by the type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004083.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004084.png" />.
+
(with a lattice $  \Lambda \simeq \mathbf Z  ^ {4} $
 +
in $  \mathbf C  ^ {2} $)  
 +
admitting a polarization. A polarization $  \lambda $
 +
on $  A $
 +
can be considered as a non-degenerate alternating form $  \Lambda \times \Lambda \rightarrow \mathbf Z $,  
 +
the elementary divisors of which are given by the type $  ( d _ {1} ,d _ {2} ) $
 +
of $  \lambda $.
  
In the sequel, the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004085.png" /> is assumed to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004086.png" />, although some of the following results are also valid for arbitrary algebraically closed fields.
+
In the sequel, the field $  k $
 +
is assumed to be $  \mathbf C $,  
 +
although some of the following results are also valid for arbitrary algebraically closed fields.
  
Suppose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004087.png" /> is of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004088.png" /> and the linear system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004089.png" /> has no fixed components. The Reider theorem states that for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004090.png" /> the invertible sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004091.png" /> is very ample if and only if there is no elliptic curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004092.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004093.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004094.png" /> (see [[#References|[a14]]] and [[#References|[a10]]]). For arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004095.png" /> there exist finitely many isogenies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004096.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004097.png" /> onto principally polarized Abelian surfaces (cf. also [[Isogeny|Isogeny]]). Suppose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004098.png" /> with a symmetric invertible sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a11004099.png" /> (i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040100.png" />) and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040101.png" /> be the unique divisor in the linear system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040102.png" />. The divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040103.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040104.png" /> defines a symmetric invertible sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040105.png" /> with class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040106.png" /> and the covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040107.png" /> is étale of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040108.png" />. One distinguishes two cases: i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040109.png" /> is smooth of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040110.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040111.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040112.png" /> is smooth of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040113.png" />; and ii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040114.png" /> is the sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040115.png" /> of two elliptic curves with intersection number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040116.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040117.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040118.png" /> is the sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040119.png" /> of two elliptic curves with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040120.png" />.
+
Suppose $  ( A, \lambda = [ L ] ) $
 +
is of type $  ( 1,d ) $
 +
and the linear system $  | L | $
 +
has no fixed components. The Reider theorem states that for $  d \geq  5 $
 +
the invertible sheaf $  L $
 +
is very ample if and only if there is no elliptic curve $  E $
 +
on $  A $
 +
with $  ( E \cdot L ) = 2 $(
 +
see [[#References|[a14]]] and [[#References|[a10]]]). For arbitrary $  d \geq  1 $
 +
there exist finitely many isogenies $  f : {( A, \lambda ) } \rightarrow {( B, \theta ) } $
 +
of degree $  d $
 +
onto principally polarized Abelian surfaces (cf. also [[Isogeny|Isogeny]]). Suppose $  \theta = [ \Theta ] $
 +
with a symmetric invertible sheaf $  \Theta $(
 +
i.e., $  ( -1 ) _ {A}  ^ {*} \Theta \simeq \Theta $)  
 +
and let $  H $
 +
be the unique divisor in the linear system $  | \Theta | $.  
 +
The divisor $  C = f ^ {- 1 } ( H ) $
 +
on $  A $
 +
defines a symmetric invertible sheaf $  L = {\mathcal O} _ {A} ( C ) $
 +
with class $  [ L ] = \lambda $
 +
and the covering $  {f \mid  _ {C} } : C \rightarrow H $
 +
is étale of degree $  d $.  
 +
One distinguishes two cases: i) $  H $
 +
is smooth of genus $  2 $,  
 +
$  B = J ( H ) $
 +
and $  C $
 +
is smooth of genus $  d + 1 $;  
 +
and ii) $  H $
 +
is the sum $  E _ {1} + E _ {2} $
 +
of two elliptic curves with intersection number $  ( E _ {1} \cdot E _ {2} ) = 1 $,  
 +
$  B = E _ {1} \times E _ {2} $
 +
and $  C $
 +
is the sum $  F _ {1} + F _ {2} $
 +
of two elliptic curves with $  ( F _ {1} \cdot F _ {2} ) = d $.
  
In the following list, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040121.png" /> is a polarized Abelian surface of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040122.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040123.png" /> admits no fixed components
+
In the following list, $  ( A, \lambda = [ L ] ) $
 +
is a polarized Abelian surface of type $  ( d _ {1} ,d _ {2} ) $
 +
such that $  | L | $
 +
admits no fixed components
  
Type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040124.png" />— The linear system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040125.png" /> has exactly <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040126.png" /> base points. The blow-up <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040127.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040128.png" /> in these points admits a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040129.png" /> induced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040130.png" />. The general fibre of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040131.png" /> is a smooth curve of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040132.png" />. The curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040133.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040134.png" /> defining <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040135.png" /> as above is double elliptic: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040136.png" /> with an elliptic curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040137.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040138.png" /> is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040139.png" /> (see [[#References|[a3]]]).
+
Type $  ( 1,2 ) $—  
 +
The linear system $  | L | $
 +
has exactly $  4 $
 +
base points. The blow-up $  {\widetilde{A}  } $
 +
of $  A $
 +
in these points admits a morphism $  { {\widetilde \varphi  } _ {L} } : { {\widetilde{A}  } } \rightarrow {\mathbf P  ^ {1} } $
 +
induced by $  \varphi _ {L} $.  
 +
The general fibre of $  {\widetilde \varphi  } _ {L} $
 +
is a smooth curve of genus $  3 $.  
 +
The curve $  C $
 +
on $  A $
 +
defining $  L $
 +
as above is double elliptic: $  C { \mathop \rightarrow \limits ^ { {2:1 }}  } E $
 +
with an elliptic curve $  E $,  
 +
and $  A $
 +
is isomorphic to $  { {J ( C ) } / E } $(
 +
see [[#References|[a3]]]).
  
Type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040140.png" /><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040141.png" /> defines a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040142.png" />-fold covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040143.png" /> ramified along a curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040144.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040145.png" />. The general divisor in the linear system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040146.png" /> is a smooth curve of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040147.png" />. There are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040148.png" /> isogenies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040149.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040150.png" /> onto principally polarized Abelian surfaces. In case i) the smooth genus-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040151.png" /> curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040152.png" /> is double elliptic: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040153.png" />, and the embedding of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040154.png" /> into the Jacobian <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040155.png" /> induces an exact sequence
+
Type $  ( 1,3 ) $—  
 +
$  L $
 +
defines a $  6 $-
 +
fold covering $  {\varphi _ {L} } : A \rightarrow {\mathbf P  ^ {2} } $
 +
ramified along a curve $  R \subset  \mathbf P  ^ {2} $
 +
of degree $  18 $.  
 +
The general divisor in the linear system $  | L | $
 +
is a smooth curve of genus $  4 $.  
 +
There are $  4 $
 +
isogenies $  f : {( A, \lambda ) } \rightarrow {( B, \theta ) } $
 +
of degree $  3 $
 +
onto principally polarized Abelian surfaces. In case i) the smooth genus- $  4 $
 +
curve $  C \in | L | $
 +
is double elliptic: $  C { \mathop \rightarrow \limits ^ { {2:1 }}  } E $,  
 +
and the embedding of $  E $
 +
into the Jacobian $  J ( C ) $
 +
induces an 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/a/a110/a110040/a110040156.png" /></td> </tr></table>
+
$$
 +
0 \rightarrow E \times E \rightarrow J ( C ) \rightarrow A \rightarrow 0.
 +
$$
  
The étale <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040157.png" />-fold covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040158.png" /> induces a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040159.png" /> with image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040160.png" />, the dual Abelian surface of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040161.png" /> (see [[#References|[a7]]]).
+
The étale $  3 $-
 +
fold covering $  {f \mid  _ {C} } : C \rightarrow H $
 +
induces a morphism $  {f  ^ {*} } : {J ( H ) } \rightarrow {J ( C ) } $
 +
with image $  {\widehat{A}  } $,  
 +
the dual Abelian surface of $  A $(
 +
see [[#References|[a7]]]).
  
Type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040162.png" />— There are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040163.png" /> isogenies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040164.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040165.png" /> onto principally polarized Abelian surfaces. If the curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040166.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040167.png" /> do not admit elliptic involutions compatible with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040168.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040169.png" /> is a birational morphism onto a singular octic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040170.png" />. In the exceptional case, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040171.png" /> is a double covering of a singular quartic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040172.png" />, which is birational to an elliptic scroll. In the first case the octic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040173.png" /> is smooth outside the four coordinate planes of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040174.png" /> and touches the coordinate planes in curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040175.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040176.png" />, of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040177.png" />. Each of the curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040178.png" /> has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040179.png" /> double points and passes through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040180.png" /> pinch points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040181.png" />. The octic is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040182.png" /> covering of a [[Kummer surface|Kummer surface]]: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040183.png" /> (see also Type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040184.png" /> below). The restrictions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040185.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040186.png" />-fold coverings of four double conics of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040187.png" /> lying on a coordinate tetrahedron. The three double points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040188.png" /> map to three double points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040189.png" /> on the conic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040190.png" /> and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040191.png" /> pinch points on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040192.png" /> map to the other three double points on the double conic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040193.png" /> (see [[#References|[a6]]]).
+
Type $  ( 1,4 ) $—  
 +
There are $  24 $
 +
isogenies $  f : {( A, \lambda ) } \rightarrow {( B, \theta ) } $
 +
of degree $  4 $
 +
onto principally polarized Abelian surfaces. If the curves $  C $
 +
and $  H $
 +
do not admit elliptic involutions compatible with $  f $,  
 +
then $  \varphi _ {L} :A \twoheadrightarrow {\overline{A}\; } \subset  \mathbf P  ^ {3} $
 +
is a birational morphism onto a singular octic $  {\overline{A}\; } $.  
 +
In the exceptional case, $  \varphi _ {L} : A \twoheadrightarrow {\overline{A}\; } \subset  \mathbf P  ^ {3} $
 +
is a double covering of a singular quartic $  {\overline{A}\; } $,  
 +
which is birational to an elliptic scroll. In the first case the octic $  {\overline{A}\; } $
 +
is smooth outside the four coordinate planes of $  \mathbf P  ^ {3} $
 +
and touches the coordinate planes in curves $  D _ {i} $,  
 +
$  i = 1 \dots 4 $,  
 +
of degree $  4 $.  
 +
Each of the curves $  D _ {i} $
 +
has $  3 $
 +
double points and passes through $  12 $
 +
pinch points of $  {\overline{A}\; } $.  
 +
The octic is a $  8:1 $
 +
covering of a [[Kummer surface|Kummer surface]]: $  p: {\overline{A}\; } \twoheadrightarrow K \subset  \mathbf P  ^ {3} $(
 +
see also Type $  ( 2,2 ) $
 +
below). The restrictions $  p \mid  _ {D _ {i}  } $
 +
are $  4 $-
 +
fold coverings of four double conics of $  K $
 +
lying on a coordinate tetrahedron. The three double points of $  D _ {i} $
 +
map to three double points of $  K $
 +
on the conic $  p ( D _ {i} ) $
 +
and the $  12 $
 +
pinch points on $  D _ {i} $
 +
map to the other three double points on the double conic $  p ( D _ {i} ) $(
 +
see [[#References|[a6]]]).
  
Type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040194.png" />— The invertible sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040195.png" /> is very ample, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040196.png" /> is an embedding if and only if the curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040197.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040198.png" /> do not admit elliptic involutions compatible with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040199.png" />. In the exceptional case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040200.png" /> is a double covering of an elliptic scroll (see [[#References|[a13]]] and [[#References|[a9]]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040201.png" /> is very ample, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040202.png" /> is a smooth surface of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040203.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040204.png" />. It is the zero locus of a section of the Horrocks–Mumford bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040205.png" /> (see [[#References|[a8]]]). Conversely, the zero set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040206.png" /> of a general section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040207.png" /> is an Abelian surface of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040208.png" />, i.e. of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040209.png" />.
+
Type $  ( 1,5 ) $—  
 +
The invertible sheaf $  L $
 +
is very ample, i.e. $  {\varphi _ {L} } : A \rightarrow {\mathbf P  ^ {4} } $
 +
is an embedding if and only if the curves $  C $
 +
and $  H $
 +
do not admit elliptic involutions compatible with $  f $.  
 +
In the exceptional case $  \varphi _ {L} $
 +
is a double covering of an elliptic scroll (see [[#References|[a13]]] and [[#References|[a9]]]). If $  L $
 +
is very ample, $  \varphi _ {L} ( A ) $
 +
is a smooth surface of degree $  10 $
 +
in $  \mathbf P  ^ {4} $.  
 +
It is the zero locus of a section of the Horrocks–Mumford bundle $  F $(
 +
see [[#References|[a8]]]). Conversely, the zero set $  \{ \sigma = 0 \} \subset  \mathbf P  ^ {4} $
 +
of a general section $  \sigma \in H  ^ {0} ( \mathbf P  ^ {4} ,F ) $
 +
is an Abelian surface of degree $  10 $,  
 +
i.e. of type $  ( 1,5 ) $.
  
Type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040210.png" /><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040211.png" /> is twice a principal polarization on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040212.png" />. The morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040213.png" /> is a double covering of the [[Kummer surface|Kummer surface]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040214.png" /> associated with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040215.png" />. It is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040216.png" />.
+
Type $  ( 2,2 ) $—  
 +
$  \lambda $
 +
is twice a principal polarization on $  A $.  
 +
The morphism $  \varphi _ {L} : A \twoheadrightarrow K _ {A} \subset  \mathbf P  ^ {3} $
 +
is a double covering of the [[Kummer surface|Kummer surface]] $  K _ {A} $
 +
associated with $  A $.  
 +
It is isomorphic to $  {A / {( - 1 ) _ {A} } } $.
  
Type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040217.png" />— The ideal sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040218.png" /> of the image of the embedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040219.png" /> is generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040220.png" /> quadrics (see [[#References|[a3]]]).
+
Type $  ( 2,4 ) $—  
 +
The ideal sheaf $  {\mathcal I} _ { {A / {\mathbf P  ^ {7}  } } } $
 +
of the image of the embedding $  \varphi _ {L} : A \hookrightarrow \mathbf P  ^ {7} $
 +
is generated by $  6 $
 +
quadrics (see [[#References|[a3]]]).
  
Type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040221.png" />— Suppose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040222.png" /> is very ample and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040223.png" /> be the associated Kummer surface. The subvector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040224.png" /> of odd sections induces an embedding of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040225.png" />, the blow-up of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040226.png" /> in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040227.png" /> double points, as a smooth quartic surface into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040228.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040229.png" /> is invariant under the action of the level-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040230.png" /> Heisenberg group (cf. also [[Heisenberg representation|Heisenberg representation]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040231.png" />. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040232.png" /> blown-up double points become skew lines on the quartic surface. Any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040233.png" />-invariant quartic surface in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040234.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040235.png" /> skew lines comes from a polarized Abelian surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040236.png" /> of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040237.png" /> in this way (see [[#References|[a5]]], [[#References|[a11]]] and [[#References|[a12]]]).
+
Type $  ( 2,6 ) $—  
 +
Suppose $  L $
 +
is very ample and let $  K _ {A} = {A / {( - 1 ) _ {A} } } $
 +
be the associated Kummer surface. The subvector space $  H  ^ {0} ( A,L )  ^ {-} \subset  H  ^ {0} ( A,L ) $
 +
of odd sections induces an embedding of $  {\widetilde{K}  } _ {A} $,  
 +
the blow-up of $  K _ {A} $
 +
in the $  16 $
 +
double points, as a smooth quartic surface into $  \mathbf P  ^ {3} $.  
 +
$  {\widetilde{K}  } _ {A} \subset  \mathbf P  ^ {3} $
 +
is invariant under the action of the level- $  2 $
 +
Heisenberg group (cf. also [[Heisenberg representation|Heisenberg representation]]) $  H ( 2,2 ) $.  
 +
The $  16 $
 +
blown-up double points become skew lines on the quartic surface. Any $  H ( 2,2 ) $-
 +
invariant quartic surface in $  \mathbf P  ^ {3} $
 +
with $  16 $
 +
skew lines comes from a polarized Abelian surface $  ( A, \lambda ) $
 +
of type $  ( 2,6 ) $
 +
in this way (see [[#References|[a5]]], [[#References|[a11]]] and [[#References|[a12]]]).
  
Type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040238.png" /><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040239.png" /> is three times a principal polarization and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040240.png" /> is an embedding. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040241.png" /> is not a product, then the quadrics <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040242.png" /> vanishing on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040243.png" /> generate the ideal sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040244.png" />. In the product case, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040245.png" /> is generated by quadrics and cubics (see [[#References|[a4]]]).
+
Type $  ( 3,3 ) $—  
 +
$  \lambda $
 +
is three times a principal polarization and $  \varphi _ {L} : A \hookrightarrow \mathbf P  ^ {8} $
 +
is an embedding. If $  ( A, \lambda ) $
 +
is not a product, then the quadrics $  Q \in H  ^ {0} ( \mathbf P  ^ {8} , {\mathcal I} _ { {A / {\mathbf P  ^ {8}  } } } ( 2 ) ) $
 +
vanishing on $  A $
 +
generate the ideal sheaf $  {\mathcal I} _ { {A / {\mathbf P  ^ {8}  } } } $.  
 +
In the product case, $  {\mathcal I} _ { {A / {\mathbf P  ^ {8}  } } } $
 +
is generated by quadrics and cubics (see [[#References|[a4]]]).
  
 
==Algebraic completely integrable systems.==
 
==Algebraic completely integrable systems.==
An algebraic completely integrable system in the sense of M. Adler and P. van Moerbeke is a completely integrable polynomial [[Hamiltonian system|Hamiltonian system]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040246.png" /> (with Casimir functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040247.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040248.png" /> independent constants of motion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040249.png" /> in involution) such that:
+
An algebraic completely integrable system in the sense of M. Adler and P. van Moerbeke is a completely integrable polynomial [[Hamiltonian system|Hamiltonian system]] on $  \mathbf C  ^ {N} $(
 +
with Casimir functions $  {H _ {1} \dots H _ {k} } : {\mathbf C  ^ {N} } \rightarrow \mathbf C $
 +
and $  m = { {( N - k ) } / 2 } $
 +
independent constants of motion $  H _ {k + 1 }  \dots H _ {k + m }  $
 +
in involution) such that:
  
a) for a general point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040250.png" /> the invariant manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040251.png" /> is an open affine part of an Abelian variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040252.png" />;
+
a) for a general point $  c = {}  ^ {t} ( c _ {1} \dots c _ {k + m }  ) \in \mathbf C ^ {k + m } $
 +
the invariant manifold $  A _ {c}  ^ {o} = \cap _ {i = 1 }  ^ {m + k } \{ H _ {i} = c _ {i} \} \subset  \mathbf C  ^ {N} $
 +
is an open affine part of an Abelian variety $  A _ {c} $;
  
b) the flows of the integrable vector fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040253.png" /> linearize on the Abelian varieties <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040254.png" /> [[#References|[a2]]].
+
b) the flows of the integrable vector fields $  X _ {u _ {i}  } $
 +
linearize on the Abelian varieties $  A _ {c} $[[#References|[a2]]].
  
The divisor at infinity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040255.png" /> defines a polarization on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040256.png" />. In this way the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040257.png" /> defines a family of polarized Abelian varieties (cf. [[Moduli problem|Moduli problem]]). Some examples of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040258.png" />-dimensional algebraic completely integrable systems and their associated Abelian surfaces are:
+
The divisor at infinity $  D _ {c} = A _ {c} - A _ {c}  ^ {o} $
 +
defines a polarization on $  A _ {c} $.  
 +
In this way the mapping $  {( H _ {1} \dots H _ {k + m }  ) } : {\mathbf C  ^ {N} } \rightarrow {\mathbf C ^ {k + m } } $
 +
defines a family of polarized Abelian varieties (cf. [[Moduli problem|Moduli problem]]). Some examples of $  2 $-
 +
dimensional algebraic completely integrable systems and their associated Abelian surfaces are:
  
 
the three-body Toda lattice and the even, respectively odd, master systems (cf. also [[Master equations in cooperative and social phenomena|Master equations in cooperative and social phenomena]]) linearize on principally polarized Abelian surfaces;
 
the three-body Toda lattice and the even, respectively odd, master systems (cf. also [[Master equations in cooperative and social phenomena|Master equations in cooperative and social phenomena]]) linearize on principally polarized Abelian surfaces;
  
the [[Kowalewski top|Kowalewski top]], the Hénon–Heiles system and the Manakov geodesic flow on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040259.png" /> linearize on Abelian surfaces of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040260.png" /> [[#References|[a1]]];
+
the [[Kowalewski top|Kowalewski top]], the Hénon–Heiles system and the Manakov geodesic flow on $  { \mathop{\rm SO} } ( 4 ) $
 +
linearize on Abelian surfaces of type $  ( 1,2 ) $[[#References|[a1]]];
  
the Garnier system linearizes on Abelian surfaces of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040261.png" /> [[#References|[a15]]].
+
the Garnier system linearizes on Abelian surfaces of type $  ( 1,4 ) $[[#References|[a15]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> M. Adler,   P. van Moerbeke,   "The Kowalewski and Hénon–Heiles motions as Manakov geodesic flows on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040262.png" />: a two-dimensional family of Lax pairs" ''Comm. Math. Phys.'' , '''113''' (1988) pp. 659–700</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> M. Adler,   P. van Moerbeke,   "The complex geometry of the Kowalewski–Painlevé analysis" ''Invent. Math.'' , '''97''' (1989) pp. 3–51</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> W. Barth,   "Abelian surfaces with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040263.png" />-polarization" , ''Algebraic Geometry, Sendai, 1985'' , ''Advanced Studies in Pure Math.'' , '''10''' (1987) pp. 41–84</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> W. Barth,   "Quadratic equations for level-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040264.png" /> abelian surfaces" , ''Abelian Varieties, Proc. Workshop Egloffstein 1993'' , de Gruyter (1995) pp. 1–18</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> W. Barth,   I. Nieto,   "Abelian surfaces of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040265.png" /> and quartic surfaces with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040266.png" /> skew lines" ''J. Algebraic Geom.'' , '''3''' (1994) pp. 173–222</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> Ch. Birkenhake,   H. Lange,   D. van Straten,   "Abelian surfaces of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040267.png" />"  ''Math. Ann.'' , '''285''' (1989) pp. 625–646</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> Ch. Birkenhake,   H. Lange,   "Moduli spaces of Abelian surfaces wih isogeny" , ''Geometry and Analysis, Bombay Colloquium 1992'' , Tata Inst. Fundam. Res. (1995) pp. 225–243</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> G. Horrocks,   D. Mumford,   "A rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040268.png" /> vector bundle on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040269.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040270.png" /> symmetries" ''Topology'' , '''12''' (1973) pp. 63–81</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> K. Hulek,   H. Lange,   "Examples of abelian surfaces in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040271.png" />"  ''J. Reine Angew. Math.'' , '''363''' (1985) pp. 200–216</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> H. Lange, Ch. Birkenhake,   "Complex Abelian varieties" , ''Grundlehren math. Wiss.'' , '''302''' , Springer (1992)</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top"> I. Naruki,   "On smooth quartic embeddings of Kummer surfaces" ''Proc. Japan Acad.'' , '''67 A''' (1991) pp. 223–224</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top"> V. V. Nikulin,   "On Kummer surfaces" ''Math USSR Izv.'' , '''9''' (1975) pp. 261–275 (In Russian)</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top"> S. Ramanan,   "Ample divisors on abelian surfaces" ''Proc. London Math. Soc.'' , '''51''' (1985) pp. 231–245</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top"> I. Reider,   "Vector bundles of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040272.png" /> and linear systems on algebraic surfaces" ''Ann. of Math.'' , '''127''' (1988) pp. 309–316</TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top"> P. Vanhaecke,   "A special case of the Garnier system, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040273.png" />-polarized Abelian surfaces and their moduli" ''Compositio Math.'' , '''92''' (1994) pp. 157–203</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top"> M. Adler, P. van Moerbeke, "The Kowalewski and Hénon–Heiles motions as Manakov geodesic flows on $SO(4)$: a two-dimensional family of Lax pairs" ''Comm. Math. Phys.'' , '''113''' (1988) pp. 659–700</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> M. Adler, P. van Moerbeke, "The complex geometry of the Kowalewski–Painlevé analysis" ''Invent. Math.'' , '''97''' (1989) pp. 3–51 {{MR|}} {{ZBL|0678.58020}} </TD></TR>
 +
<TR><TD valign="top">[a3]</TD> <TD valign="top"> W. Barth, "Abelian surfaces with $(1,2)$-polarization" , ''Algebraic Geometry, Sendai, 1985'' , ''Advanced Studies in Pure Math.'' , '''10''' (1987) pp. 41–84 {{MR|946234}} {{ZBL|}} </TD></TR>
 +
<TR><TD valign="top">[a4]</TD> <TD valign="top"> W. Barth, "Quadratic equations for level-$3$ abelian surfaces" , ''Abelian Varieties, Proc. Workshop Egloffstein 1993'' , de Gruyter (1995) pp. 1–18 {{MR|1336597}} {{ZBL|}} </TD></TR>
 +
<TR><TD valign="top">[a5]</TD> <TD valign="top"> W. Barth, I. Nieto, "Abelian surfaces of type $(1,3)$ and quartic surfaces with $16$ skew lines" ''J. Algebraic Geom.'' , '''3''' (1994) pp. 173–222 {{MR|1257320}} {{ZBL|0809.14027}} </TD></TR>
 +
<TR><TD valign="top">[a6]</TD> <TD valign="top"> Ch. Birkenhake, H. Lange, D. van Straten, "Abelian surfaces of type $(1,4)$" ''Math. Ann.'' , '''285''' (1989) pp. 625–646 {{MR|1027763}} {{ZBL|0714.14028}} </TD></TR>
 +
<TR><TD valign="top">[a7]</TD> <TD valign="top"> Ch. Birkenhake, H. Lange, "Moduli spaces of Abelian surfaces wih isogeny" , ''Geometry and Analysis, Bombay Colloquium 1992'' , Tata Inst. Fundam. Res. (1995) pp. 225–243 {{MR|1351509}} {{ZBL|}} </TD></TR>
 +
<TR><TD valign="top">[a8]</TD> <TD valign="top"> G. Horrocks, D. Mumford, "A rank $2$ vector bundle on $\mathbb{P}^4$ with $15000$ symmetries" ''Topology'' , '''12''' (1973) pp. 63–81 {{MR|382279}} {{ZBL|0255.14017}} </TD></TR>
 +
<TR><TD valign="top">[a9]</TD> <TD valign="top"> K. Hulek, H. Lange, "Examples of abelian surfaces in $\mathbb{P}^4$" ''J. Reine Angew. Math.'' , '''363''' (1985) pp. 200–216 {{MR|0814021}} {{ZBL|0593.14027}} </TD></TR>
 +
<TR><TD valign="top">[a10]</TD> <TD valign="top"> H. Lange, Ch. Birkenhake, "Complex Abelian varieties" , ''Grundlehren math. Wiss.'' , '''302''' , Springer (1992) {{MR|1217487}} {{ZBL|0779.14012}} </TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top"> I. Naruki, "On smooth quartic embeddings of Kummer surfaces" ''Proc. Japan Acad.'' , '''67 A''' (1991) pp. 223–224 {{MR|1137912}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top"> V. V. Nikulin, "On Kummer surfaces" ''Math USSR Izv.'' , '''9''' (1975) pp. 261–275 (In Russian) {{MR|429917}} {{ZBL|0325.14015}} </TD></TR>
 +
<TR><TD valign="top">[a13]</TD> <TD valign="top"> S. Ramanan, "Ample divisors on abelian surfaces" ''Proc. London Math. Soc.'' , '''51''' (1985) pp. 231–245 {{MR|0794112}} {{ZBL|0603.14013}} </TD></TR>
 +
<TR><TD valign="top">[a14]</TD> <TD valign="top"> I. Reider, "Vector bundles of rank $2$ and linear systems on algebraic surfaces" ''Ann. of Math.'' , '''127''' (1988) pp. 309–316 {{MR|0932299}} {{ZBL|0663.14010}} </TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top"> P. Vanhaecke, "A special case of the Garnier system, $(1,4)$-polarized Abelian surfaces and their moduli" ''Compositio Math.'' , '''92''' (1994) pp. 157–203 {{MR|1283227}} {{ZBL|}} </TD></TR></table>

Latest revision as of 06:25, 26 March 2023


An Abelian variety of dimension two, i.e. a complete connected group variety of dimension two over a field $ k $. The group law is commutative. In the sequel, $ k $ is assumed to be algebraically closed (cf. Algebraically closed field).

In the classification of algebraic surfaces, Abelian surfaces are exactly the smooth complete surfaces $ A $ with Kodaira dimension $ \kappa = 0 $, geometric genus $ p _ {g} = h ^ {2} ( A, {\mathcal O} _ {A} ) =1 $ and irregularity $ q = h ^ {1} ( A, {\mathcal O} _ {A} ) = 2 $.

For an Abelian surface $ A $, the dual Abelian variety $ {\widehat{A} } = { \mathop{\rm Pic} } ^ {0} ( A ) $ is again an Abelian surface. An invertible sheaf $ L $ on $ A $ defines the homomorphism $ {\phi _ {L} } : A \rightarrow { {\widehat{A} } } $, $ a \mapsto t _ {a} ^ {*} L \otimes L ^ {- 1 } $. The homomorphism $ \phi _ {L} $ depends only on the algebraic equivalence class of $ L $. The invertible sheaf $ L $ is ample (cf. Ample sheaf) if and only if $ \phi _ {L} $ is an isogeny (i.e., $ \phi _ {L} $ is surjective and has finite kernel) and $ h ^ {0} ( A,L ) \neq0 $. In this case, $ { \mathop{\rm deg} } \phi _ {L} = d ^ {2} $ with a positive integer $ d $ and the Riemann–Roch theorem says that

$$ h ^ {0} ( A,L ) = { \frac{1}{2} } ( L ^ {2} ) = d, $$

where $ ( L ^ {2} ) $ denotes the self-intersection number. Every Abelian surface admits an ample invertible sheaf and hence is projective (cf. Projective scheme).

A polarization $ \lambda $ on $ A $ is the algebraic equivalence class $ [ L ] $ of an ample invertible sheaf $ L $. The degree $ { \mathop{\rm deg} } \lambda $ of $ \lambda $ is by definition $ d = \sqrt { { \mathop{\rm deg} } \phi _ {L} } $. An Abelian surface $ A $ together with a polarization $ \lambda $ is a polarized Abelian surface. A principal polarization is a polarization of degree $ 1 $. A principally polarized Abelian surface $ ( A, \lambda ) $ is either the Jacobi variety $ J ( H ) $ of a smooth projective curve $ H $ of genus $ 2 $, and $ \lambda = \theta $ is the class of the theta divisor, or $ A $ is the product of two elliptic curves (Abelian varieties of dimension one, cf. also Elliptic curve) with $ \lambda $ the product polarization.

If the degree of $ \lambda = [ L ] $ is prime to $ { \mathop{\rm char} } ( k ) $, then $ \lambda $ is said to be a separable polarization and the kernel of $ \phi _ {L} $ is isomorphic to the group $ ( \mathbf Z/d _ {1} \mathbf Z ) ^ {2} \times ( \mathbf Z/d _ {2} \mathbf Z ) ^ {2} $, where $ d _ {1} $ and $ d _ {2} $ are positive integers such that $ d _ {1} $ divides $ d _ {2} $ and $ d _ {1} d _ {2} = { \mathop{\rm deg} } \lambda $. The pair $ ( d _ {1} ,d _ {2} ) $ is the type of the polarized Abelian surface $ ( A, \lambda ) $.

A polarization $ \lambda = [ L ] $ of type $ ( d _ {1} ,d _ {2} ) $ on $ A $ defines a polarization $ {\widehat \lambda } = [ {\widehat{L} } ] $ on the dual Abelian surface $ {\widehat{A} } $. The polarization $ {\widehat \lambda } $ is again of type $ ( d _ {1} ,d _ {2} ) $ and it is characterized by each of the following two equivalent properties:

$$ \phi _ {L} ^ {*} {\widehat \lambda } = d _ {1} d _ {2} \lambda \iff \phi _ { {\widehat{L} } } \phi _ {L} = d _ {1} d _ {2} { \mathop{\rm id} } _ {A} . $$

For a polarized Abelian surface $ ( A, \lambda = [ L ] ) $ of type $ ( d _ {1} ,d _ {2} ) $, the assignment $ A \ni a \mapsto \{ {\sigma \in H ^ {0} ( A,L ) } : {\sigma ( a ) = 0 } \} \subset H ^ {0} ( A,L ) $ defines a rational mapping from $ A $ into the projective space of hyperplanes in $ H ^ {0} ( A,L ) $:

$$ {\varphi _ {L} } : A \rightarrow {\mathbf P ( H ^ {0} ( A,L ) ^ {*} ) } \simeq \mathbf P _ {k} ^ {d _ {1} d _ {2} - 1 } . $$

If $ d _ {1} \geq 2 $, then $ \varphi _ {L} $ is everywhere defined. The Lefschetz theorem says that for $ d _ {1} \geq 3 $ the morphism $ \varphi _ {L} $ is an embedding. Suppose $ d _ {1} = 2 $; then $ \lambda = 2 \mu $ with a polarization $ \mu = [ M ] $ of type $ ( 1, { {d _ {2} } / 2 } ) $. If the linear system $ | M | $ has no fixed components, then $ \varphi _ {L} $ is an embedding.

Complex Abelian surfaces.

An Abelian surface over the field $ \mathbf C $ of complex numbers is a complex torus

$$ A = { {\mathbf C ^ {2} } / \Lambda } $$

(with a lattice $ \Lambda \simeq \mathbf Z ^ {4} $ in $ \mathbf C ^ {2} $) admitting a polarization. A polarization $ \lambda $ on $ A $ can be considered as a non-degenerate alternating form $ \Lambda \times \Lambda \rightarrow \mathbf Z $, the elementary divisors of which are given by the type $ ( d _ {1} ,d _ {2} ) $ of $ \lambda $.

In the sequel, the field $ k $ is assumed to be $ \mathbf C $, although some of the following results are also valid for arbitrary algebraically closed fields.

Suppose $ ( A, \lambda = [ L ] ) $ is of type $ ( 1,d ) $ and the linear system $ | L | $ has no fixed components. The Reider theorem states that for $ d \geq 5 $ the invertible sheaf $ L $ is very ample if and only if there is no elliptic curve $ E $ on $ A $ with $ ( E \cdot L ) = 2 $( see [a14] and [a10]). For arbitrary $ d \geq 1 $ there exist finitely many isogenies $ f : {( A, \lambda ) } \rightarrow {( B, \theta ) } $ of degree $ d $ onto principally polarized Abelian surfaces (cf. also Isogeny). Suppose $ \theta = [ \Theta ] $ with a symmetric invertible sheaf $ \Theta $( i.e., $ ( -1 ) _ {A} ^ {*} \Theta \simeq \Theta $) and let $ H $ be the unique divisor in the linear system $ | \Theta | $. The divisor $ C = f ^ {- 1 } ( H ) $ on $ A $ defines a symmetric invertible sheaf $ L = {\mathcal O} _ {A} ( C ) $ with class $ [ L ] = \lambda $ and the covering $ {f \mid _ {C} } : C \rightarrow H $ is étale of degree $ d $. One distinguishes two cases: i) $ H $ is smooth of genus $ 2 $, $ B = J ( H ) $ and $ C $ is smooth of genus $ d + 1 $; and ii) $ H $ is the sum $ E _ {1} + E _ {2} $ of two elliptic curves with intersection number $ ( E _ {1} \cdot E _ {2} ) = 1 $, $ B = E _ {1} \times E _ {2} $ and $ C $ is the sum $ F _ {1} + F _ {2} $ of two elliptic curves with $ ( F _ {1} \cdot F _ {2} ) = d $.

In the following list, $ ( A, \lambda = [ L ] ) $ is a polarized Abelian surface of type $ ( d _ {1} ,d _ {2} ) $ such that $ | L | $ admits no fixed components

Type $ ( 1,2 ) $— The linear system $ | L | $ has exactly $ 4 $ base points. The blow-up $ {\widetilde{A} } $ of $ A $ in these points admits a morphism $ { {\widetilde \varphi } _ {L} } : { {\widetilde{A} } } \rightarrow {\mathbf P ^ {1} } $ induced by $ \varphi _ {L} $. The general fibre of $ {\widetilde \varphi } _ {L} $ is a smooth curve of genus $ 3 $. The curve $ C $ on $ A $ defining $ L $ as above is double elliptic: $ C { \mathop \rightarrow \limits ^ { {2:1 }} } E $ with an elliptic curve $ E $, and $ A $ is isomorphic to $ { {J ( C ) } / E } $( see [a3]).

Type $ ( 1,3 ) $— $ L $ defines a $ 6 $- fold covering $ {\varphi _ {L} } : A \rightarrow {\mathbf P ^ {2} } $ ramified along a curve $ R \subset \mathbf P ^ {2} $ of degree $ 18 $. The general divisor in the linear system $ | L | $ is a smooth curve of genus $ 4 $. There are $ 4 $ isogenies $ f : {( A, \lambda ) } \rightarrow {( B, \theta ) } $ of degree $ 3 $ onto principally polarized Abelian surfaces. In case i) the smooth genus- $ 4 $ curve $ C \in | L | $ is double elliptic: $ C { \mathop \rightarrow \limits ^ { {2:1 }} } E $, and the embedding of $ E $ into the Jacobian $ J ( C ) $ induces an exact sequence

$$ 0 \rightarrow E \times E \rightarrow J ( C ) \rightarrow A \rightarrow 0. $$

The étale $ 3 $- fold covering $ {f \mid _ {C} } : C \rightarrow H $ induces a morphism $ {f ^ {*} } : {J ( H ) } \rightarrow {J ( C ) } $ with image $ {\widehat{A} } $, the dual Abelian surface of $ A $( see [a7]).

Type $ ( 1,4 ) $— There are $ 24 $ isogenies $ f : {( A, \lambda ) } \rightarrow {( B, \theta ) } $ of degree $ 4 $ onto principally polarized Abelian surfaces. If the curves $ C $ and $ H $ do not admit elliptic involutions compatible with $ f $, then $ \varphi _ {L} :A \twoheadrightarrow {\overline{A}\; } \subset \mathbf P ^ {3} $ is a birational morphism onto a singular octic $ {\overline{A}\; } $. In the exceptional case, $ \varphi _ {L} : A \twoheadrightarrow {\overline{A}\; } \subset \mathbf P ^ {3} $ is a double covering of a singular quartic $ {\overline{A}\; } $, which is birational to an elliptic scroll. In the first case the octic $ {\overline{A}\; } $ is smooth outside the four coordinate planes of $ \mathbf P ^ {3} $ and touches the coordinate planes in curves $ D _ {i} $, $ i = 1 \dots 4 $, of degree $ 4 $. Each of the curves $ D _ {i} $ has $ 3 $ double points and passes through $ 12 $ pinch points of $ {\overline{A}\; } $. The octic is a $ 8:1 $ covering of a Kummer surface: $ p: {\overline{A}\; } \twoheadrightarrow K \subset \mathbf P ^ {3} $( see also Type $ ( 2,2 ) $ below). The restrictions $ p \mid _ {D _ {i} } $ are $ 4 $- fold coverings of four double conics of $ K $ lying on a coordinate tetrahedron. The three double points of $ D _ {i} $ map to three double points of $ K $ on the conic $ p ( D _ {i} ) $ and the $ 12 $ pinch points on $ D _ {i} $ map to the other three double points on the double conic $ p ( D _ {i} ) $( see [a6]).

Type $ ( 1,5 ) $— The invertible sheaf $ L $ is very ample, i.e. $ {\varphi _ {L} } : A \rightarrow {\mathbf P ^ {4} } $ is an embedding if and only if the curves $ C $ and $ H $ do not admit elliptic involutions compatible with $ f $. In the exceptional case $ \varphi _ {L} $ is a double covering of an elliptic scroll (see [a13] and [a9]). If $ L $ is very ample, $ \varphi _ {L} ( A ) $ is a smooth surface of degree $ 10 $ in $ \mathbf P ^ {4} $. It is the zero locus of a section of the Horrocks–Mumford bundle $ F $( see [a8]). Conversely, the zero set $ \{ \sigma = 0 \} \subset \mathbf P ^ {4} $ of a general section $ \sigma \in H ^ {0} ( \mathbf P ^ {4} ,F ) $ is an Abelian surface of degree $ 10 $, i.e. of type $ ( 1,5 ) $.

Type $ ( 2,2 ) $— $ \lambda $ is twice a principal polarization on $ A $. The morphism $ \varphi _ {L} : A \twoheadrightarrow K _ {A} \subset \mathbf P ^ {3} $ is a double covering of the Kummer surface $ K _ {A} $ associated with $ A $. It is isomorphic to $ {A / {( - 1 ) _ {A} } } $.

Type $ ( 2,4 ) $— The ideal sheaf $ {\mathcal I} _ { {A / {\mathbf P ^ {7} } } } $ of the image of the embedding $ \varphi _ {L} : A \hookrightarrow \mathbf P ^ {7} $ is generated by $ 6 $ quadrics (see [a3]).

Type $ ( 2,6 ) $— Suppose $ L $ is very ample and let $ K _ {A} = {A / {( - 1 ) _ {A} } } $ be the associated Kummer surface. The subvector space $ H ^ {0} ( A,L ) ^ {-} \subset H ^ {0} ( A,L ) $ of odd sections induces an embedding of $ {\widetilde{K} } _ {A} $, the blow-up of $ K _ {A} $ in the $ 16 $ double points, as a smooth quartic surface into $ \mathbf P ^ {3} $. $ {\widetilde{K} } _ {A} \subset \mathbf P ^ {3} $ is invariant under the action of the level- $ 2 $ Heisenberg group (cf. also Heisenberg representation) $ H ( 2,2 ) $. The $ 16 $ blown-up double points become skew lines on the quartic surface. Any $ H ( 2,2 ) $- invariant quartic surface in $ \mathbf P ^ {3} $ with $ 16 $ skew lines comes from a polarized Abelian surface $ ( A, \lambda ) $ of type $ ( 2,6 ) $ in this way (see [a5], [a11] and [a12]).

Type $ ( 3,3 ) $— $ \lambda $ is three times a principal polarization and $ \varphi _ {L} : A \hookrightarrow \mathbf P ^ {8} $ is an embedding. If $ ( A, \lambda ) $ is not a product, then the quadrics $ Q \in H ^ {0} ( \mathbf P ^ {8} , {\mathcal I} _ { {A / {\mathbf P ^ {8} } } } ( 2 ) ) $ vanishing on $ A $ generate the ideal sheaf $ {\mathcal I} _ { {A / {\mathbf P ^ {8} } } } $. In the product case, $ {\mathcal I} _ { {A / {\mathbf P ^ {8} } } } $ is generated by quadrics and cubics (see [a4]).

Algebraic completely integrable systems.

An algebraic completely integrable system in the sense of M. Adler and P. van Moerbeke is a completely integrable polynomial Hamiltonian system on $ \mathbf C ^ {N} $( with Casimir functions $ {H _ {1} \dots H _ {k} } : {\mathbf C ^ {N} } \rightarrow \mathbf C $ and $ m = { {( N - k ) } / 2 } $ independent constants of motion $ H _ {k + 1 } \dots H _ {k + m } $ in involution) such that:

a) for a general point $ c = {} ^ {t} ( c _ {1} \dots c _ {k + m } ) \in \mathbf C ^ {k + m } $ the invariant manifold $ A _ {c} ^ {o} = \cap _ {i = 1 } ^ {m + k } \{ H _ {i} = c _ {i} \} \subset \mathbf C ^ {N} $ is an open affine part of an Abelian variety $ A _ {c} $;

b) the flows of the integrable vector fields $ X _ {u _ {i} } $ linearize on the Abelian varieties $ A _ {c} $[a2].

The divisor at infinity $ D _ {c} = A _ {c} - A _ {c} ^ {o} $ defines a polarization on $ A _ {c} $. In this way the mapping $ {( H _ {1} \dots H _ {k + m } ) } : {\mathbf C ^ {N} } \rightarrow {\mathbf C ^ {k + m } } $ defines a family of polarized Abelian varieties (cf. Moduli problem). Some examples of $ 2 $- dimensional algebraic completely integrable systems and their associated Abelian surfaces are:

the three-body Toda lattice and the even, respectively odd, master systems (cf. also Master equations in cooperative and social phenomena) linearize on principally polarized Abelian surfaces;

the Kowalewski top, the Hénon–Heiles system and the Manakov geodesic flow on $ { \mathop{\rm SO} } ( 4 ) $ linearize on Abelian surfaces of type $ ( 1,2 ) $[a1];

the Garnier system linearizes on Abelian surfaces of type $ ( 1,4 ) $[a15].

References

[a1] M. Adler, P. van Moerbeke, "The Kowalewski and Hénon–Heiles motions as Manakov geodesic flows on $SO(4)$: a two-dimensional family of Lax pairs" Comm. Math. Phys. , 113 (1988) pp. 659–700
[a2] M. Adler, P. van Moerbeke, "The complex geometry of the Kowalewski–Painlevé analysis" Invent. Math. , 97 (1989) pp. 3–51 Zbl 0678.58020
[a3] W. Barth, "Abelian surfaces with $(1,2)$-polarization" , Algebraic Geometry, Sendai, 1985 , Advanced Studies in Pure Math. , 10 (1987) pp. 41–84 MR946234
[a4] W. Barth, "Quadratic equations for level-$3$ abelian surfaces" , Abelian Varieties, Proc. Workshop Egloffstein 1993 , de Gruyter (1995) pp. 1–18 MR1336597
[a5] W. Barth, I. Nieto, "Abelian surfaces of type $(1,3)$ and quartic surfaces with $16$ skew lines" J. Algebraic Geom. , 3 (1994) pp. 173–222 MR1257320 Zbl 0809.14027
[a6] Ch. Birkenhake, H. Lange, D. van Straten, "Abelian surfaces of type $(1,4)$" Math. Ann. , 285 (1989) pp. 625–646 MR1027763 Zbl 0714.14028
[a7] Ch. Birkenhake, H. Lange, "Moduli spaces of Abelian surfaces wih isogeny" , Geometry and Analysis, Bombay Colloquium 1992 , Tata Inst. Fundam. Res. (1995) pp. 225–243 MR1351509
[a8] G. Horrocks, D. Mumford, "A rank $2$ vector bundle on $\mathbb{P}^4$ with $15000$ symmetries" Topology , 12 (1973) pp. 63–81 MR382279 Zbl 0255.14017
[a9] K. Hulek, H. Lange, "Examples of abelian surfaces in $\mathbb{P}^4$" J. Reine Angew. Math. , 363 (1985) pp. 200–216 MR0814021 Zbl 0593.14027
[a10] H. Lange, Ch. Birkenhake, "Complex Abelian varieties" , Grundlehren math. Wiss. , 302 , Springer (1992) MR1217487 Zbl 0779.14012
[a11] I. Naruki, "On smooth quartic embeddings of Kummer surfaces" Proc. Japan Acad. , 67 A (1991) pp. 223–224 MR1137912
[a12] V. V. Nikulin, "On Kummer surfaces" Math USSR Izv. , 9 (1975) pp. 261–275 (In Russian) MR429917 Zbl 0325.14015
[a13] S. Ramanan, "Ample divisors on abelian surfaces" Proc. London Math. Soc. , 51 (1985) pp. 231–245 MR0794112 Zbl 0603.14013
[a14] I. Reider, "Vector bundles of rank $2$ and linear systems on algebraic surfaces" Ann. of Math. , 127 (1988) pp. 309–316 MR0932299 Zbl 0663.14010
[a15] P. Vanhaecke, "A special case of the Garnier system, $(1,4)$-polarized Abelian surfaces and their moduli" Compositio Math. , 92 (1994) pp. 157–203 MR1283227
How to Cite This Entry:
Abelian surface. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Abelian_surface&oldid=16005
This article was adapted from an original article by Ch. Birkenhake (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article