View source for Abelian surface

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]]).

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" />.

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

<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>

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]]).

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.

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" />.

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:

<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>

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" />:

<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>

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.

==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]]

<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>

(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" />.

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.

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" />.

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

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 <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

<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>

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]]]).

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 <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 <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 <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 <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 <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]]]).

==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:

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" />;

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]]].

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 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 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]]].

====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>