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− | An algebraic or analytic complete non-singular surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e0355501.png" /> having a fibration of elliptic curves (cf. [[Elliptic curve|Elliptic curve]]), that is, a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e0355502.png" /> onto a non-singular curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e0355503.png" /> whose generic fibre is a non-singular elliptic curve. Every elliptic surface is birationally (bimeromorphically) equivalent over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e0355504.png" /> to a unique minimal model, which is characterized by the fact that the fibre of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e0355505.png" /> does not contain exceptional curves of arithmetic genus 1. In what follows an elliptic surface is assumed to be minimal. Minimal elliptic surfaces have a more complicated structure than ruled surfaces. They can have singular fibres <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e0355506.png" /> (that is, fibres that are not non-singular elliptic curves). There is a classification
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− | of the singular fibres of elliptic surfaces. A singular fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e0355507.png" /> is called multiple if the greatest common divisor of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e0355508.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e0355509.png" />, and then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555011.png" /> is called the multiplicity of the fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555012.png" />.
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− | On a minimal elliptic surface the canonical class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555013.png" /> contains a divisor that is a rational combination of fibres, in particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555014.png" />. Moreover, the following formula holds for the canonical class (see [[#References|[1]]], ):
| + | An algebraic or analytic complete non-singular surface $ X $ |
| + | having a fibration of elliptic curves (cf. [[Elliptic curve|Elliptic curve]]), that is, a morphism $ \pi : X \rightarrow B $ |
| + | onto a non-singular curve $ B $ |
| + | whose generic fibre is a non-singular elliptic curve. Every elliptic surface is birationally (bimeromorphically) equivalent over $ B $ |
| + | to a unique minimal model, which is characterized by the fact that the fibre of $ \pi $ |
| + | does not contain exceptional curves of arithmetic genus 1. In what follows an elliptic surface is assumed to be minimal. Minimal elliptic surfaces have a more complicated structure than ruled surfaces. They can have singular fibres $ X _ {t} = \pi ^ {- 1} ( t) $ (that is, fibres that are not non-singular elliptic curves). There is a classification |
| | | |
− | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555015.png" /></td> </tr></table>
| + | of the singular fibres of elliptic surfaces. A singular fibre $ X _ {t} = \sum n _ {i} E _ {i} $ |
| + | is called multiple if the greatest common divisor of the $ n _ {i} $ |
| + | is $ m \geq 2 $, |
| + | and then $ X _ {t} = m F $ |
| + | and $ m $ |
| + | is called the multiplicity of the fibre $ X _ {t} $. |
| | | |
− | where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555016.png" /> are all the multiple fibres of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555018.png" /> is a divisor on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555019.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555020.png" />. The topological [[Euler characteristic|Euler characteristic]] satisfies the formula
| + | On a minimal elliptic surface the canonical class $ K _ {X} $ |
| + | contains a divisor that is a rational combination of fibres, in particular, $ ( K _ {X} ^ {2} ) = 0 $. |
| + | Moreover, the following formula holds for the canonical class (see [[#References|[1]]], ): |
| | | |
− | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555021.png" /></td> </tr></table>
| + | $$ |
| + | K _ {X} = \pi ^ {*} ( K _ {B} - d ) + |
| + | \sum ( m _ {i} - 1 ) F _ {i} , |
| + | $$ |
| + | |
| + | where $ X _ {t _ {i} } = m _ {i} F _ {i} $ |
| + | are all the multiple fibres of $ \pi $ |
| + | and $ d $ |
| + | is a divisor on $ B $ |
| + | of degree $ - \chi ( {\mathcal O} _ {X} ) $. |
| + | The topological [[Euler characteristic|Euler characteristic]] satisfies the formula |
| + | |
| + | $$ |
| + | e ( X) = \sum e ( X _ {t _ {i} } ) . |
| + | $$ |
| | | |
| ==The classification of elliptic fibrations.== | | ==The classification of elliptic fibrations.== |
− | A fibration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555022.png" /> can be regarded as an elliptic curve over the function field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555023.png" />. This curve, generally speaking, does not have the structure of an Abelian variety over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555024.png" />. For this to happen it is necessary that it has a rational point over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555025.png" /> (and then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555026.png" /> is birationally isomorphic to the surface defined in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555027.png" /> by the Weierstrass equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555028.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555029.png" />). The specification of a rational point is equivalent to that of a section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555030.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555031.png" />; a necessary condition for the existence of a section is the absence of multiple fibres. Fibrations without multiple fibres are called reduced. Every fibration, after a suitable ramified covering of the base, has a section (that is, is reduced) . Every fibration can also be made reduced by a sequence of transformations inverse to logarithmic ones | + | A fibration $ \pi : X \rightarrow B $ |
| + | can be regarded as an elliptic curve over the function field $ k ( B) $. |
| + | This curve, generally speaking, does not have the structure of an Abelian variety over $ k ( B) $. |
| + | For this to happen it is necessary that it has a rational point over $ k ( B) $ (and then $ X $ |
| + | is birationally isomorphic to the surface defined in $ B \times A ^ {2} $ |
| + | by the Weierstrass equation $ y ^ {2} = x ^ {3} - g _ {2} x - g _ {3} $, |
| + | where $ g _ {2} , g _ {3} \in k ( B) $). |
| + | The specification of a rational point is equivalent to that of a section $ e : B \rightarrow X $ |
| + | such that $ \pi e = \mathop{\rm id} $; |
| + | a necessary condition for the existence of a section is the absence of multiple fibres. Fibrations without multiple fibres are called reduced. Every fibration, after a suitable ramified covering of the base, has a section (that is, is reduced) . Every fibration can also be made reduced by a sequence of transformations inverse to logarithmic ones |
| | | |
| — local surgery of the fibration in neighbourhoods of fibres. | | — local surgery of the fibration in neighbourhoods of fibres. |
| | | |
− | Reduced elliptic fibrations may be described as follows. To every such fibration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555032.png" /> corresponds a unique fibration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555033.png" /> that is a [[Group object|group object]] and is such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555034.png" /> is a [[Principal homogeneous space|principal homogeneous space]] over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555035.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555036.png" /> is the Jacobi fibration for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555037.png" />; it characterizes the existence of a section. For a given Jacobi fibration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555038.png" />, the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555039.png" /> of isomorphism classes of fibrations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555040.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555041.png" /> has a cohomology description analogous to that of invertible sheaves (cf. [[Invertible sheaf|Invertible sheaf]]). Here the role of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555042.png" /> is played by the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555043.png" /> of local sections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555044.png" />. There is a natural one-to-one correspondence | + | Reduced elliptic fibrations may be described as follows. To every such fibration $ \pi : X \rightarrow B $ |
| + | corresponds a unique fibration $ {\mathcal J} _ {B} ( X) \rightarrow B $ |
| + | that is a [[Group object|group object]] and is such that $ X / B $ |
| + | is a [[Principal homogeneous space|principal homogeneous space]] over $ {\mathcal J} _ {B} ( X) / B $; |
| + | $ {\mathcal J} _ {B} ( X) / B $ |
| + | is the Jacobi fibration for $ X / B $; |
| + | it characterizes the existence of a section. For a given Jacobi fibration $ {\mathcal J} / B $, |
| + | the set $ I ( {\mathcal J} / B ) $ |
| + | of isomorphism classes of fibrations $ X / B $ |
| + | for which $ {\mathcal J} _ {B} ( X) \cong J $ |
| + | has a cohomology description analogous to that of invertible sheaves (cf. [[Invertible sheaf|Invertible sheaf]]). Here the role of $ {\mathcal O} _ {B} ^ {*} $ |
| + | is played by the sheaf $ {\mathcal H} ^ {0} ( {\mathcal J} / B ) $ |
| + | of local sections $ \tau : {\mathcal J} \rightarrow B $. |
| + | There is a natural one-to-one correspondence |
| | | |
− | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555045.png" /></td> </tr></table>
| + | $$ |
| + | \theta : I ( {\mathcal J} / B ) \rightarrow H ^ {1} |
| + | ( B , {\mathcal H} ^ {0} ( {\mathcal J} / B ) ) , |
| + | $$ |
| | | |
− | under which the Jacobi fibration corresponds to the zero element. By means of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555046.png" /> one can distinguish between algebraic and non-algebraic fibrations: For a reduced fibration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555047.png" /> the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555048.png" /> is algebraic if and only if the element corresponding to it in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555049.png" /> is of finite order. The analogy with invertible sheaves can be pursued further. The analogue of the exact sequence | + | under which the Jacobi fibration corresponds to the zero element. By means of $ \theta $ |
| + | one can distinguish between algebraic and non-algebraic fibrations: For a reduced fibration $ \pi : X \rightarrow B $ |
| + | the surface $ X $ |
| + | is algebraic if and only if the element corresponding to it in $ H ^ {1} ( B , {\mathcal H} ^ {0} ( {\mathcal J} / B )) $ |
| + | is of finite order. The analogy with invertible sheaves can be pursued further. The analogue of the exact sequence |
| | | |
− | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555050.png" /></td> </tr></table>
| + | $$ |
| + | 0 \rightarrow \mathbf Z \rightarrow {\mathcal O} _ {B} \mathop \rightarrow \limits ^ {\mathop {\rm exp}} {\mathcal O} _ {B} ^ {*} \rightarrow 1 |
| + | $$ |
| | | |
| is the exact sequence | | is the exact sequence |
| | | |
− | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555051.png" /></td> </tr></table>
| + | $$ |
| + | 0 \rightarrow R ^ {1} \tau _ {0} \mathbf Z \rightarrow {\mathcal H} ^ {0} |
| + | ( T ( {\mathcal J} ) / B ) \rightarrow {\mathcal H} ^ {0} ( {\mathcal J} / B ) \rightarrow 0 , |
| + | $$ |
| | | |
− | where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555052.png" /> is the sheaf of local sections of the bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555053.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555054.png" /> is the tangent space to the fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555055.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555056.png" />. The boundary homomorphism | + | where $ {\mathcal H} ^ {0} ( T ( {\mathcal J} ) / B ) $ |
| + | is the sheaf of local sections of the bundle $ T ( {\mathcal J} )/ B $ |
| + | and $ T ( {\mathcal J} ) $ |
| + | is the tangent space to the fibre $ \tau ^ {- 1 }( b) $ |
| + | at $ e ( b) $. |
| + | The boundary homomorphism |
| | | |
− | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555057.png" /></td> </tr></table>
| + | $$ |
| + | \delta : H ^ {1} ( B , {\mathcal H} ^ {0} |
| + | ( {\mathcal J} / B ) ) \rightarrow H ^ {2} ( B , R ^ {1} \tau _ {*} \mathbf Z ) |
| + | $$ |
| | | |
− | allows one to recognize when one fibration is a deformation of another. For that it is necessary and sufficient that the elements corresponding to these fibrations have one and the same image under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555058.png" /> (see ). | + | allows one to recognize when one fibration is a deformation of another. For that it is necessary and sufficient that the elements corresponding to these fibrations have one and the same image under $ \delta $ (see ). |
| | | |
| ==The classification of algebraic elliptic surfaces.== | | ==The classification of algebraic elliptic surfaces.== |
− | Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555059.png" />. For an elliptic surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555060.png" /> the canonical dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555061.png" />, that is, it is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555062.png" />, 0 or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555063.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555064.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555065.png" /> is said to be an elliptic surface of general type. These are characterized by the conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555066.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555067.png" />. Elliptic surfaces with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555068.png" /> or, more generally, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555069.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555070.png" />, are of general type. | + | Suppose that $ \mathop{\rm char} k = 0 $. |
| + | For an elliptic surface $ X $ |
| + | the canonical dimension $ k ( X) \leq 1 $, |
| + | that is, it is equal to $ - 1 $, |
| + | 0 or $ 1 $. |
| + | If $ k ( X) = 1 $, |
| + | $ X $ |
| + | is said to be an elliptic surface of general type. These are characterized by the conditions $ 12 K _ {X} \neq 0 $ |
| + | and $ | 12 K _ {X} | \neq \emptyset $. |
| + | Elliptic surfaces with $ p _ {g} \geq 2 $ |
| + | or, more generally, with $ P _ {m} \geq 2 $ |
| + | for some $ m $, |
| + | are of general type. |
| | | |
− | Elliptic surfaces with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555071.png" /> are characterized by the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555072.png" />. In this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555073.png" /> can take the three values 2, 1 or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555074.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555075.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555076.png" /> is an elliptic [[K3-surface|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555077.png" />-surface]] (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555078.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555079.png" />). In this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555080.png" /> is isomorphic to the projective line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555081.png" />, the fibration has no multiple fibres and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555082.png" /> has the invariants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555083.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555084.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555085.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555086.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555087.png" /> is an Enriques surface, that is, a surface with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555088.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555089.png" />. (Every Enriques surface is elliptic.) In this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555090.png" />, the fibration has two fibres of multiplicity 2, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555091.png" /> has the invariants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555092.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555093.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555094.png" />, then two cases are possible. Either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555095.png" /> is an Abelian variety (and then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555096.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555097.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555098.png" />); or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e03555099.png" /> is a hyper-elliptic surface, that is, a surface that has a finite unramified covering — the product of two elliptic curves. In that case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550100.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550101.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550102.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550103.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550104.png" /> has 3 or 4 multiple fibres with four possibilities for their multiplicity: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550105.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550106.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550107.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550108.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550109.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550110.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550111.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550112.png" />, respectively. | + | Elliptic surfaces with $ k ( X) = 0 $ |
| + | are characterized by the condition $ 12 K _ {X} = 0 $. |
| + | In this case $ \chi ( {\mathcal O} _ {X} ) $ |
| + | can take the three values 2, 1 or $ 0 $. |
| + | If $ \chi ( {\mathcal O} _ {X} )= 2 $, |
| + | then $ X $ |
| + | is an elliptic [[K3-surface| $ K 3 $-surface]] ( $ q = 0 $, |
| + | $ K _ {X} = 0 $). |
| + | In this case $ B $ |
| + | is isomorphic to the projective line $ P ^ {1} $, |
| + | the fibration has no multiple fibres and $ X $ |
| + | has the invariants $ p _ {g} = 1 $, |
| + | $ q = 0 $, |
| + | $ b _ {2} = 22 $. |
| + | If $ \chi ( {\mathcal O} _ {X} ) = 1 $, |
| + | then $ X $ |
| + | is an Enriques surface, that is, a surface with $ p _ {g} = q = 0 $, |
| + | $ 2 K _ {X} = 0 $. |
| + | (Every Enriques surface is elliptic.) In this case $ B \simeq P ^ {1} $, |
| + | the fibration has two fibres of multiplicity 2, and $ X $ |
| + | has the invariants $ p _ {g} = q = 0 $, |
| + | $ b _ {2} = 10 $. |
| + | If $ \chi ( {\mathcal O} _ {X} ) = 0 $, |
| + | then two cases are possible. Either $ X $ |
| + | is an Abelian variety (and then $ p _ {g} = 1 $, |
| + | $ q = 2 $, |
| + | $ b _ {2} = 6 $); |
| + | or $ X $ |
| + | is a hyper-elliptic surface, that is, a surface that has a finite unramified covering — the product of two elliptic curves. In that case $ p _ {g} = 0 $, |
| + | $ b _ {1} = 2 $, |
| + | $ b _ {2} = 2 $, |
| + | $ B = P ^ {1} $, |
| + | and $ \pi $ |
| + | has 3 or 4 multiple fibres with four possibilities for their multiplicity: $ ( 3 , 3 , 3 ) $, |
| + | $ ( 2 , 4 , 4 ) $, |
| + | $ ( 2 , 3 , 6 ) $, |
| + | and $ ( 2 , 2 , 2 , 2 ) $, |
| + | and $ 3 K _ {X} = 0 $, |
| + | $ 4 K _ {X} = 0 $, |
| + | $ 6 K _ {X} = 0 $, |
| + | and $ 2 K _ {X} = 0 $, |
| + | respectively. |
| | | |
− | An elliptic surface with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550113.png" /> is ruled (cf. [[Ruled surface|Ruled surface]]). It is characterized by the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550114.png" />. Here two cases are possible: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550115.png" /> is a surface with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550116.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550117.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550118.png" /> has no multiple fibres or one; moreover, a surface without multiple fibres can be obtained as follows: one has to take a rational mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550119.png" /> determined by two cubics <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550120.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550121.png" /> and blow up their 9 points of intersection; or 2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550122.png" /> is a surface with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550123.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550124.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550125.png" />, and the multiplicities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550126.png" /> are subject to the inequality | + | An elliptic surface with $ k ( X) = - 1 $ |
| + | is ruled (cf. [[Ruled surface|Ruled surface]]). It is characterized by the condition $ | 12 K _ {X} | = \emptyset $. |
| + | Here two cases are possible: 1) $ X $ |
| + | is a surface with $ p _ {g} = q = 0 $, |
| + | $ b _ {2} = 10 $, |
| + | and $ \pi $ |
| + | has no multiple fibres or one; moreover, a surface without multiple fibres can be obtained as follows: one has to take a rational mapping $ P ^ {2} \rightarrow P ^ {1} $ |
| + | determined by two cubics $ F _ {0} $ |
| + | and $ F _ {1} $ |
| + | and blow up their 9 points of intersection; or 2) $ X $ |
| + | is a surface with $ p _ {g} = 0 $, |
| + | $ q = 1 $, |
| + | $ b _ {2} = 2 $, |
| + | and the multiplicities $ m _ {i} $ |
| + | are subject to the inequality |
| | | |
− | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550127.png" /></td> </tr></table>
| + | $$ |
| + | \sum \left ( 1 - |
| + | \frac{1}{m _ {i} } |
| + | \right ) < 2 . |
| + | $$ |
| | | |
| The formula for the canonical class and the classification of elliptic surfaces can also be generalized to the case of a field of finite characteristic [[#References|[5]]], [[#References|[6]]]. | | The formula for the canonical class and the classification of elliptic surfaces can also be generalized to the case of a field of finite characteristic [[#References|[5]]], [[#References|[6]]]. |
| | | |
| ==The classification of non-algebraic elliptic surfaces.== | | ==The classification of non-algebraic elliptic surfaces.== |
− | The classification of non-algebraic elliptic surfaces. For non-algebraic surfaces the algebraic dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550128.png" /> is 1 or 0. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550129.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550130.png" /> is non-elliptic. All surfaces with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550131.png" /> are elliptic. Here the structure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550132.png" /> is determined almost canonically: Every such fibration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550133.png" /> is necessarily elliptic. The classification by the canonical dimension is precisely the same as for algebraic elliptic surfaces: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550134.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550135.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550136.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550137.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550138.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550139.png" />; and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550140.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550141.png" /> is of basic type) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550142.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550143.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550144.png" />. | + | The classification of non-algebraic elliptic surfaces. For non-algebraic surfaces the algebraic dimension $ a ( X) = \mathop{\rm tr} \mathop{\rm deg} M ( X) $ |
| + | is 1 or 0. If $ a ( X) = 0 $, |
| + | then $ X $ |
| + | is non-elliptic. All surfaces with $ a ( X) = 1 $ |
| + | are elliptic. Here the structure of $ \pi : X \rightarrow B $ |
| + | is determined almost canonically: Every such fibration $ \pi $ |
| + | is necessarily elliptic. The classification by the canonical dimension is precisely the same as for algebraic elliptic surfaces: $ k ( X) = - 1 $ |
| + | $ \iff $ |
| + | $ | 12 K _ {X} | = \emptyset $; |
| + | $ k ( X) = 0 $ |
| + | $ \iff $ |
| + | $ 12K _ {X} = 0 $; |
| + | and $ k ( X) = 1 $ ($ X $ |
| + | is of basic type) $ \iff $ |
| + | $ | 12 K _ {X} | = \emptyset $, |
| + | $ 12 K _ {X} \neq 0 $. |
| | | |
− | Non-algebraic elliptic surfaces with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550145.png" /> belong to one of the following classes: 1) the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550146.png" />-surfaces (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550147.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550148.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550149.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550150.png" /> is simply connected); 2) the complex tori (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550151.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550152.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550153.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550154.png" />); 3) the Kodaira surfaces (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550155.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550156.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550157.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550158.png" />). Primary Kodaira surfaces are holomorphically locally trivial fibrations over an elliptic curve with an elliptic curve as typical fibre, and from the point of view of differentiability, bundles over a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550159.png" />-dimensional torus with a circle as fibre; or 4) the surfaces with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550160.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550161.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550162.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550163.png" />. For them <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550164.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550165.png" />, (analogous to hyper-elliptic surfaces). They have Kodaira surfaces as finite unramified coverings. In the cases 2), 3) and 4) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550166.png" /> is the universal covering of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550167.png" />. | + | Non-algebraic elliptic surfaces with $ k ( X) = 0 $ |
| + | belong to one of the following classes: 1) the $ K 3 $-surfaces ( $ \chi ( {\mathcal O} _ {X} )= 2 $, |
| + | $ b _ {1} = 0 $, |
| + | $ b _ {2} = 22 $, |
| + | $ X $ |
| + | is simply connected); 2) the complex tori ( $ K _ {X} = 0 $, |
| + | $ \chi ( {\mathcal O} _ {X} ) = 0 $, |
| + | $ b _ {1} = 4 $, |
| + | $ b _ {2} = 6 $); |
| + | 3) the Kodaira surfaces ( $ K _ {X} = 0 $, |
| + | $ \chi ( {\mathcal O} _ {X} ) = 0 $, |
| + | $ b _ {1} = 3 $, |
| + | $ b _ {2} = 4 $). |
| + | Primary Kodaira surfaces are holomorphically locally trivial fibrations over an elliptic curve with an elliptic curve as typical fibre, and from the point of view of differentiability, bundles over a $ 3 $-dimensional torus with a circle as fibre; or 4) the surfaces with $ \chi ( {\mathcal O} _ {X} ) = 0 $, |
| + | $ p _ {g} = 0 $, |
| + | $ b _ {1} = 1 $, |
| + | $ b _ {2} = 0 $. |
| + | For them $ m K _ {X} = 0 $ |
| + | with $ m = 2 , 3 , 4 $, |
| + | (analogous to hyper-elliptic surfaces). They have Kodaira surfaces as finite unramified coverings. In the cases 2), 3) and 4) $ \mathbf C ^ {2} $ |
| + | is the universal covering of $ X $. |
| | | |
− | Non-algebraic elliptic surfaces with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550168.png" /> are Hopf surfaces, that is, their universal covering is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550169.png" />. For them <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550170.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550171.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550172.png" />. The proper Hopf surfaces are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550173.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550174.png" /> are real generators of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550175.png" />. They are homeomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550176.png" /> and are characterized by this property. Arbitrary elliptic Hopf surfaces are quotients of proper Hopf surfaces . | + | Non-algebraic elliptic surfaces with $ k ( X) = - 1 $ |
| + | are Hopf surfaces, that is, their universal covering is $ \mathbf C ^ {2} \setminus 0 $. |
| + | For them $ \chi ( {\mathcal O} _ {X} ) = 0 $, |
| + | $ b _ {1} = 1 $, |
| + | $ b _ {2} = 0 $. |
| + | The proper Hopf surfaces are $ ( \mathbf C ^ {2} \setminus 0 ) / T $, |
| + | where $ T ( z _ {1} , z _ {2} ) = ( \alpha _ {1} z _ {1} , \alpha _ {2} z _ {2} ) $ |
| + | are real generators of $ T $. |
| + | They are homeomorphic to $ S ^ {1} \times S ^ {2} $ |
| + | and are characterized by this property. Arbitrary elliptic Hopf surfaces are quotients of proper Hopf surfaces . |
| | | |
| ====References==== | | ====References==== |
− | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> "Algebraic surfaces" ''Proc. Steklov Inst. Math.'' , '''75''' (1967) ''Trudy Mat. Inst. Steklov.'' , '''75''' (1965)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> D. Husemoller, "Classification and embeddings of surfaces" R. Hartshorne (ed.) , ''Algebraic geometry (Arcata, 1974)'' , ''Proc. Symp. Pure Math.'' , '''29''' , Amer. Math. Soc. (1982) pp. 329–420</TD></TR><TR><TD valign="top">[3a]</TD> <TD valign="top"> K. Kodaira, "On compact complex analytic surfaces I" ''Ann. of Math. (2)'' , '''71''' (10) pp. 111–152</TD></TR><TR><TD valign="top">[3b]</TD> <TD valign="top"> K. Kodaira, "On compact complex analytic surfaces II" ''Ann. of Math. (2)'' , '''77''' (1963) pp. 563–626</TD></TR><TR><TD valign="top">[3c]</TD> <TD valign="top"> K. Kodaira, "On compact complex analytic surfaces III" ''Ann. of Math. (2)'' , '''78''' (1963) pp. 1–40</TD></TR><TR><TD valign="top">[4a]</TD> <TD valign="top"> K. Kodaira, "On the structure of compact complex analytic surfaces I" ''Amer. J. Math.'' , '''86''' (1964) pp. 751–798</TD></TR><TR><TD valign="top">[4b]</TD> <TD valign="top"> K. Kodaira, "On the structure of compact complex analytic surfaces II" ''Amer. J. Math.'' , '''88''' (1966) pp. 682–721</TD></TR><TR><TD valign="top">[4c]</TD> <TD valign="top"> K. Kodaira, "On the structure of compact complex analytic surfaces III" ''Amer. J. Math.'' , '''90''' (1968) pp. 55–83</TD></TR><TR><TD valign="top">[4d]</TD> <TD valign="top"> K. Kodaira, "On the structure of compact complex analytic surfaces IV" ''Amer. J. Math.'' , '''90''' (1968) pp. 1048–1066</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> D. Mumford, "Enriques' classification of surfaces in char <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550177.png" />. I" D.C. Spencer (ed.) S. Iyanaga (ed.) , ''Global analysis (papers in honor of K. Kodaira)'' , Princeton Univ. Press (1969) pp. 325–339</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> E. Bombieri, D. Mumford, "Enriques' classification of surfaces in char <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550178.png" />. II" W.L. Baily jr. (ed.) T. Shioda (ed.) , ''Complex Analysis and Algebraic geometry'' , Cambridge Univ. Press & Iwanami Shoten (1977) pp. 23–42</TD></TR></table> | + | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> "Algebraic surfaces" ''Proc. Steklov Inst. Math.'' , '''75''' (1967) ''Trudy Mat. Inst. Steklov.'' , '''75''' (1965) {{MR|}} {{ZBL|0172.37901}} {{ZBL|0153.22401}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> D. Husemoller, "Classification and embeddings of surfaces" R. Hartshorne (ed.) , ''Algebraic geometry (Arcata, 1974)'' , ''Proc. Symp. Pure Math.'' , '''29''' , Amer. Math. Soc. (1982) pp. 329–420 {{MR|0506292}} {{ZBL|0326.14009}} </TD></TR><TR><TD valign="top">[3a]</TD> <TD valign="top"> K. Kodaira, "On compact complex analytic surfaces I" ''Ann. of Math. (2)'' , '''71''' (10) pp. 111–152 {{MR|0132556}} {{ZBL|0137.17501}} {{ZBL|0098.13004}} </TD></TR><TR><TD valign="top">[3b]</TD> <TD valign="top"> K. Kodaira, "On compact complex analytic surfaces II" ''Ann. of Math. (2)'' , '''77''' (1963) pp. 563–626 {{MR|165541}} {{ZBL|0193.37701}} {{ZBL|0133.16505}} </TD></TR><TR><TD valign="top">[3c]</TD> <TD valign="top"> K. Kodaira, "On compact complex analytic surfaces III" ''Ann. of Math. (2)'' , '''78''' (1963) pp. 1–40 {{MR|165541}} {{ZBL|0193.37701}} </TD></TR><TR><TD valign="top">[4a]</TD> <TD valign="top"> K. Kodaira, "On the structure of compact complex analytic surfaces I" ''Amer. J. Math.'' , '''86''' (1964) pp. 751–798 {{MR|0187255}} {{ZBL|0137.17501}} </TD></TR><TR><TD valign="top">[4b]</TD> <TD valign="top"> K. Kodaira, "On the structure of compact complex analytic surfaces II" ''Amer. J. Math.'' , '''88''' (1966) pp. 682–721 {{MR|0205280}} {{ZBL|0193.37701}} </TD></TR><TR><TD valign="top">[4c]</TD> <TD valign="top"> K. Kodaira, "On the structure of compact complex analytic surfaces III" ''Amer. J. Math.'' , '''90''' (1968) pp. 55–83 {{MR|0228019}} {{ZBL|0193.37701}} </TD></TR><TR><TD valign="top">[4d]</TD> <TD valign="top"> K. Kodaira, "On the structure of compact complex analytic surfaces IV" ''Amer. J. Math.'' , '''90''' (1968) pp. 1048–1066 {{MR|239114}} {{ZBL|}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> D. Mumford, "Enriques' classification of surfaces in char <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550177.png" />. I" D.C. Spencer (ed.) S. Iyanaga (ed.) , ''Global analysis (papers in honor of K. Kodaira)'' , Princeton Univ. Press (1969) pp. 325–339 {{MR|0491719}} {{MR|0491720}} {{ZBL|}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> E. Bombieri, D. Mumford, "Enriques' classification of surfaces in char <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550178.png" />. II" W.L. Baily jr. (ed.) T. Shioda (ed.) , ''Complex Analysis and Algebraic geometry'' , Cambridge Univ. Press & Iwanami Shoten (1977) pp. 23–42 {{MR|0491719}} {{MR|0491720}} {{ZBL|}} </TD></TR></table> |
− | | |
− | | |
| | | |
| ====Comments==== | | ====Comments==== |
| A secondary Kodaira surface is a surface other than a primary one, admitting a primary Kodaira surface as an unramified covering. They are elliptic fibrations over a rational curve. Their first Betti number is 1. | | A secondary Kodaira surface is a surface other than a primary one, admitting a primary Kodaira surface as an unramified covering. They are elliptic fibrations over a rational curve. Their first Betti number is 1. |
| | | |
− | The canonical dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550179.png" /> mentioned at the start of the section on classification of algebraic elliptic surfaces is the [[Kodaira dimension|Kodaira dimension]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550180.png" /> (with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550181.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035550/e035550182.png" />). | + | The canonical dimension $ k ( X) $ |
| + | mentioned at the start of the section on classification of algebraic elliptic surfaces is the [[Kodaira dimension|Kodaira dimension]] $ \mathop{\rm Kod} ( X) $ (with $ k ( X) = - 1 $ |
| + | if $ \mathop{\rm Kod} ( X) = - \infty $). |
| | | |
| ====References==== | | ====References==== |
− | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> W. Barth, C. Peters, A. van der Ven, "Compact complex surfaces" , Springer (1984)</TD></TR></table> | + | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> W. Barth, C. Peters, A. van der Ven, "Compact complex surfaces" , Springer (1984) {{MR|0749574}} {{ZBL|0718.14023}} </TD></TR></table> |
An algebraic or analytic complete non-singular surface $ X $
having a fibration of elliptic curves (cf. Elliptic curve), that is, a morphism $ \pi : X \rightarrow B $
onto a non-singular curve $ B $
whose generic fibre is a non-singular elliptic curve. Every elliptic surface is birationally (bimeromorphically) equivalent over $ B $
to a unique minimal model, which is characterized by the fact that the fibre of $ \pi $
does not contain exceptional curves of arithmetic genus 1. In what follows an elliptic surface is assumed to be minimal. Minimal elliptic surfaces have a more complicated structure than ruled surfaces. They can have singular fibres $ X _ {t} = \pi ^ {- 1} ( t) $ (that is, fibres that are not non-singular elliptic curves). There is a classification
of the singular fibres of elliptic surfaces. A singular fibre $ X _ {t} = \sum n _ {i} E _ {i} $
is called multiple if the greatest common divisor of the $ n _ {i} $
is $ m \geq 2 $,
and then $ X _ {t} = m F $
and $ m $
is called the multiplicity of the fibre $ X _ {t} $.
On a minimal elliptic surface the canonical class $ K _ {X} $
contains a divisor that is a rational combination of fibres, in particular, $ ( K _ {X} ^ {2} ) = 0 $.
Moreover, the following formula holds for the canonical class (see [1], ):
$$
K _ {X} = \pi ^ {*} ( K _ {B} - d ) +
\sum ( m _ {i} - 1 ) F _ {i} ,
$$
where $ X _ {t _ {i} } = m _ {i} F _ {i} $
are all the multiple fibres of $ \pi $
and $ d $
is a divisor on $ B $
of degree $ - \chi ( {\mathcal O} _ {X} ) $.
The topological Euler characteristic satisfies the formula
$$
e ( X) = \sum e ( X _ {t _ {i} } ) .
$$
The classification of elliptic fibrations.
A fibration $ \pi : X \rightarrow B $
can be regarded as an elliptic curve over the function field $ k ( B) $.
This curve, generally speaking, does not have the structure of an Abelian variety over $ k ( B) $.
For this to happen it is necessary that it has a rational point over $ k ( B) $ (and then $ X $
is birationally isomorphic to the surface defined in $ B \times A ^ {2} $
by the Weierstrass equation $ y ^ {2} = x ^ {3} - g _ {2} x - g _ {3} $,
where $ g _ {2} , g _ {3} \in k ( B) $).
The specification of a rational point is equivalent to that of a section $ e : B \rightarrow X $
such that $ \pi e = \mathop{\rm id} $;
a necessary condition for the existence of a section is the absence of multiple fibres. Fibrations without multiple fibres are called reduced. Every fibration, after a suitable ramified covering of the base, has a section (that is, is reduced) . Every fibration can also be made reduced by a sequence of transformations inverse to logarithmic ones
— local surgery of the fibration in neighbourhoods of fibres.
Reduced elliptic fibrations may be described as follows. To every such fibration $ \pi : X \rightarrow B $
corresponds a unique fibration $ {\mathcal J} _ {B} ( X) \rightarrow B $
that is a group object and is such that $ X / B $
is a principal homogeneous space over $ {\mathcal J} _ {B} ( X) / B $;
$ {\mathcal J} _ {B} ( X) / B $
is the Jacobi fibration for $ X / B $;
it characterizes the existence of a section. For a given Jacobi fibration $ {\mathcal J} / B $,
the set $ I ( {\mathcal J} / B ) $
of isomorphism classes of fibrations $ X / B $
for which $ {\mathcal J} _ {B} ( X) \cong J $
has a cohomology description analogous to that of invertible sheaves (cf. Invertible sheaf). Here the role of $ {\mathcal O} _ {B} ^ {*} $
is played by the sheaf $ {\mathcal H} ^ {0} ( {\mathcal J} / B ) $
of local sections $ \tau : {\mathcal J} \rightarrow B $.
There is a natural one-to-one correspondence
$$
\theta : I ( {\mathcal J} / B ) \rightarrow H ^ {1}
( B , {\mathcal H} ^ {0} ( {\mathcal J} / B ) ) ,
$$
under which the Jacobi fibration corresponds to the zero element. By means of $ \theta $
one can distinguish between algebraic and non-algebraic fibrations: For a reduced fibration $ \pi : X \rightarrow B $
the surface $ X $
is algebraic if and only if the element corresponding to it in $ H ^ {1} ( B , {\mathcal H} ^ {0} ( {\mathcal J} / B )) $
is of finite order. The analogy with invertible sheaves can be pursued further. The analogue of the exact sequence
$$
0 \rightarrow \mathbf Z \rightarrow {\mathcal O} _ {B} \mathop \rightarrow \limits ^ {\mathop {\rm exp}} {\mathcal O} _ {B} ^ {*} \rightarrow 1
$$
is the exact sequence
$$
0 \rightarrow R ^ {1} \tau _ {0} \mathbf Z \rightarrow {\mathcal H} ^ {0}
( T ( {\mathcal J} ) / B ) \rightarrow {\mathcal H} ^ {0} ( {\mathcal J} / B ) \rightarrow 0 ,
$$
where $ {\mathcal H} ^ {0} ( T ( {\mathcal J} ) / B ) $
is the sheaf of local sections of the bundle $ T ( {\mathcal J} )/ B $
and $ T ( {\mathcal J} ) $
is the tangent space to the fibre $ \tau ^ {- 1 }( b) $
at $ e ( b) $.
The boundary homomorphism
$$
\delta : H ^ {1} ( B , {\mathcal H} ^ {0}
( {\mathcal J} / B ) ) \rightarrow H ^ {2} ( B , R ^ {1} \tau _ {*} \mathbf Z )
$$
allows one to recognize when one fibration is a deformation of another. For that it is necessary and sufficient that the elements corresponding to these fibrations have one and the same image under $ \delta $ (see ).
The classification of algebraic elliptic surfaces.
Suppose that $ \mathop{\rm char} k = 0 $.
For an elliptic surface $ X $
the canonical dimension $ k ( X) \leq 1 $,
that is, it is equal to $ - 1 $,
0 or $ 1 $.
If $ k ( X) = 1 $,
$ X $
is said to be an elliptic surface of general type. These are characterized by the conditions $ 12 K _ {X} \neq 0 $
and $ | 12 K _ {X} | \neq \emptyset $.
Elliptic surfaces with $ p _ {g} \geq 2 $
or, more generally, with $ P _ {m} \geq 2 $
for some $ m $,
are of general type.
Elliptic surfaces with $ k ( X) = 0 $
are characterized by the condition $ 12 K _ {X} = 0 $.
In this case $ \chi ( {\mathcal O} _ {X} ) $
can take the three values 2, 1 or $ 0 $.
If $ \chi ( {\mathcal O} _ {X} )= 2 $,
then $ X $
is an elliptic $ K 3 $-surface ( $ q = 0 $,
$ K _ {X} = 0 $).
In this case $ B $
is isomorphic to the projective line $ P ^ {1} $,
the fibration has no multiple fibres and $ X $
has the invariants $ p _ {g} = 1 $,
$ q = 0 $,
$ b _ {2} = 22 $.
If $ \chi ( {\mathcal O} _ {X} ) = 1 $,
then $ X $
is an Enriques surface, that is, a surface with $ p _ {g} = q = 0 $,
$ 2 K _ {X} = 0 $.
(Every Enriques surface is elliptic.) In this case $ B \simeq P ^ {1} $,
the fibration has two fibres of multiplicity 2, and $ X $
has the invariants $ p _ {g} = q = 0 $,
$ b _ {2} = 10 $.
If $ \chi ( {\mathcal O} _ {X} ) = 0 $,
then two cases are possible. Either $ X $
is an Abelian variety (and then $ p _ {g} = 1 $,
$ q = 2 $,
$ b _ {2} = 6 $);
or $ X $
is a hyper-elliptic surface, that is, a surface that has a finite unramified covering — the product of two elliptic curves. In that case $ p _ {g} = 0 $,
$ b _ {1} = 2 $,
$ b _ {2} = 2 $,
$ B = P ^ {1} $,
and $ \pi $
has 3 or 4 multiple fibres with four possibilities for their multiplicity: $ ( 3 , 3 , 3 ) $,
$ ( 2 , 4 , 4 ) $,
$ ( 2 , 3 , 6 ) $,
and $ ( 2 , 2 , 2 , 2 ) $,
and $ 3 K _ {X} = 0 $,
$ 4 K _ {X} = 0 $,
$ 6 K _ {X} = 0 $,
and $ 2 K _ {X} = 0 $,
respectively.
An elliptic surface with $ k ( X) = - 1 $
is ruled (cf. Ruled surface). It is characterized by the condition $ | 12 K _ {X} | = \emptyset $.
Here two cases are possible: 1) $ X $
is a surface with $ p _ {g} = q = 0 $,
$ b _ {2} = 10 $,
and $ \pi $
has no multiple fibres or one; moreover, a surface without multiple fibres can be obtained as follows: one has to take a rational mapping $ P ^ {2} \rightarrow P ^ {1} $
determined by two cubics $ F _ {0} $
and $ F _ {1} $
and blow up their 9 points of intersection; or 2) $ X $
is a surface with $ p _ {g} = 0 $,
$ q = 1 $,
$ b _ {2} = 2 $,
and the multiplicities $ m _ {i} $
are subject to the inequality
$$
\sum \left ( 1 -
\frac{1}{m _ {i} }
\right ) < 2 .
$$
The formula for the canonical class and the classification of elliptic surfaces can also be generalized to the case of a field of finite characteristic [5], [6].
The classification of non-algebraic elliptic surfaces.
The classification of non-algebraic elliptic surfaces. For non-algebraic surfaces the algebraic dimension $ a ( X) = \mathop{\rm tr} \mathop{\rm deg} M ( X) $
is 1 or 0. If $ a ( X) = 0 $,
then $ X $
is non-elliptic. All surfaces with $ a ( X) = 1 $
are elliptic. Here the structure of $ \pi : X \rightarrow B $
is determined almost canonically: Every such fibration $ \pi $
is necessarily elliptic. The classification by the canonical dimension is precisely the same as for algebraic elliptic surfaces: $ k ( X) = - 1 $
$ \iff $
$ | 12 K _ {X} | = \emptyset $;
$ k ( X) = 0 $
$ \iff $
$ 12K _ {X} = 0 $;
and $ k ( X) = 1 $ ($ X $
is of basic type) $ \iff $
$ | 12 K _ {X} | = \emptyset $,
$ 12 K _ {X} \neq 0 $.
Non-algebraic elliptic surfaces with $ k ( X) = 0 $
belong to one of the following classes: 1) the $ K 3 $-surfaces ( $ \chi ( {\mathcal O} _ {X} )= 2 $,
$ b _ {1} = 0 $,
$ b _ {2} = 22 $,
$ X $
is simply connected); 2) the complex tori ( $ K _ {X} = 0 $,
$ \chi ( {\mathcal O} _ {X} ) = 0 $,
$ b _ {1} = 4 $,
$ b _ {2} = 6 $);
3) the Kodaira surfaces ( $ K _ {X} = 0 $,
$ \chi ( {\mathcal O} _ {X} ) = 0 $,
$ b _ {1} = 3 $,
$ b _ {2} = 4 $).
Primary Kodaira surfaces are holomorphically locally trivial fibrations over an elliptic curve with an elliptic curve as typical fibre, and from the point of view of differentiability, bundles over a $ 3 $-dimensional torus with a circle as fibre; or 4) the surfaces with $ \chi ( {\mathcal O} _ {X} ) = 0 $,
$ p _ {g} = 0 $,
$ b _ {1} = 1 $,
$ b _ {2} = 0 $.
For them $ m K _ {X} = 0 $
with $ m = 2 , 3 , 4 $,
(analogous to hyper-elliptic surfaces). They have Kodaira surfaces as finite unramified coverings. In the cases 2), 3) and 4) $ \mathbf C ^ {2} $
is the universal covering of $ X $.
Non-algebraic elliptic surfaces with $ k ( X) = - 1 $
are Hopf surfaces, that is, their universal covering is $ \mathbf C ^ {2} \setminus 0 $.
For them $ \chi ( {\mathcal O} _ {X} ) = 0 $,
$ b _ {1} = 1 $,
$ b _ {2} = 0 $.
The proper Hopf surfaces are $ ( \mathbf C ^ {2} \setminus 0 ) / T $,
where $ T ( z _ {1} , z _ {2} ) = ( \alpha _ {1} z _ {1} , \alpha _ {2} z _ {2} ) $
are real generators of $ T $.
They are homeomorphic to $ S ^ {1} \times S ^ {2} $
and are characterized by this property. Arbitrary elliptic Hopf surfaces are quotients of proper Hopf surfaces .
References
[1] | "Algebraic surfaces" Proc. Steklov Inst. Math. , 75 (1967) Trudy Mat. Inst. Steklov. , 75 (1965) Zbl 0172.37901 Zbl 0153.22401 |
[2] | D. Husemoller, "Classification and embeddings of surfaces" R. Hartshorne (ed.) , Algebraic geometry (Arcata, 1974) , Proc. Symp. Pure Math. , 29 , Amer. Math. Soc. (1982) pp. 329–420 MR0506292 Zbl 0326.14009 |
[3a] | K. Kodaira, "On compact complex analytic surfaces I" Ann. of Math. (2) , 71 (10) pp. 111–152 MR0132556 Zbl 0137.17501 Zbl 0098.13004 |
[3b] | K. Kodaira, "On compact complex analytic surfaces II" Ann. of Math. (2) , 77 (1963) pp. 563–626 MR165541 Zbl 0193.37701 Zbl 0133.16505 |
[3c] | K. Kodaira, "On compact complex analytic surfaces III" Ann. of Math. (2) , 78 (1963) pp. 1–40 MR165541 Zbl 0193.37701 |
[4a] | K. Kodaira, "On the structure of compact complex analytic surfaces I" Amer. J. Math. , 86 (1964) pp. 751–798 MR0187255 Zbl 0137.17501 |
[4b] | K. Kodaira, "On the structure of compact complex analytic surfaces II" Amer. J. Math. , 88 (1966) pp. 682–721 MR0205280 Zbl 0193.37701 |
[4c] | K. Kodaira, "On the structure of compact complex analytic surfaces III" Amer. J. Math. , 90 (1968) pp. 55–83 MR0228019 Zbl 0193.37701 |
[4d] | K. Kodaira, "On the structure of compact complex analytic surfaces IV" Amer. J. Math. , 90 (1968) pp. 1048–1066 MR239114 |
[5] | D. Mumford, "Enriques' classification of surfaces in char . I" D.C. Spencer (ed.) S. Iyanaga (ed.) , Global analysis (papers in honor of K. Kodaira) , Princeton Univ. Press (1969) pp. 325–339 MR0491719 MR0491720 |
[6] | E. Bombieri, D. Mumford, "Enriques' classification of surfaces in char . II" W.L. Baily jr. (ed.) T. Shioda (ed.) , Complex Analysis and Algebraic geometry , Cambridge Univ. Press & Iwanami Shoten (1977) pp. 23–42 MR0491719 MR0491720 |
A secondary Kodaira surface is a surface other than a primary one, admitting a primary Kodaira surface as an unramified covering. They are elliptic fibrations over a rational curve. Their first Betti number is 1.
The canonical dimension $ k ( X) $
mentioned at the start of the section on classification of algebraic elliptic surfaces is the Kodaira dimension $ \mathop{\rm Kod} ( X) $ (with $ k ( X) = - 1 $
if $ \mathop{\rm Kod} ( X) = - \infty $).
References