# Elliptic surface

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 .

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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 $\mathop{\rm Kod} ( X) = - \infty$).