Inmathematics, anelliptic surface is a surface that has an ellipticfibration, in other words aproper morphism with connectedfibers to analgebraic curve such that almost all fibers aresmooth curves ofgenus 1. (Over an algebraically closed field such as thecomplex numbers, these fibers areelliptic curves, perhaps without a chosen origin.) This is equivalent to thegeneric fiber being a smooth curve of genus one. This follows fromproper base change.

The surface and the base curve are assumed to be non-singular (complex manifolds orregular schemes, depending on the context). The fibers that are not elliptic curves are called thesingular fibers and were classified byKunihiko Kodaira. Both elliptic and singular fibers are important instring theory, especially inF-theory.

Elliptic surfaces form a large class of surfaces that contains many of the interesting examples of surfaces, and are relatively well understood in the theories of complex manifolds andsmooth4-manifolds. They are similar to (have analogies with, that is), elliptic curves overnumber fields.

• The product of any elliptic curve with any curve is an elliptic surface (with no singular fibers).
• All surfaces ofKodaira dimension 1 are elliptic surfaces.
• Every complexEnriques surface is elliptic, and has an elliptic fibration over the projective line.
• Kodaira surfaces
• Dolgachev surfaces
• Shioda modular surfaces

Most of the fibers of an elliptic fibration are (non-singular) elliptic curves. The remaining fibers are called singular fibers: there are a finite number of them, and each one consists of a union of rational curves, possibly with singularities or non-zero multiplicities (so the fibers may be non-reduced schemes). Kodaira and Néron independently classified the possible fibers, andTate's algorithm can be used to find the type of the fibers of an elliptic curve over a number field.

The following table lists the possible fibers of aminimal elliptic fibration. ("Minimal" means roughly one that cannot be factored through a "smaller" one; precisely, the singular fibers should contain no smooth rational curves with self-intersection number −1.) It gives:

• Kodaira's symbol for the fiber,
• André Néron's symbol for the fiber,
• The number of irreducible components of the fiber (all rational except for type I₀)
• The intersection matrix of the components. This is either a 1×1zero matrix, or anaffine Cartan matrix, whoseDynkin diagram is given.
• The multiplicities of each fiber are indicated in the Dynkin diagram.

Kodaira	Néron	Components	Intersection matrix	Dynkin diagram	Fiber
I0	A	1 (elliptic)	0
I1	B1	1 (with double point)	0
I2	B2	2 (2 distinct intersection points)	affine A1
Iv (v≥2)	Bv	v (v distinct intersection points)	affine Av-1
mIv (v≥0,m≥2)		Iv with multiplicitym
II	C1	1 (with cusp)	0
III	C2	2 (meet at one point of order 2)	affine A1
IV	C3	3 (all meet in 1 point)	affine A2
I0*	C4	5	affine D4
Iv* (v≥1)	C5,v	5+v	affine D4+v
IV*	C6	7	affine E6
III*	C7	8	affine E7
II*	C8	9	affine E8

This table can be found as follows. Geometric arguments show that the intersection matrix of the components of the fiber must be negative semidefinite, connected, symmetric, and have no diagonal entries equal to −1 (by minimality). Such a matrix must be 0 or a multiple of the Cartan matrix of an affine Dynkin diagram of typeADE.

The intersection matrix determines the fiber type with three exceptions:

• If the intersection matrix is 0 the fiber can be either an elliptic curve (type I₀), or have a double point (type I₁), or a cusp (type II).
• If the intersection matrix is affine A₁, there are 2 components with intersection multiplicity 2. They can meet either in 2 points with order 1 (type I₂), or at one point with order 2 (type III).
• If the intersection matrix is affine A₂, there are 3 components each meeting the other two. They can meet either in pairs at 3 distinct points (type I₃), or all meet at the same point (type IV).

Themonodromy around each singular fiber is a well-definedconjugacy class in the group SL(2,Z) of 2 × 2 integer matrices withdeterminant 1. The monodromy describes the way the firsthomology group of a smooth fiber (which is isomorphic toZ²) changes as we go around a singular fiber. Representatives for these conjugacy classes associated to singular fibers are given by:^[1]

Fiber	Intersection matrix	Monodromy	j-invariant	Group structure on smooth locus
Iν	affine Aν-1		${\displaystyle \mathrm{\infty }}$	${\displaystyle \mathbf{Z}/\nu \times {\mathbf{C}}^{\ast }}$
II	0		0	${\displaystyle \mathbf{C}}$
III	affine A1		1728	${\displaystyle \mathbf{Z}/2\times \mathbf{C}}$
IV	affine A2		0	${\displaystyle \mathbf{Z}/3\times \mathbf{C}}$
I0*	affine D4		in${\displaystyle \mathbf{C}}$	${\displaystyle (\mathbf{Z}/2{)}^2\times \mathbf{C}}$
Iν* (ν≥1)	affine D4+ν		${\displaystyle \mathrm{\infty }}$	${\displaystyle (\mathbf{Z}/2{)}^2\times \mathbf{C}}$ if ν is even,${\displaystyle \mathbf{Z}/4\times \mathbf{C}}$ if ν is odd
IV*	affine E6		0	${\displaystyle \mathbf{Z}/3\times \mathbf{C}}$
III*	affine E7		1728	${\displaystyle \mathbf{Z}/2\times \mathbf{C}}$
II*	affine E8		0	${\displaystyle \mathbf{C}}$

For singular fibers of type II, III, IV, I₀^*, IV^*, III^*, or II^*, the monodromy has finite order in SL(2,Z). This reflects the fact that an elliptic fibration haspotential good reduction at such a fiber. That is, after a ramified finite covering of the base curve, the singular fiber can be replaced by a smooth elliptic curve. Which smooth curve appears is described by thej-invariant in the table. Over the complex numbers, the curve withj-invariant 0 is the unique elliptic curve with automorphism group of order 6, and the curve withj-invariant 1728 is the unique elliptic curve with automorphism group of order 4. (All other elliptic curves have automorphism group of order 2.)

For an elliptic fibration with asection, called aJacobian elliptic fibration, the smooth locus of each fiber has a group structure. For singular fibers, this group structure on the smooth locus is described in the table, assuming for convenience that the base field is the complex numbers. (For a singular fiber with intersection matrix given by an affine Dynkin diagram${\displaystyle \tilde{\mathrm{\Gamma }}}$, the group of components of the smooth locus is isomorphic to the center of the simply connected simple Lie group with Dynkin diagram${\displaystyle \mathrm{\Gamma }}$, as listedhere.) Knowing the group structure of the singular fibers is useful for computing theMordell-Weil group of an elliptic fibration (the group of sections), in particular its torsion subgroup.

To understand how elliptic surfaces fit into theclassification of surfaces, it is important to compute thecanonical bundle of a minimal elliptic surfacef:X →S. Over the complex numbers, Kodaira proved the followingcanonical bundle formula:^[2]

${\displaystyle K_X=f^{\ast }(L)\otimes O_S(\sum _i(m_i-1)D_i).}$

Here the multiple fibers off (if any) are written as${\displaystyle f^{\ast }(p_i)=m_i{D}_i}$, for an integerm_i at least 2 and a divisorD_i whose coefficients have greatest common divisor equal to 1, andL is some line bundle on the smooth curveS. IfS is projective (or equivalently, compact), then thedegree ofL is determined by theholomorphic Euler characteristics ofX andS: deg(L) = χ(X,O_X) − 2χ(S,O_S). The canonical bundle formula implies thatK_X isQ-linearly equivalent to the pullback of someQ-divisor onS; it is essential here that the elliptic surfaceX →S is minimal.

Building on work ofKenji Ueno, Takao Fujita (1986) gave a useful variant of the canonical bundle formula, showing howK_X depends on the variation of the smooth fibers.^[3] Namely, there is aQ-linear equivalence

${\displaystyle K_X{\sim }_{\mathbf{Q}}{f}^{\ast }(K_S+B_S+M_S),}$

where thediscriminant divisorB_S is an explicit effectiveQ-divisor onS associated to the singular fibers off, and themoduli divisorM_S is${\displaystyle (1/12)j^{\ast }O(1)}$, wherej:S →P¹ is the function giving thej-invariant of the smooth fibers. (ThusM_S is aQ-linear equivalence class ofQ-divisors, using the identification between thedivisor class group Cl(S) and thePicard group Pic(S).) In particular, forS projective, the moduli divisorM_S has nonnegative degree, and it has degree zero if and only if the elliptic surface is isotrivial, meaning that all the smooth fibers are isomorphic.

The discriminant divisor in Fujita's formula is defined by

${\displaystyle B_S=\sum _{p\in S}(1-c(p))[p]}$,

wherec(p) is thelog canonical threshold${\displaystyle \text{lct}(X,f^{\ast }(p))}$. This is an explicit rational number between 0 and 1, depending on the type of singular fiber. Explicitly, the lct is 1 for a smooth fiber or type${\displaystyle I_{\nu }}$, and it is 1/m for a multiple fiber${\displaystyle {}_m{I}_{\nu }}$, 1/2 for${\displaystyle I_{\nu }^{\ast }}$, 5/6 for II, 3/4 for III, 2/3 for IV, 1/3 for IV*, 1/4 for III*, and 1/6 for II*.

The canonical bundle formula (in Fujita's form) has been generalized byYujiro Kawamata and others to families ofCalabi–Yau varieties of any dimension.^[4]

Alogarithmic transformation (of orderm with centerp) of an elliptic surface or fibration turns a fiber of multiplicity 1 over a pointp of the base space into a fiber of multiplicitym. It can be reversed, so fibers of high multiplicity can all be turned into fibers of multiplicity 1, and this can be used to eliminate all multiple fibers.

Logarithmic transformations can be quite violent: they can change the Kodaira dimension, and can turn algebraic surfaces into non-algebraic surfaces.

Example:LetL be the latticeZ+iZ ofC, and letE be the elliptic curveC/L. Then the projection map fromE×C toC is an elliptic fibration. We will show how to replace the fiber over 0 with a fiber of multiplicity 2.

There is an automorphism ofE×C of order 2 that maps (c,s) to (c+1/2,−s). We letX be the quotient ofE×C by this group action. We makeX into a fiber space overC by mapping (c,s) tos². We construct an isomorphism fromX minus the fiber over 0 toE×C minus the fiber over 0 by mapping (c,s) to (c-log(s)/2πi,s²). (The two fibers over 0 are non-isomorphic elliptic curves, so the fibrationX is certainly not isomorphic to the fibrationE×C over all ofC.)

Then the fibrationX has a fiber of multiplicity 2 over 0, and otherwise looks likeE×C. We say thatX is obtained by applying a logarithmic transformation of order 2 toE×C with center 0.

• Enriques–Kodaira classification
• Néron minimal model

• Barth, Wolf P.;Hulek, Klaus; Peters, Chris A.M.; Van de Ven, Antonius (2004) [1984],Compact Complex Surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, vol. 4,Springer,doi:10.1007/978-3-642-57739-0,ISBN 978-3-540-00832-3,MR 2030225
• Cossec, François;Dolgachev, Igor (1989).Enriques Surfaces. Boston:Birkhäuser.doi:10.1007/978-1-4612-3696-2.ISBN 3-7643-3417-7.MR 0986969.
• Kodaira, Kunihiko (1963). "On compact analytic surfaces. II".Ann. of Math.77 (3):563–626.doi:10.2307/1970131.JSTOR 1970131.MR 0184257.Zbl 0118.15802.
• Kodaira, Kunihiko (1964). "On the structure of compact complex analytic surfaces. I".Am. J. Math.86 (4):751–798.doi:10.2307/2373157.JSTOR 2373157.MR 0187255.Zbl 0137.17501.
• Kollár, János (2007), "Kodaira's canonical bundle formula and adjunction",Flips for 3-folds and 4-folds,Oxford University Press, pp. 134–162,doi:10.1093/acprof:oso/9780198570615.003.0008,ISBN 978-0-19-857061-5,MR 2359346
• Néron, André (1964)."Modèles minimaux des variétés abéliennes sur les corps locaux et globaux".Publications Mathématiques de l'IHÉS (in French).21:5–128.doi:10.1007/BF02684271.MR 0179172.Zbl 0132.41403.

• Complex surfaces
• Birational geometry
• Algebraic surfaces
• Mathematical classification systems

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