Difference between revisions of "Elliptic surface"
(Importing text file) |
Ulf Rehmann (talk | contribs) m (MR/ZBL numbers added) |
||
Line 53: | Line 53: | ||
====References==== | ====References==== | ||
− | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> | + | <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> |
Line 63: | Line 63: | ||
====References==== | ====References==== | ||
− | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> | + | <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> |
Revision as of 21:52, 30 March 2012
An algebraic or analytic complete non-singular surface having a fibration of elliptic curves (cf. Elliptic curve), that is, a morphism
onto a non-singular curve
whose generic fibre is a non-singular elliptic curve. Every elliptic surface is birationally (bimeromorphically) equivalent over
to a unique minimal model, which is characterized by the fact that the fibre of
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
(that is, fibres that are not non-singular elliptic curves). There is a classification
of the singular fibres of elliptic surfaces. A singular fibre is called multiple if the greatest common divisor of the
is
, and then
and
is called the multiplicity of the fibre
.
On a minimal elliptic surface the canonical class contains a divisor that is a rational combination of fibres, in particular,
. Moreover, the following formula holds for the canonical class (see [1], ):
![]() |
where are all the multiple fibres of
and
is a divisor on
of degree
. The topological Euler characteristic satisfies the formula
![]() |
The classification of elliptic fibrations.
A fibration can be regarded as an elliptic curve over the function field
. This curve, generally speaking, does not have the structure of an Abelian variety over
. For this to happen it is necessary that it has a rational point over
(and then
is birationally isomorphic to the surface defined in
by the Weierstrass equation
, where
). The specification of a rational point is equivalent to that of a section
such that
; 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 corresponds a unique fibration
that is a group object and is such that
is a principal homogeneous space over
;
is the Jacobi fibration for
; it characterizes the existence of a section. For a given Jacobi fibration
, the set
of isomorphism classes of fibrations
for which
has a cohomology description analogous to that of invertible sheaves (cf. Invertible sheaf). Here the role of
is played by the sheaf
of local sections
. There is a natural one-to-one correspondence
![]() |
under which the Jacobi fibration corresponds to the zero element. By means of one can distinguish between algebraic and non-algebraic fibrations: For a reduced fibration
the surface
is algebraic if and only if the element corresponding to it in
is of finite order. The analogy with invertible sheaves can be pursued further. The analogue of the exact sequence
![]() |
is the exact sequence
![]() |
where is the sheaf of local sections of the bundle
and
is the tangent space to the fibre
at
. The boundary homomorphism
![]() |
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 (see ).
The classification of algebraic elliptic surfaces.
Suppose that . For an elliptic surface
the canonical dimension
, that is, it is equal to
, 0 or
. If
,
is said to be an elliptic surface of general type. These are characterized by the conditions
and
. Elliptic surfaces with
or, more generally, with
for some
, are of general type.
Elliptic surfaces with are characterized by the condition
. In this case
can take the three values 2, 1 or
. If
, then
is an elliptic
-surface (
,
). In this case
is isomorphic to the projective line
, the fibration has no multiple fibres and
has the invariants
,
,
. If
, then
is an Enriques surface, that is, a surface with
,
. (Every Enriques surface is elliptic.) In this case
, the fibration has two fibres of multiplicity 2, and
has the invariants
,
. If
, then two cases are possible. Either
is an Abelian variety (and then
,
,
); or
is a hyper-elliptic surface, that is, a surface that has a finite unramified covering — the product of two elliptic curves. In that case
,
,
,
, and
has 3 or 4 multiple fibres with four possibilities for their multiplicity:
,
,
, and
, and
,
,
, and
, respectively.
An elliptic surface with is ruled (cf. Ruled surface). It is characterized by the condition
. Here two cases are possible: 1)
is a surface with
,
, and
has no multiple fibres or one; moreover, a surface without multiple fibres can be obtained as follows: one has to take a rational mapping
determined by two cubics
and
and blow up their 9 points of intersection; or 2)
is a surface with
,
,
, and the multiplicities
are subject to the inequality
![]() |
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 is 1 or 0. If
, then
is non-elliptic. All surfaces with
are elliptic. Here the structure of
is determined almost canonically: Every such fibration
is necessarily elliptic. The classification by the canonical dimension is precisely the same as for algebraic elliptic surfaces:
;
; and
(
is of basic type)
,
.
Non-algebraic elliptic surfaces with belong to one of the following classes: 1) the
-surfaces (
,
,
,
is simply connected); 2) the complex tori (
,
,
,
); 3) the Kodaira surfaces (
,
,
,
). 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
-dimensional torus with a circle as fibre; or 4) the surfaces with
,
,
,
. For them
with
, (analogous to hyper-elliptic surfaces). They have Kodaira surfaces as finite unramified coverings. In the cases 2), 3) and 4)
is the universal covering of
.
Non-algebraic elliptic surfaces with are Hopf surfaces, that is, their universal covering is
. For them
,
,
. The proper Hopf surfaces are
, where
are real generators of
. They are homeomorphic to
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 ![]() |
[6] | E. Bombieri, D. Mumford, "Enriques' classification of surfaces in char ![]() |
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.
The canonical dimension mentioned at the start of the section on classification of algebraic elliptic surfaces is the Kodaira dimension
(with
if
).
References
[a1] | W. Barth, C. Peters, A. van der Ven, "Compact complex surfaces" , Springer (1984) MR0749574 Zbl 0718.14023 |
Elliptic surface. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Elliptic_surface&oldid=17304