Cent. Eur. J. Math. • 12(12) • 2014 • 1772-1795
DOI: 10.2478/s11533-014-0446-6
Central European Journal of Mathematics
Automorphism groups of rational elliptic surfaces
with section and constant J-map
Research Article
Tolga Karayayla1∗
1 Department of Mathematics, Middle East Technical University, Cankaya, 06800, Ankara, Turkey
Received 22 November 2013; accepted 13 February 2014
Abstract: In this paper, the automorphism groups of relatively minimal rational elliptic surfaces with section which have
constant J-maps are classified. The ground field is C. The automorphism group of such a surface β : B → P1 ,
denoted by Aut(B), consists of all biholomorphic maps on the complex manifold B. The group Aut(B) is isomorphic
to the semi-direct product MW (B)oAutσ (B) of the Mordell-Weil group MW (B) (the group of sections of B), and the
subgroup Autσ (B) of the automorphisms preserving a fixed section σ of B which is called the zero section on B.
The Mordell-Weil group MW (B) is determined by the configuration of singular fibers on the elliptic surface B due
to Oguiso and Shioda [9]. In this work, the subgroup Autσ (B) is determined with respect to the configuration of
singular fibers of B. Together with a previous paper [4] where the case with non-constant J-maps was considered,
this completes the classification of automorphism groups of relatively minimal rational elliptic surfaces with section.
MSC:
14J50, 14J27, 14J26
Keywords: Elliptic surface • Rational elliptic surface • Automorphism group • Mordell-Weil group • J map • Singular fiber
© Versita Sp. z o.o.
1.
Introduction
The goal of this study is to give a description of the automorphisms, and the structure of the automorphism groups, of
relatively minimal rational elliptic surfaces with section over the field C. For such a surface B, Aut(B) denotes the group
of regular isomorphisms on B, or equivalently the group of biholomorphic maps on the complex surface B. Note that by
Aut(B) we do not mean the birational automorphism group of B.
The motivation for this study comes from the work of Bouchard and Donagi [2], On a Class of Non-simply Connected
Calabi-Yau 3-folds. In that paper, they obtain non-simply connected Calabi-Yau threefolds as the quotients of Schoen
threefolds (which are the fibered products of two rational elliptic surfaces) by free actions of finite abelian groups. This
is done by studying pairs of automorphisms of rational elliptic surfaces that induce a free action on the fibered product of
∗
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E-mail: [email protected]
T. Karayayla
the elliptic surfaces. The classification of such pairs of automorphisms is given in their paper, and the general question of
classifying all automorphisms of rational elliptic surfaces is asked. Achieving this more general classification can extend
the results to non-simply connected Calabi-Yau threefolds with finite non-abelian fundamental groups. Non-simply
connected Calabi-Yau threefolds have a significance in string theory, particularly in the study of finding low energy
limits of the theory.
Given a relatively minimal rational elliptic surface β : B → P1 with section, fixing a section σ of B and declaring it
as the zero section, the generic fiber of β is an elliptic curve (whose zero point is the intersection of that fiber with the
zero section σ ), and there are singular fibers which are of one of the Kodaira types In (n > 0), In∗ (n ≥ 0), II, III, IV , II ∗ ,
III ∗ or IV ∗ (described in the next section). These singular fibers play an important role in determining the automorphism
group Aut(B) of the surface B since singular fibers are permuted under automorphisms of B. To study how Aut(B) is
affected by the singular fibers of B, we use the work of Persson [10] and Miranda [6] where they produce the list of all
possible configurations of singular fibers on relatively minimal rational elliptic surfaces with section. The Mordell-Weil
group MW (B) of an elliptic surface B with section is the group of sections of the elliptic surface, and it can be identified
with a subgroup of Aut(B) by defining the action of a section as follows: If ζ is a section of B, then let tζ ∈ Aut(B)
be the automorphism of B which acts on every smooth fiber F as the translation by ζ ∩ F determined by the group law
on the elliptic curve F . Oguiso and Shioda [9] have shown that MW (B) is determined by the configuration of singular
fibers on B for a relatively minimal rational elliptic surface B with section, and they have calculated MW (B) for each
configuration of singular fibers. MW (B) is a finitely generated abelian group of rank at most 8, and the order of its
torsion subgroup is at most 9.
In the previous work [4], it was proved that Aut(B) is the semi-direct product MW (B) o Autσ (B) of the Mordell-Weil
group of B and the subgroup Autσ (B) of the automorphisms of B which preserve the zero section σ of B (Theorem 3.0.1).
Together with the classification of MW (B) by Oguiso and Shioda [9], Theorem 6.0.1 in the previous work of the author
[4], where the subgroup Autσ (B) is listed with respect to the configuration of singular fibers on B, gives the classification
of Aut(B) for surfaces B which have non-constant J-maps.
In general, |Autσ (B)| = |Aut(B) : MW (B)| ≤ 24 if the J-map of the surface B is not constant. If the J-map of the surface
is constant, one may expect to have more symmetry, hence larger Autσ (B) groups since in this case all smooth fibers
are isomorphic as elliptic curves rather than having finitely many (degree of J) isomorphic elliptic curve fibers for each
different j-invariant as in the non-constant J-map case. As it is shown in this paper, it turns out that for the constant
J-map case, the order of the group Autσ (B) may be infinite and for those surfaces where Autσ (B) is finite, the order is
at most144. One other reason for having more symmetry if the J-map is constant is that we have automorphisms of B
which act on each smooth fiber as complex multiplication of order 4 when the J-map is equal to 1, and of order 6 when
the J-map is equal to 0.
The main result of this paper is stated in Theorem 2.1 and it lists all distinct Autσ (B) groups and the corresponding
configurations of singular fibers of B for relatively minimal rational elliptic surfaces B with section. The organization
of the paper is as follows. In Section 3 we give some background information for elliptic surfaces. In Section 4 we
begin discussing the rational elliptic surfaces with constant J-maps and some properties of their automorphism groups.
In this section we concentrate on the group AutB (P1 ) of the automorphisms induced on the base curve P1 , and we give a
list of candidates for these groups for each distinct configuration of singular fibers that can occur on a rational elliptic
surface with a constant J-map. In Section 5 we prove the existence of all finite Autσ (B) groups. This is done in several
steps. First, we discuss the existence of α ∈ Autσ (B) with specified orders in 5.1. Using some geometric constructions,
the existence of such automorphisms are proved in 5.2. After proving which of the candidate groups AutB (P1 ) do exist
for each distinct configuration of singular fibers in 5.3 and 5.4, we calculate the corresponding Autσ (B) groups in 5.5.
Section 6 discusses the cases where Autσ (B) is an infinite group separately. All the information obtained throughout
the paper is presented in advance in Section 2 in the form of a table as the main theorem of the paper.
2.
Results
We can collect all the information obtained throughout the paper in the following theorem.
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Automorphism groups of rational elliptic surfaces with section and constant J-map
Theorem 2.1.
For a relatively minimal rational elliptic surface B with a section σ and which has a constant J-map, the automorphism
group Aut(B) is isomorphic to the semi-direct product
Aut(B) = MW (B) o Autσ (B)
of the Mordell-Weil group MW(B) and the subgroup Autσ (B) of the automorphisms preserving the zero section σ . All
distinct cases that occur are listed in Table 1 where the configuration of singular fibers of B, the group Autσ (B) and the
group AutB (P1 ) of the induced automorphisms on P are shown. Together with the classification of MW(B) with respect
to the configuration of singular fibers of B (which is due to Oguiso and Shioda), this completes the classification of
Aut(B) for such surfaces.
Table 1.
The table of results
AutB (P1 ) Autσ (B)
0
Z/2Z
Configuration
Z/4Z
III 4
Z/6Z
I0∗ IV II, I0∗ II 3 , IV 2 II 2 , IV II 4 , II 6
Z/4Z × Z/2Z
I0∗ III 2 , III 4
Z/6Z × Z/2Z
IV ∗ II 2 , IV 2 II 2 , IV II 4 , II 6
Z/12Z
I0∗ II 3 , II 6
Z/6Z × Z/3Z
I0∗ II 3 , II 6
Z/18Z
IV II 4
Z/4Z
Z/6Z × Z/4Z
IV II 4
Z/5Z
Z/30Z
II 6
D2
Z/3Z × D4
IV 2 II 2 , II 6
G1
III 4
D3
Z/6Z × D3
IV 3 , II 6
D4
G2
III 4
D6
G3
II 6
A4
G4
II 6
G5
III 4
S4
G6
II 6
DC∗
(DC∗ × C M(B))/h(−z, −I)i I0∗ I0∗
Z/3Z
C∗
(C∗ × Z/4Z)/h(i, 1)i = C∗
III ∗ III
(C∗ × Z/6Z)/h(µ6 , 1)i = C∗ II ∗ II
C∗ × Z/2Z
IV ∗ IV
G1 = hA, R, S | A4 = R 2 = S 2 = 1, AR = RA, AS = SA, RSRS = A2 i
G2 = hR, S | R 4 = S 8 = 1, RS 2 = S 2 R, RSRS = 1i
G3 = hA, R, S | A6 = R 6 = S 2 = 1, AS = SA, AR = RA, RSRS = Ai
G4 = hA, R, S | A6 = R 3 = 1, S 2 = A3 , AR = RA, AS = SA, (RS)3 = 1i
G5 = hA, R, S | A4 = R 3 = S 2 = 1, AR = RA, AS = SA, (RS)3 = Ai
G6 = hA, R, S | A6 = R 2 = 1, S 4 = A, AR = RA, (SR)3 = 1i
DC∗ = {τ ∈ Aut(P1 )|τ = cz or τ = c/z for some c ∈ C∗ }
C M(B) is a cyclic group of order 6,4 or 2 depending on whether the J-map equals 0,1 or c ∈ C − {0, 1}, respectively.
The order 2 element of C M(B) is −I.
3.
Preliminaries
In this section, we give some basic information about elliptic surfaces relevant to the following sections. The reader can
consult the book [7] for more details.
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T. Karayayla
An elliptic surface is a smooth algebraic surface B together with a holomorphic map β : B → C to a smooth algebraic
curve C such that the generic fiber is a smooth genus 1 curve. If there is a section of the map β, then every smooth
genus 1 fiber has a marked point, hence it is an elliptic curve. An elliptic surface is relatively minimal if it has no smooth
rational (−1) curves in the fibers. By blowing down such (−1) curves, every elliptic surface can be transformed to a
birational relatively minimal elliptic surface. A rational elliptic surface is an elliptic surface birational to P2 . A rational
elliptic surface with section is necessarily fibered over the base curve C = P1 . A relatively minimal rational elliptic
surface with section is the blow-up of P2 at the 9 base points of a pencil of generically smooth cubics.
Singular Fibers of Elliptic Surfaces: While the generic fiber is a smooth genus 1 curve, elliptic surfaces usually have
singular fibers. Kodaira [5] has shown that fibers of a relatively minimal elliptic surface are of one of the following types:
Name
I0
I1
I2
I3
In , (n ≥ 4)
m In
In∗ ,
, (n ≥ 0)
(n ≥ 0)
II
III
IV
II ∗
III ∗
IV ∗
Description
Smooth genus 1 curve
Nodal rational curve
2 copies of P1 meeting at 2 distinct points transversally
3 copies of P1 meeting at 3 distinct points with dual graph Ā2
n copies of P1 meeting in a cycle,
i.e. meeting with dual graph Ān−1
Multiple fiber, In with multiplicity m
n + 5 copies of P1 meeting with dual graph D̄n+4
Cuspidal rational curve
2 copies of P1 meeting at a single point to order 2
3 copies of P1 all meeting at a single point
9 copies of P1 meeting with dual graph Ē8
8 copies of P1 meeting with dual graph Ē7
7 copies of P1 meeting with dual graph Ē6
The graphs referred to in the above descriptions are the extended Dynkin diagrams given below. Each graph describes
a singular fiber where the singular fiber has a P1 component with self intersection (−2) corresponding to each vertex
of the graph, and two components have intersection number k if there are k edges between the corresponding vertices.
A singular fiber is a divisor of the elliptic surface and the multiplicities of each component are denoted next to the graphs
below.
Note that if an elliptic surface has a section, then the intersection number of the section with each fiber is 1, hence
elliptic surfaces with section do not have singular fibers of type m In for m > 1.
Name
Ān−1
D̄n
Ē6
Ē7
Graph
◦
◦
Multiplicities
A cycle of n vertices
> ◦ − ◦ − · · · − ◦ <◦◦ n + 1 vertices
◦−◦
◦−◦ > ◦ − ◦ − ◦
◦−◦−◦−◦−◦−◦−◦
|
1 for each vertex
> 2 − 2 − · · · − 2 <11
1−2
1−2 > 3 − 2 − 1
1−2−3−4−3−2−1
1
1
|
◦
Ē8
2
◦−◦−◦−◦−◦−◦−◦−◦
|
2−4−6−5−4−3−2−1
|
◦
3
Configurations of Singular Fibers: Since the Euler characteristic of a genus 1 curve is zero, by the additivity of the
Euler characteristic, for any elliptic surface B we have:
X
χ(S) = χ(B).
S singular fiber
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Automorphism groups of rational elliptic surfaces with section and constant J-map
But this equation does not suffice to determine the configurations of singular fibers of elliptic surfaces. Persson [10] and
Miranda [6] have classified all possible configurations of singular fibers of relatively minimal rational elliptic surfaces
with section.
Notation and Ordering: The notation IV II 2 I2 I12 indicates that there is one singular fiber of type IV , one of type I2 ,
two of type II and two of type I1 . The exponents in the notation are the numbers of each type of singular fiber in the
configuration. When writing configurations and ordering them in the lists and tables below, a decreasing lexicographic
order is followed according to the following:
S n+1 > S n for any fiber S , and
and
T > Sn
if
T >S
∗
In+1
> In∗ > IV ∗ > III ∗ > II ∗ > IV > III > II > In+1 > In .
The J-map: Given a relatively minimal rational elliptic surface β : B → P1 with a fixed section σ , the Weierstrass
fibration B0 of B is obtained by collapsing all the components of singular fibers which do not intersect the section σ . B0
may be a singular surface. Its fibers are either elliptic curves, cuspidal rational curves or nodal rational curves. There
are two sections D and E of the line bundles OP1 (4H) and OP1 (6H), respectively (where H is the hyperplane), such that
B0 is a divisor on the P2 bundle P(OP1 ⊕ OP1 (−2H) ⊕ OP1 (−3H)) given by the Weierstrass equation
Y 2 = X 3 + DX + E.
Then
J=
4D 3
4D 3 + 27E 2
defines a meromorphic map on P1 . If the fiber over z ∈ P1 is a smooth elliptic curve, then J(z) is the j-invariant of that
elliptic curve. Note that there is a singular fiber over z if the section 4D 3 + 27E 2 vanishes at z. Since this is a section of
a line bundle with degree 12, there are at most 12 singular fibers. The type of the singular fiber of B can be determined
by the orders of vanishing of the sections D, E and ∆ = 4D 3 + 27E 2 , which is known as Tate’s Algorithm (p.41 in [7]).
Double Cover of F2 : If the Weierstrass fibration B0 is given by Y 2 = X 3 + DX + E in the P2 bundle over P1 as above,
then (X , Y ) 7→ X maps B0 to the rational ruled surface P(OP1 ⊕ OP1 (−2H)) = F2 as a double cover branched over the
minimal section of the F2 and a trisection T given by the equation X 3 + DX + E = 0 in F2 .
Mordell-Weil Group: The set of sections of an elliptic surface B with section is a group called the Mordell-Weil group
of B, and denoted by MW (B). If ζ1 and ζ2 are two sections, using the group law on each smooth fiber, which is an
elliptic curve, one can define ζ1 + ζ2 over the open set of the base curve corresponding to smooth fibers. Then by taking
the closure in B, this can be extended uniquely to a section of B. For a relatively minimal rational elliptic surface B with
section, Oguiso and Shioda [9] have shown that MW (B) (even its lattice structure) is determined by the configuration
of singular fibers on the surface, and they have listed the lattice structure of MW (B) corresponding to each possible
configuration. Their results show that MW (B) is a finitely generated abelian group of rank at most 8 and torsion group
of size at most 9.
4.
Rational elliptic surfaces with constant J-maps and their automorphisms
The goal of this paper is to give a classification of the automorphism groups of relatively minimal rational elliptic surfaces
with section which have constant J-maps. In the rest of the paper, assume thatx β : B → P1 is such a surface and σ is
a fixed section of B called the zero section such that every smooth fiber F of B is an elliptic curve whose zero point
is F ∩ σ . According to the classification of singular fibers of relatively minimal rational elliptic surfaces by Miranda [6]
and Persson [10], the configuration of singular fibers of B and the value of the constant J-map are as shown in Table 2.
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T. Karayayla
Table 2.
Configurations of singular fibers of a relatively minimal rational elliptic surface B with section which has a constant J-map
J
Configuration of singular fibers of B
1
I0∗ III 2 , III ∗ III , III 4
I0∗ IV II , I0∗ II 3 , IV ∗ IV , IV ∗ II 2
0
II ∗ II , IV 3 , IV 2 II 2 , IV II 4 , II 6
any c ∈ C I0∗ I0∗
It has been shown in Theorem 3.0.1 in [4] that
Aut(B) = MW (B) o Autσ (B)
(1)
where MW (B) is the Mordell-Weil group (the group of sections) of the elliptic surface B, σ is a given section on B
which is taken as the zero section (the identity of MW (B)), and Autσ (B) is the subgroup of the automorphisms of B
which preserve the zero section σ setwise:
Autσ (B) = {τ ∈ Aut(B)|τ(σ ) = σ }.
(2)
The Mordell-Weil groups of relatively minimal rational elliptic surfaces have been classified by Oguiso and Shioda [9].
The group MW (B) for such a surface is determined by the configuration of singular fibers. To determine the group
Aut(B), we must determine the group Autσ (B) which is established in this paper.
Since the canonical divisor KB of B is given by −F in the Picard group of B where F denotes the divisor class of the
fiber of the elliptic surface, and the linear system |F | is a pencil, any automorphism τ ∈ Aut(B) maps fibers to fibers.
This induces an automorphism τP1 on the base curve P1 by τP1 ◦ β = β ◦ τ. The map
φ : Aut(B) → Aut(P1 )
τ 7→ τP1
(3)
is a group homomorphism, and we denote its image, which is the group of induced automorphisms on P1 , by
AutB (P1 ) = φ(Aut(B)).
(4)
For an orbit of the action of the group of induced automorphisms AutB (P1 ) on P1 , the fibers over the points in that orbit
are either all elliptic curves with the same j-invariant, or all singular fibers of the same type (Section 4 in [4]). Since
the points of P1 with singular fibers are permuted by the action of AutB (P1 ) and any automorphism of P1 (which can be
viewed as a Möbius transformation) is determined by the images of three fixed points, we can conclude that if there are
n ≥ 3 singular fibers on B, then AutB (P1 ) ≤ Sn , where Sn is the symmetric group of order n!.
The above argument shows that if we exclude the surfaces B with the singular fiber configurations I0∗ I0∗ , IV ∗ IV , III ∗ III,
or II ∗ II which have only two singular fibers, then AutB (P1 ) is a finite subgroup of Aut(P1 ). The only finite subgroups of
Aut(P1 ) are Z/nZ, Dn (Dihedral group of order 2n, n ≥ 2), or the groups of rotations of the regular polyhedra (p. 184 in
[1], the proof for SO(3) can be modified for Aut(P1 )). Table 3 shows the finite subgroups of Aut(P1 ) and the orbit sizes
of their actions on P1 . The number after the semicolon in the second column indicates the size of the generic orbit. If a
number n is repeated k times before the semicolon, then there are k orbits of size n for the action of the corresponding
group.
Considering the orbits of the actions of the induced automorphisms, we can prove the following proposition about the
possible orders of the induced automorphisms and possible finite groups AutB (P1 ).
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Automorphism groups of rational elliptic surfaces with section and constant J-map
Table 3.
Finite subgroups of Aut(P1 ) and the orbit sizes of their actions on P1
Group
Orbit sizes
Z/nZ
1, 1; n
Dn (n ≥ 2)
2, n, n; 2n
A4 (Tetrahedral) 4, 4, 6; 12
S4 (Octahedral) 6, 8, 12; 24
Icosahedral
12, 20, 30; 60
Proposition 4.1.
If B is a relatively minimal rational elliptic surface with section which has a constant J-map, and if B has more than two
singular fibers, then, corresponding to each different configuration of singular fibers of B, the possible orders of induced
automorphisms and possible induced automorphism groups AutB (P1 ) on the base curve P1 are as shown in Table 4.
Table 4.
Candidate groups for AutB (P1 ) and orders n of induced automorphisms.
Configuration
n
AutB (P1 )
I0∗ IV II
1
0
I0∗ II 3
1, 2, 3
0, Z/2Z, Z/3Z
IV II 4
1, 2, 3, 4
0, Z/2Z, Z/3Z, Z/4Z
IV 2 II 2
1, 2
0, Z/2Z, D2
IV 3
1, 2, 3
0, Z/2Z, Z/3Z, D3
III 4
1, 2, 3, 4
0, Z/2Z, D2 , Z/3Z
I0∗ III 2 or IV ∗ II 2 1, 2
0, Z/2Z
Z/4Z, D4 , A4
II 6
1, 2, 3, 4, 5, 6 0, Z/2Z, D2 , Z/3Z, D3 , Z/4Z
D4 , A4 , Z/5Z, Z/6Z, D6 , S4
Proof.
If an induced automorphism in Aut(P1 ) has order n, then the group Z/nZ acts on P1 where there are two
orbits of size 1 (fixed points) and all the other orbits have size n. All finite subgroups of Aut(P1 ) and the orbit sizes
of their actions are as shown on Table 3. For a given configuration of singular fibers, the points in P1 corresponding
to the singular fibers of the same type are partitioned into the orbits of the action of AutB (P1 ). Then a case by case
combinatorial analysis shows the candidate groups for AutB (P1 ) and the orders of induced automorphisms. For example,
if we take the configuration IV II 4 , there is only one singular fiber of type IV , hence the point whose fiber is IV is fixed
by all induced automorphisms (there is an orbit of size 1). Then AutB (P1 ) is cyclic. The 4 points whose fibers are the
singular fibers of type II are partitioned into orbits of the actions of this cyclic group. If m is the order of this cyclic
group, then m = 2 is possible with the partition of 4 as 2 + 2 (the second orbit of size 1 should correspond to a smooth
fiber in this case), m = 3 is possible for the partition of 4 as 3 + 1 and m = 4 is possible for the partition of 4 as 4 (again
the second orbit of size 1 corresponds to a smooth fiber). For the configuration II 6 , the partition 2 + 2 + 2 gives Z/2Z
(where the two orbits of size 1 correspond to smooth fibers) or D2 , the partition 3 + 3 gives Z/3Z or D3 , the partition
1 + 1 + 4 gives Z/4Z, the partition 2 + 4 gives D4 , the partition 1 + 5 gives Z/5Z, and the partition 6 gives the Tetrahedral
group (which is isomorphic to the Alternating group A4 ), Z/6Z, D6 and the Octahedral group (which is isomorphic to the
Symmetric group S4 ) as candidates for AutB (P1 ). The other configurations can be checked similarly.
When the J-map is identically 0 or 1, the smooth fibers of the surface are elliptic curves which can be obtained by the
quotient of C by the hexagonal lattice or the square lattice, respectively. In this case the complex multiplications by µ6
and by i on C induce automorphisms of orders 6 or 4 on the elliptic curves with the j-invariants 0 and 1, respectively.
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T. Karayayla
When the J-map is 0, there is an automorphism µ6 I on B, called the complex multiplication map of order 6, on B which
acts on each smooth fiber as the order 6 automorphism of that elliptic curve induced by the multiplication by the 6th root
of one µ6 . In this case we define the complex multiplication group C M(B) as the group generated by µ6 I. If the J-map is
identically 1, then there is an automorphism iI, the complex multiplication of order 4, on B which acts on each smooth
fiber by multiplication by i. In this case the complex multiplication group C M(B) is generated by iI. If the J-map is not
0 or 1, then C M(B) is generated by −I which acts on smooth fibers by the inversion of the group law on the elliptic
curve (multiplication by −1).
C M(B) = Ker (φ|Autσ (B) )

∼


hµ6 Ii = Z/6Z
∼
= hiIi = Z/4Z


h−Ii ∼
= Z/2Z
if J ≡ 0
if J ≡ 1
(5)
if J ≡ c ∈ C − {0, 1}
Proposition 4.2.
The sequence
φ
0 → C M(B) ,→ Autσ (B) →
− AutB (P1 ) → 0
is short exact. Hence, Autσ (B) is a group extension of AutB (P1 ) by CM(B). Moreover, CM(B) is in the center of Autσ (B).
C M(B) ≤ Z (Autσ (B)).
(6)
Proof.
If τ ∈ Ker(φ|Autσ (B) ), then τ induces the identity on P1 and τ sends the zero section σ to σ . Hence, every fiber
is mapped to itself by τ and the zero section σ is fixed pointwise. Any biholomorphic map on an elliptic curve which
fixes the zero of the elliptic curve is given by a complex multiplication. Therefore, τ is a complex multiplication map on
B. Thus, C M(B) = Ker(φ|Autσ (B) ). Any α ∈ Autσ (B) acts as an elliptic curve isomorphism when restricted to smooth
fibers since it maps the zero of an elliptic curve to the zero of another elliptic curve. Then, α is given by a complex
multiplication between two smooth fibers, and it commutes with any global complex multiplication map τ ∈ C M(B).
Hence, C M(B) ≤ Z (Autσ (B)).
5. The group Autσ (B) of automorphisms preserving the zero section – finite
case
In this section, we prove which finite groups can arise as the group Autσ (B) of automorphisms preserving the zero
section σ for a relatively minimal rational elliptic surface B with a constant J-map. We exclude the surfaces B with
the configurations I0∗ I0∗ , IV ∗ IV , III ∗ III and II ∗ II from the discussion of this section since the group Autσ (B) for such
surfaces are infinite (as it will be shown in Section 6).In the previous section, we have shown that Autσ (B) is a C M(B)
(a cyclic group of order 2,4 or 6) extension of the group AutB (P1 ) of the induced automorphisms on P1 (Proposition
4.2). In Proposition 4.1 we have listed the candidates for AutB (P1 ) with respect to the configurations of singular fibers
of B (excluding four configurations with only two singular fibers which will be considered in the next section). We
first consider the existence of automorphisms α ∈ Autσ (B) with specified orders. The existence is shown via geometric
constructions. Then we show which groups actually arise as Autσ (B).
5.1.
Orders of automorphisms α ∈ Autσ (B)
If α ∈ Autσ (B) and the order of the induced automorphism φ(α) is n, then the order of α is kn where k divides the order
of the group C M(B) since α n ∈ C M(B).
If ord(φ(α)) = n, then ord(α) = kn, and k| |C M(B)| ∈ {2, 4, 6}.
(7)
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Automorphism groups of rational elliptic surfaces with section and constant J-map
It is shown in Theorem 5.1.1 in [4] that a relatively minimal rational elliptic surface B with section has an automorphism
α ∈ Autσ (B) with ord(α) = ord(φ(α)) = n if and only if B is obtained as the Kodaira Model of a fibered product
P1 ×P1 B0 of a relatively minimal rational elliptic surface B0 and the map gn : P1 → P1 , z 7→ z n . In this case B is called
a pull-back of B0 by the map gn . The fibered product P1 ×P1 B0 is always an elliptic surface. The singular fibers of the
Kodaira Model of this product (which is the relatively minimal elliptic surface birational to it) are easily determined if
one knows the singular fibers of B0 . If there is a singular fiber of B0 over a point p 6= 0, ∞ (0 and ∞ are the branch
points of the map gn ), then B has n singular fibers of the same type over the n points which map to p by gn . The
fibers of B over 0 or ∞ depend on n and the singular fibers of B0 as shown in Table 5 (Table 7.1 p.555 in [8]). The
Kodaira Model of the fibered product is rational if and only if the sum of the Euler characteristics of its singular fibers
is 12. Using this construction we can list all relatively minimal rational elliptic surfaces with constant J-maps which are
pull-backs of other such surfaces, hence have α ∈ Autσ (B) with ord(α) = ord(φ(α)).
Table 5.
Transformation of the fibers over the ramified points in the pull-back process.
Fiber of B0 over gn (p) Fiber of B over p (p = 0, ∞)
I0
I0
IM
InM
∗
IM
InM if n even
II
I0 if n = 0 mod 6
∗ if n odd
InM
IV if n = 2 mod 6
IV ∗ if n = 4 mod 6
III
IV
II if n = 1 mod 6
I0∗ if n = 3 mod 6
II ∗ if n = 5 mod 6
I0 if n = 0 mod 4
III if n = 1 mod 4
I0∗ if n = 2 mod 4
III ∗ if n = 3 mod 4
I0 if n = 0 mod 3
IV if n = 1 mod 3
IV ∗ if n = 2 mod 3
IV ∗
I0 if n = 0 mod 3
IV ∗ if n = 1 mod 3
IV if n = 2 mod 3
III ∗
II ∗
I0 if n = 0 mod 4
I0∗
III ∗ if n = 1 mod 4
if n = 2 mod 4
III if n = 3 mod 4
I0 if n = 0 mod 6
II ∗ if n = 1 mod 6
IV ∗ if n = 2 mod 6
IV if n = 4 mod 6
I0∗ if n = 3 mod 6
II if n = 5 mod 6
Proposition 5.1.
The configurations of singular fibers of the relatively minimal rational elliptic surfaces B with section and constant
J-maps for which there is an automorphism α ∈ Autσ (B) such that ord(α) = ord(φ(α)) = n are as shown in Table 6.
The third column shows the configurations of singular fibers of rational elliptic surfaces B0 where B is obtained as the
pull-back of B0 by the map gn .
If we consider the case ord(α) = 2 · ord(φ(α)) = 2n, the existence of such α ∈ Autσ (B) is equivalent to the existence
of α ∗ ∈ Autσ (B) with ord(α ∗ ) = ord(φ(α ∗ )) = n when n is odd (take α ∗ = −I ◦ α). The existence of such α ∗ was
characterized above in Proposition 5.1. If we consider even n, the singular fiber of B over the fixed points of φ(α) cannot
be of type IV or IV ∗ (Lemma 5.2.1 in [4]) and this singular fiber cannot be a smooth fiber unless n ≡ 2 mod 4 and the
j-invariant of the fiber is 1 (Lemma 5.2.2 in [4]). Due to these facts, we can eliminate the cases from Table 4 except for
I0∗ III 2 , I0∗ II 3 , IV 2 II 2 , III 4 and II 6 for n = 2, and the case II 6 for n = 4. Among these cases, for I0∗ III 2 and III 4 , the existence
of α is equivalent to ord(iI ◦ α) = ord(φ(iI ◦ α)) = 2 when n = 2. The remaining cases will be checked later in this
section.
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T. Karayayla
Table 6.
Configurations of singular fibers of B and B0 where B is a pull-back of B0 by the map gn and there exists α ∈ Autσ (B) such that
ord(α) = ord(φ(α)) = n
n Configuration of B Configuration of B0
6 II 6
5 II 6
4
IV II 4
III 4
3
2
I0∗ II 3
IV 3
I0∗ IV II, IV ∗ II 2 , II ∗ II
II ∗ II
I0∗ IV II, II ∗ II
I0∗ III 2 , III ∗ III
I0∗ IV II, IV ∗ II 2 , II ∗ II
IV ∗ IV , IV 3
III 4
III ∗ III
II 6
IV ∗ II 2
I0∗ III 2
IV ∗ II 2
I0∗ III 2 , III ∗ III
IV 3
I0∗ IV II, IV ∗ IV
IV 2 II 2
IV II 4
III 4
II 6
I0∗ IV II, II ∗ II
I0∗ IV II, IV ∗ II 2
I0∗ II 3 , IV ∗ II 2
I0∗ III 2
I0∗ II 3
If we consider the case ord(α) = 4 · ord(φ(α)) = 4n which may be possible when J ≡ 1, the existence of such an
α ∈ Autσ (B) is equivalent to the existence of α ∗ ∈ Autσ (B) with ord(α ∗ ) = ord(φ(α ∗ )) = n for odd n (take α ∗ = ±iI ◦ α).
Then we need to check the cases from Table 4 with even n. For the configuration I0∗ III 2 and n = 2, such an α cannot
exist since otherwise there is a smooth fiber over a fixed point of φ(α) and α restricts to an automorphism of this elliptic
curve such that its square is the complex multiplication by ±i which is not possible. Similarly, for the configuration III 4
and n = 4, α should act on the smooth fibers over the fixed points of φ(α) as an elliptic curve automorphism whose
fourth power is multiplication by ±i, which is not possible. Therefore, the only case that remains to be checked is the
configuration III 4 and n = 2.
If there exists an element α ∈ Autσ (B) with ord(α) = 6 · ord(φ(α)) = 6n, then J ≡ 0 and α n = µ6 I. If B has a smooth
fiber over a fixed point of φ(α), then α restricts to an elliptic curve isomorphism on that fiber whose n-th power is an
automorphism of order 6. This is not possible unless n ≡ ±1 mod 6. Also, if n is even, α 3n = −I hence B cannot have
a singular fiber of type IV or IV ∗ over a fixed point of φ(α) (Lemma 5.2.1 in [4]). Due to these facts, the remaining cases
from Table 4 to be checked are I0∗ II 3 , IV 2 II 2 for n = 2, II 6 for n = 2, 4 or 5, and IV II 4 for n = 3. But the existence of
such an α in any of those cases is equivalent to the existence of α ∗ ∈ Autσ (B) such that ord(α ∗ ) = k · ord(φ(α ∗ )) = kn
for some k = 1, 2 or 3 instead of k = 6.
The last possibility in terms of orders of α ∈ Autσ (B) and φ(α) is that ord(α) = 3 · ord(φ(α)) = 3n when J ≡ 0. If n
and 3 are coprime, the existence of such an α is equivalent to the existence of an α ∗ with ord(α ∗ ) = ord(φ(α ∗ )) = n.
Thus, we need to check only n = 3 or 6. In this case, α does not exist if B has a smooth fiber over a fixed point of φ(α)
since in this case α cannot act on this smooth fiber so that α n = µ3 I. Due to these facts, the only case that remains to
be checked is IV II 4 and n = 3.
Table 7 lists all the cases to be checked after the discussion above. Note that only one case is listed among equivalent
cases.
5.2.
Constructions
Now we prove the existence of the automorphism groups Autσ (B) by means of a geometric construction.
Note that the Weierstrass fibration of B is a double cover of the rational ruled surface F2 branched over the minimal
section of F2 and a trisection T (a divisor with intersection number 3 with the fiber of F2 ) disjoint from the minimal
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Automorphism groups of rational elliptic surfaces with section and constant J-map
Table 7.
The configurations for which the existence of α ∈ Autσ (B) with ord(α) = k · ord(φ(α)) = kn should be checked
n k Configuration
2 2 I0∗ II 3 , IV 2 II 2 , II 6
4 III 4
3 3 IV II 4
4 2 II 6
section [7]. The involution of this double cover is the map on the Weierstrass fibration induced by the automorphism
−I on B. Since −I is in the center of Autσ (B), any automorphism α ∈ Autσ (B) induces an automorphism on F2 which
preserves the branch locus (hence the trisection T ) of the double cover. This gives an epimorphism to a subgroup of
Aut(F2 ) with kernel h−Ii. Conversely any subgroup of Aut(F2 ) which preserves a trisection T lifts to a subgroup of
Autσ (B) where B is the rational elliptic surface corresponding to the double cover of F2 branched over the minimal
section and the trisection T (Section 5.2.1 in [4]). Therefore, to prove the existence of Autσ (B) groups, it suffices to study
subgroups of Aut(F2 ) which preserve a trisection T of F2 .
The rational ruled surface F2 maps onto the quadric cone Q ⊂ P3 given by Y 2 = X Z . The minimal section of F2 is
blown down to the vertex of the cone. The map gives an isomorphism away from the minimal section. Since every
automorphism of F2 preserves the minimal section, Aut(F2 ) can be identified with the automorphisms of the quadric
cone Q. Automorphisms of Q are induced by automorphisms of P3 which is the group PGL4 (C). If v is the vertex of
the cone, then Q − v is isomorphic to the line bundle OP1 (2H) [3]. On a local chart, this isomorphism is given by
([1, t], w) 7→ [1, t, t 2 , w]. The automorphisms of F2 induced by the automorphisms of the elliptic surface B in Autσ (B)
map the fibers of F2 to fibers. Such automorphisms of F2 correspond to the automorphisms of OP1 (2H) which are of the
following form on an affine chart:
([1, t], w) 7→ ([1, at], b + ct + dt 2 + ew).
Such automorphisms correspond to the automorphisms of the quadric cone Q which are given by automorphisms of P3
of the form:
f : P3 → P3
[X , Y , Z , W ] 7→ [f1 (X , Y , Z ), f2 (X , Y , Z ), f3 (X , Y , Z ), f4 (X , Y , Z , W )]
(8)
where fi are linear and f22 = f1 f3 whenever Y 2 = X Z .
If we consider the case J ≡ 1, the complex multiplication map iI induces an order 2 automorphism on F2 . On each
fiber of F2 , this induced map fixes two points, one on the minimal section and one on the trisection T . Without loss
of generality, we may assume that the fixed section of F2 is given by W = 0 in P3 and hence the trisection T on the
quadric cone Q is given by the equation
(
T :
Y 2 = XZ
W 3 + A(X , Y , Z )W = 0
(9)
where A(X , Y , Z ) is homogeneous of degree 2. Then the complex multiplication maps ±iI on B induce the map on Q
which is given by
[X , Y , Z , W ] 7→ [X , Y , Z , −W ].
(10)
For the case J ≡ 0, the complex multiplication map µ6 I induces an order 3 map on F2 . In each fiber of F2 , this induced
map fixes two points (one on the minimal section) and it permutes the three points (generically distinct) of T by a 3-cycle.
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T. Karayayla
Without loss of generality, we may assume that the fixed section (other than the minimal section) of this induced map
is given by W = 0 on the quadric cone. Then, if [x, y, z, w] is on T , [x, y, z, µ3 w] is also on T . Hence, T is given by:
(
T :
Y 2 = XZ
W 3 + B(X , Y , Z ) = 0
(11)
where B(X , Y , Z ) is a homogeneous cubic. Then the maps of order 3 induced by the complex multiplications ±µ6 I and
±µ62 I respectively are given by:
[X , Y , Z , W ] 7→ [X , Y , Z , µ3 W ]
[X , Y , Z , W ] 7→ [X , Y , Z , µ32 W ]
(12)
In both cases (J ≡ 1 or 0), the automorphisms of Q preserving the trisection T must be of the form
[X , Y , Z , W ] 7→ [f1 (X , Y , Z ), f2 (X , Y , Z ), f3 (X , Y , Z ), cW ].
(13)
5.2.1. ord(α) = 2 · ord(φ(α)) = 4 case
According to Table 7, we need to check the existence of such α for the configurations I0∗ II 3 , IV 2 II 2 and II 6 . We consider
the action of the map
Γ2 : P3 → P3 , [X , Y , Z , W ] 7→ [X , −Y , Z , −W ]
(14)
on the quadric cone Q (Y 2 = X Z ) whose action on P1 is by z 7→ −z. If we exclude the line of the cone Q over the point
∞ (which is the line [0, 0, 1, w], w ∈ C), then the local chart ([1, t], w) 7→ [1, t, t 2 , w] can be used. The trisections on Q
which are preserved by the action of Γ2 and which do not pass through the vertex [0, 0, 0, 1] of the cone are of the form
T :
(
Y 2 = XZ
W 3 + (cX 2 + dX Z + eZ 2 )W + fX 2 Y + gX Y Z + hY Z 2 = 0
(15)
In local coordinates we get
w 3 + a(t)w + b(t) = 0
(16)
where
a(t) = c + dt 2 + et 4 ,
b(t) = ft + gt 3 + ht 5 .
(17)
When we homogenize the polynomials a and b as polynomials of degrees 4 and 6, respectively, then they will be the
Weierstrass coefficients of the elliptic surface B whose Weierstrass fibration is a double cover of F2 branched over the
minimal section and the trisection T . Tate’s Algorithm can be used to determine the singular fibers of B [7]. The type
of a singular fiber over the point t ∈ P1 is determined by the order of vanishing of a, b and ∆ = 4a3 + 27b2 .
For the three cases we want to check, we have J ≡ 0, hence c = d = e = 0. Then a(t) = 0, b(t) = ft + gt 3 + ht 5 , and
∆(t) = 27b2 (t) = 27t 2 (f + gt 2 + ht 4 )2 .
It can be checked by Tate’s Algorithm that I0∗ II 3 occurs when f = 0, gh 6= 0, or when h = 0 and fg 6= 0. The configuration
IV 2 II 2 occurs when f 6= 0, h 6= 0 and g2 − 4fh = 0. Finally II 6 occurs when f 6= 0, h 6= 0 and g2 − 4fh 6= 0.
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Automorphism groups of rational elliptic surfaces with section and constant J-map
5.2.2. ord(α) = 4 · ord(φ(α)) = 8 case
To show the existence of such an α ∈ Autσ (B) for an elliptic surface B with the configuration of singular fibers III 4 , we
consider the action of
Γ3 : P3 → P3 , [X , Y , Z , W ] 7→ [X , −Y , Z , iW ]
(18)
on the quadric cone Q. Note that if α exists, then α 2 = ±iI ∈ C M(B), hence the automorphism induced on F2 by α
should have order 4 (since ±I induces the identity on F2 ) and it should also induce an order 2 automorphism on P1 . The
automorphism Γ3 has these properties. The trisections on Q which are preserved by this action and which do not pass
through the vertex of the cone are given by
(
Y 2 = XZ,
T :
W 3 + cX Y W + eY Z W = 0.
(19)
In local coordinates we get
w 3 + (ct + et 3 )w = 0.
It can be checked by Tate’s Algorithm that whenever ce 6= 0, this trisection gives an elliptic surface B with the
configuration III 4 . The automorphism Γ3 lifts to two automorphisms of B, both with order 8 and both have induced
automorphisms of order 2 on P1 .
5.2.3. ord(α) = 3 · ord(φ(α)) = 9 case
To prove the existence of such an α ∈ Autσ (B) for an elliptic surface B with the configuration of singular fibers IV II 4 ,
consider the action of
Ω : P3 → P3 , [X , Y , Z , W ] 7→ [X , µ3 Y , µ32 Z , µ9 W ]
(20)
on the quadric cone Q, where µ9 is a primitive 9th root of 1 such that µ93 = µ3 . Note that ord(Ω) = 9 and the induced
automorphism on the base section P1 of F2 by Ω has order 3. The trisections T on Q which are preserved by this action
and which do not pass through the vertex of the cone are given by
(
T :
Y 2 = XZ
W 3 + fX 2 Y + gX Z 2 = 0.
(21)
In local coordinates we get
w 3 + ft + gt 4 = 0.
Then ∆(t) = 27t 2 (f + gt 3 )2 . It can be checked by Tate’s Algorithm that whenever fg 6= 0, this trisection corresponds to
an elliptic surface B with the configuration IV II 4 and the automorphism Ω lifts to two automorphisms on B, one with
order 9 and the other with order 18, and both have induced automorphisms of order 3 on P1 .
5.2.4. ord(α) = 2 · ord(φ(α)) = 8 case
To show the existence of such an α ∈ Autσ (B) for an elliptic surface B with the configuration II 6 , consider the action of
∆2 : P3 → P3 ,
[X , Y , Z , W ] 7→ [X , iY , −Z , −iW ]
(22)
on the quadric cone Q. The trisections T on Q which are preserved by the action of ∆2 and which do not pass through
the vertex of the cone are given by
(
T :
1784
Y 2 = XZ
W 3 + cX Z W + dX 2 Y + eY Z 2 = 0.
(23)
T. Karayayla
In local coordinates we get
w 3 + ct 2 w + dt + et 5 = 0
and by Tate’s Algorithm, it can be checked that whenever c = 0 and de 6= 0, this trisection T gives rise to an elliptic
surface B with the configuration II 6 . In this case, ∆2 lifts to two automorphisms on B, both with order 8, and both have
induced automorphisms of order 4 on P1 .
Thus, all the cases in Table 7 have been checked and their existence is confirmed.
5.3.
Cyclic AutB (P1 ) case
In the sections above we have shown what the orders of α ∈ Autσ (B) and the orders of their induced automorphisms φ(α)
can be. Using this information, we can now prove what the group Autσ (B) is if the group AutB (P1 ) is cyclic. We know
that Autσ (B) is a group extension of AutB (P1 ) by the complex multiplication group C M(B) which is a cyclic subgroup of
the center Z (Autσ (B)). Thus, if AutB (P1 ) is cyclic, then Autσ (B) is abelian. We analyze each configuration of singular
fibers separately below.
• I0∗ III 2 : For a rational elliptic surface B with such a configuration, the only possibilities for AutB (P1 ) are Z/2Z or the
trivial group. We have shown that α ∈ Autσ (B) with ord(α) = ord(φ(α)) = 2 exists for some B (which are obtained by
a pull-back process). Indeed for all such B we can show the existence of such an α (if one chooses coordinates on P1
such that III fibers are over ±1 and I0∗ is over 0, then it can be shown that the trisection of the double cover over F2
is preserved by an action of Z/2Z). We have shown that α ∈ Autσ (B) with ord(α) = 4 · ord(φ(α)) = 8 does not exist.
Therefore AutB (P1 ) = Z/2Z and Autσ (B) = Z/4Z × Z/2Z (since C M(B) = Z/4Z).
• IV ∗ II 2 : For a rational elliptic surface B with this configuration we have J ≡ 0 and C M(B) = Z/6Z. It can be shown
that AutB (P1 ) = Z/2Z for all such B (if one chooses coordinates on P1 such that IV ∗ is over 0 and the two II fibers are
over ±1, then the trisection of the double cover is preserved under the action of Z/2Z). We have shown that α ∈ Autσ (B)
with ord(α) = ord(φ(α)) = 2 exists, and ord(α) = 2 · ord(φ(α)) = 4 does not exist. The existence of the other cases
about the orders of α and φ(α) are equivalent to one of these. Therefore, AutB (P1 ) = Z/2Z and Autσ (B) = Z/6Z × Z/2Z.
• I0∗ II 3 : For an elliptic surface with this configuration of singular fibers, we have J ≡ 0 and C M(B) = Z/6Z. There are
three candidates for AutB (P1 ), which all exist, and these are Z/3Z, Z/2Z and 0. If the singular fibers are fibered over
such a configuration of points on P1 which does not allow a Z/2z or Z/3Z action (fixing the point corresponding to the
I0∗ fiber and permuting the other three points), then AutB (P1 ) = 0 and Autσ (B) = C M(B) = Z/6Z. We have shown that
for some B, there exists α ∈ Autσ (B) with ord(α) = ord(φ(α)) = 3. We have shown that there is no α ∈ Autσ (B) with
ord(α) = 3 · ord(φ(α)) = 9. There are no other (inequivalent) cases to check in terms of order of α if ord(φ(α)) = 3.
Therefore we have Autσ (B) = Z/6Z × Z/3Z whenever AutB (P1 ) = Z/3Z. We have also shown that it is possible to have
α ∈ Autσ (B) such that ord(α) = 2 · ord(φ(α)) = 4 or equivalently ord(α) = 6 · ord(φ(α)) = 12 for some B. We have
also shown that there is no α ∈ Autσ (B) with ord(α) = ord(φ(α)) = 2. Therefore we have Autσ (B) = Z/12Z whenever
AutB (P1 ) = Z/2Z.
• IV II 4 : For a rational elliptic surface B with this configuration, we have J ≡ 0 and C M(B) = Z/6Z. We have shown
that AutB (P1 ) can be one of the groups Z/nZ (n = 4, 3, 2), or the trivial group 0. All four possibilities occur as
follows. If the points on P1 corresponding to the singular fibers are such that there is no cyclic action fixing the point
corresponding to the fiber IV and permuting the other four points, then AutB (P1 ) = 0 and Autσ (B) = Z/6Z. We have
shown that for some B there exists α ∈ Autσ (B) with ord(α) = ord(φ(α)) = 2, and there exists no α ∈ Autσ (B)
with ord(α) = 2 · ord(φ(α)) = 4. There are no other inequivalent cases to consider if ord(φ(α)) = 2. Therefore
Autσ (B) = Z/6Z × Z/2Z whenever AutB (P1 ) = Z/2Z. We have shown that for some B there exists α ∈ Autσ (B) with
ord(α) = 3 · ord(φ(α)) = 9 (or equivalently ord(α) = 6 · ord(φ(α)) = 18), and there exists no α ∈ Autσ (B) with
ord(α) = ord(φ(α)) = 3. Therefore Autσ (B) = Z/18Z whenever AutB (P1 ) = Z/3Z. We have shown that for some B
there exists α ∈ Autσ (B) with ord(α) = ord(φ(α)) = 4, and there exists no α ∈ Autσ (B) with ord(α) = 2 · ord(α) = 8.
Therefore, Autσ (B) = Z/6Z × Z/4Z whenever AutB (P1 ) = Z/4Z.
• IV 2 II 2 : For a rational elliptic surface B with this configuration, we have J ≡ 0 and C M(B) = Z/6Z. The candidates for
AutB (P1 ) are D2 = Z/2Z × Z/2Z, Z/2Z and the trivial group 0. If the points on P1 corresponding to the singular fibers
are positioned such that no Z/2Z action permutes them, then AutB P1 = 0 and Autσ (B) = Z/6Z. We have shown that
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Automorphism groups of rational elliptic surfaces with section and constant J-map
α ∈ Autσ (B) with ord(α) = ord(φ(α)) = 2 exists for some B (the fixed points of φ(α) are either the points corresponding
to the two IV fibers or points with smooth fibers). The subgroup of AutB (P1 ) generated by φ(α) in this case (which is
isomorphic to Z/2Z) lifts to a subgroup of Autσ (B) isomorphic to Z/6Z × Z/2Z. If AutB P1 6= D2 , then Autσ (B) equals this
subgroup. If the fixed points of φ(α) correspond to smooth fibers of B we can have cases where there is no D2 action.
But if the fixed points correspond to the IV fibers, then we will see in the next section that AutB (P1 ) = D2 . We have also
shown that for some B there exists α ∈ Autσ (B) with ord(α) = 2 · ord(φ(α)) = 4 and the fixed points of φ(α) correspond
to the two II fibers. In this case the subgroup of AutB (P1 ) generated by φ(α) (which is isomorphic to Z/2Z) lifts to a
subgroup of Autσ (B) isomorphic to Z/12Z. But if such an α exists, we will see in the next section that AutB (P1 ) = D2 .
• IV 3 : For a rational elliptic surface B with this configuration, we have J ≡ 0 and C M(B) = Z/6Z. The candidates
for AutB (P1 ) are D3 , Z/3Z, Z/2Z and the trivial group 0. We will see in the next section that for all such B we have
AutB (P1 ) = D3 . We have shown that there exists α ∈ Autσ (B) with ord(α) = ord(φ(α)) = 2 and there is no other
inequivalent case if ord(φ(α)) = 2. In this case the Z/2Z subgroup of AutB (P1 ) generated by φ(α) lifts to a Z/6Z × Z/2Z
subgroup of Autσ (B). We have also shown that there exists α ∈ Autσ (B) with ord(α) = ord(φ(α)) = 3 and no other
inequivalent cases exist when ord(φ(α)) = 3. In that case the Z/3Z subgroup of AutB (P1 ) generated by φ(α) lifts to a
Z/6Z × Z/3Z subgroup of Autσ (B).
• III 4 : For a rational elliptic surface B with this configuration, we have J ≡ 1 and C M(B) = Z/4Z. The candidate
groups for AutB (P1 ) are A4 , Dn (n = 2, 4),Z/nZ (n = 4, 3, 2) and the trivial group 0. The points on P1 corresponding
to the four III fibers can be positioned so that no Z/nZ (n = 4, 3, 2) action can permute them. In such a case we have
AutB (P1 ) = 0 and Autσ (B) = Z/4Z. We have seen that for some B there exists α ∈ Autσ (B) with ord(α) = ord(φ(α)) = 4
and there is no other inequivalent condition for ord(φ(α)) = 4. In this case the Z/4Z subgroup of AutB (P1 ) generated
by φ(α) lifts to a Z/4Z × Z/4Z subgroup of Autσ (B). We have shown that for some B there exists α ∈ Autσ (B) with
ord(α) = ord(φ(α)) = 3 and no other inequivalent case exists if ord(φ(α)) = 3. In this case the Z/3Z subgroup of
AutB (P1 ) generated by φ(α) lifts to a Z/12Z subgroup of Autσ (B). We have also shown that for some B there exists
α ∈ Autσ (B) with ord(α) = ord(φ(α)) = 2. In this case the Z/2Z subgroup of AutB (P1 ) generated by φ(α) lifts to a
Z/4Z × Z/2Z subgroup of Autσ (B). Another case which occurs if ord(φ(α)) = 2 is that ord(α) = 8. In this case the Z/2Z
subgroup of AutB (P1 ) lifts to a Z/8Z subgroup of Autσ (B). The case AutB (P1 ) = Z/2Z occurs if ord(α) = ord(φ(α)) = 2
and the points of P1 are positioned so that there is no Z/4Z or D2 action on P1 permuting these four points and also
permuting the fixed points of φ(α). Whether we may have other cases where AutB (P1 ) is cyclic will be discussed when
non-cyclic AutB (P1 ) groups are considered in the next section.
• II 6 : For a rational elliptic surface B with this configuration we have J ≡ 0 and C M(B) = Z/6Z. The cyclic candidates
for AutB (P1 ) are Z/nZ (n = 6, 5, 4, 3, 2). Depending on the position of the points on P1 corresponding to the six II
fibers, there exists B with AutB (P1 ) = 0 and Autσ (B) = Z/6Z. Let G be a cyclic subgroup of AutB (P1 ) and let H be the
lift of G to Autσ (B) under the homomorphism φ (i.e. φ(H) = G). If G = Z/6Z, then H = Z/6Z × Z/6Z. If G = Z/5Z,
then H = Z/30Z. If G = Z/4Z, then H = Z/24Z. If G = Z/3Z, then H = Z/6Z × Z/3Z. Finally, if G = Z/2Z, then
H = Z/6Z × Z/2Z for some B, and H = Z/12Z for some B. Whether AutB (P1 ) can be cyclic for any B will be discussed
in the next section.
We can collect all the information obtained in Section 5.2 and Section 5.3 together as in Proposition 5.2 below.
Proposition 5.2.
For a relatively minimal rational elliptic surface B with section which has a constant J-map and more than two singular
fibers, let α ∈ Autσ (B) induce a nontrivial automorphism φ(α) ∈ AutB (P1 ). Then, the singular fiber configuration of B,
the orders of φ(α) and α, the fibers of B that are fibered over the fixed points of φ(α), and the subgroup φ−1 (hφ(α)i) of
Autσ (B) which is the lift of the cyclic group generated by φ(α) to the group Autσ (B) by the group homomorphism φ are
as shown in Table 8. All the cases listed in that table exist and all existing cases are listed.
5.4.
Non-cyclic AutB (P1 ) case
We now prove the existence of the non-cyclic AutB (P1 ) groups which were predicted on Table 4. The strategy is based
on presenting a finite group action on the rational ruled surface F2 such that the induced action on the base section P1
1786
T. Karayayla
Table 8.
Cyclic subgroups of AutB (P1 ) and their lifts to Autσ (B) by the homomorphism φ. If α ∈ Autσ (B), the 4th column lists the fibers of B fibered
over the two fixed points of the induced automorphism φ(α). Note that I0 denotes a smooth fiber
Configuration ord(φ(α)) ord(α)
Fibers over fixed points φ−1 (hφ(α)i)
I0∗ III 2
I0∗ , I0
IV ∗ II 2
I0∗ II 3
IV 3
IV 2 II 2
IV II 4
III 4
II 6
Z/4Z × Z/2Z
2
2 or 4
2
2 or 6
IV ∗ , I0
Z/6Z × Z/2Z
3
3 or 6
Z/6Z × Z/3Z
2
4 or 12
I0∗ , I0
I0∗ , II
3
3 or 6
I0 , I0
Z/6Z × Z/3Z
2
2 or 6
IV , I0
Z/6Z × Z/2Z
2
2 or 6
I0 , I0
Z/6Z × Z/2Z
IV , IV
Z/6Z × Z/2Z
4 or 12
II, II
Z/12Z
4
4 or 12
IV , I0
Z/6Z × Z/4Z
3
9 or 18
IV , II
Z/18Z
2
2 or 6
IV , I0
Z/6Z × Z/2Z
4
4
I0 , I0
Z/4Z × Z/4Z
3
3,6 or 12
III, I0
Z/12Z
2
2 or 4
I0 , I0
Z/4Z × Z/2Z
8
III, III
Z/8Z
6
6
I0 , I0
Z/6Z × Z/6Z
5
5,10,15 or 30 II, I0
Z/30Z
4
8 or 24
II, II
Z/24Z
3
3 or 6
I0 , I0
Z/6Z × Z/3Z
2
2 or 6
I0 , I0
Z/6Z × Z/2Z
4 or 12
II, II
Z/12Z
Z/12Z
is one of the non-cyclic groups in Table 4. Then any trisection T on F2 disjoint from the minimal section, and which is
preserved under that action on F2 gives rise to a rational elliptic surface B for which AutB (P1 ) contains that non-cyclic
group from Table 4. The surface B is the surface whose Weierstrass fibration is the double cover of F2 branched over
the trisection T and the minimal section of F2 . To simplify the calculations we work on the quadric cone Q ⊂ P3 given
by Y 2 = X Z . The rational ruled surface F2 maps onto Q by blowing down the minimal section to the vertex of the cone.
The trisections of F2 that are disjoint from the minimal section are then given as the zeros of homogeneous cubics in X ,
Y , z and W which do not vanish on the vertex [0, 0, 0, 1] of the cone. Note that since the involution of the double cover
of F2 is induced by the map −I on B, the induced automorphism group on F2 is isomorphic to Autσ (B)/h−Ii. The maps
induced on F2 by the complex multiplication maps ±iI, and ±µ6 I are given as in (10) and (12) by
[X , Y , Z , W ] 7→ [X , Y , Z , −W ]
[X , Y , Z , W ] 7→ [X , Y , Z , µ3 Z ]
respectively. When writing down the generators of a finite group action on the quadric cone Q in the cases below, we
implicitly assume that the first map above is also among the generators if we are searching for a trisection T such that
the J-map is 1, and similarly we assume that the second map above is among the generators when we have J ≡ 0.
• AutB (P1 ) = D2 case: Consider the action of D2 = Z/2Z × Z/2Z on the quadric cone Q ⊂ P3 (where Q is given by
Y 2 = X Z ) generated by the automorphisms
Γ2 : P3 → P3
[X , Y , Z , W ] 7→ [X , −Y , Z , −W ]
Ω1 : P → P
[X , Y , Z , W ] 7→ [Z , Y , X , W ]
3
3
(24)
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Automorphism groups of rational elliptic surfaces with section and constant J-map
of P3 . The induced action of Γ2 on P1 which embeds into Q by [1, t] 7→ [1, t, t 2 , 0] is by t 7→ −t, and the action of Ω1 is
by t 7→ 1/t, hence AutB (P1 ) = D2 . The trisections T on Q which are preserved under this action are given by
(
T :
Y 2 = XZ
W 3 + (cX 2 + dX Z + cZ 2 )W + fX 2 Y + gX Y Z + fY Z 2 = 0
(25)
In local coordinates this gives
w 3 + (c + dt 2 + ct 4 )w + ft + gt 3 + ft 5 = 0.
(26)
It can be checked by Tate’s Algorithm that whenever c = d = 0 and g = ±2f 6= 0, then T corresponds to an elliptic
surface B with the configuration IV 2 II 2 , whenever c = d = 0, f 6= 0 and g 6= ±2f, then T corresponds to an elliptic
surface B with the configuration II 6 , and whenever f = g = 0, c 6= 0 and d 6= ±2c, then T corresponds to the
configuration III 4 . In all of those cases, we have AutB (P1 ) = D2 (except for two isolated cases where a larger Autσ (B)
action is possible. These cases are c = d = g = 0, f 6= 0 which gives the configuration II 6 , and d = f = g = 0, c 6= 0
which gives the configuration III 4 . In these two cases an induced automorphism of order 4 exists). If instead of the action
of Γ2 and Ω1 above we consider the action of Γ2 and Ω2 where
Ω2 : P3 → P3
[X , Y , Z , W ] 7→ [Z , Y , X , −W ],
(27)
then the trisections T on Q which are preserved by this D2 action are given by
(
T :
Y 2 = XZ
W 3 + (cX 2 + dX Z + cZ 2 )W + fX 2 Y − fZ 2 Y = 0
(28)
In the local coordinates we get
w 3 + (c + dt 2 + ct 4 )w + f(t − t 5 ).
(29)
In this case whenever c = d = 0 and f 6= 0 we get the configuration II 6 . Here the II fibers are fibered over t = 0,, ∞,
±1, ±i. We will see that in that case Autσ (B) is the Octahedral group. Whenever f = g = 0, c 6= 0 and d 6= ±2c, then
we get the configuration III 4 by Tate’s Algorithm. In this case AutB (P1 ) = D2 unless d = 0 as above.
• AutB (P1 ) = D3 case: If we consider the action of D3 on the quadric cone Q generated by
Θ1 : P3 → P3
[X , Y , Z , W ] 7→ [X , µ3 Y , µ32 Z , µ3 W ]
Ω1 : P3 → P3
[X , Y , Z , W ] 7→ [Z , Y , X , W ],
(30)
then the induced action on the base section P1 which embeds in Q by [1, t] 7→ [1, t, t 2 , 0] is generated by t 7→ µ3 t and
t 7→ 1/t, hence the induced group on P1 is D3 . The trisections T on Q preserved under this action are given by
(
T :
Y 2 = XZ
W 3 + aY W 2 + bX Z W + cX 3 + dX Y Z + cZ 3 = 0
(31)
In local coordinates we get
w 3 + atw 2 + bt 2 w + c + dt 3 + ct 6 = 0.
(32)
It can be checked by Tate’s Algorithm that whenever a = b = 0 and d = ±2c 6= 0 this corresponds to the configuration
IV 3 and whenever a = b = 0, c 6= 0 and d 6= ±2c this corresponds to the configuration II 6 . In all of these cases we get
AutB (P1 ) = D3 (except for the isolated case a = b = d = 0 for the configuration II 6 where we will get AutB (P1 ) = D6 ).
• AutB (P1 ) = D4 case: If we consider the action of D4 on the quadric cone Q generated by
∆2 : P3 → P3
[X , Y , Z , W ] 7→ [X , iY , −Z , −iW ]
Ω1 : P → P
[]X , Y , Z , W 7→ []Z , Y , X , W ,
3
1788
3
(33)
T. Karayayla
then the induced action on the base section P1 is generated by t 7→ it and t 7→ 1/t, hence the induced automorphism
group on P1 is D4 . The trisections T on Q preserved under this action are given by
(
T :
Y 2 = XZ
W 3 + aX Z W + bX 2 Y + bY Z 2 = 0
(34)
In local coordinates we get
w 3 + at 2 w + b(t + t 5 ) = 0.
(35)
By Tate’s Algorithm, if a = 0 and b 6= 0, this corresponds to the configuration II 6 . In this case we get D4 ⊂ AutB (P1 ).
Indeed AutB (P1 ) is a larger group in this case and equals the Octahedral group.
If we now consider the group action on the quadric cone Q by the order 16 group generated by
∆3 : P3 → P3
[X , Y , Z , W ] 7→ [X , iY , −Z , −W ]
Ω3 : P → P
[X , Y , Z , W ] 7→ [Z , Y , X , iW ],
3
3
(36)
then the induced action on P1 is generated by t 7→ it and t 7→ 1/t, hence we get AutB (P1 ) = D4 . The trisections T of
Q which are preserved under this action are given by
T :
(
Y 2 = XZ
W 3 + a(X 2 − Z 2 )W = 0
(37)
In local coordinates we get
w 3 + a(1 − t 4 )w = 0.
If a 6= 0 this trisection corresponds to the configuration III 4 by Tate’s Algorithm and for this case we have AutB (P1 ) = D4 .
• AutB (P1 ) = D6 case: If we consider the action of D6 on the quadric cone Q generated by
Θ3 : P 3 → P 3
[X , Y , Z , W ] 7→ [X , µ6 Y , µ62 Z , −µ6 W ]
Ω1 : P 3 → P 3
[X , Y , Z , W ] 7→ [Z , Y , X , W ],
(38)
then the induced action on P1 is generated by t 7→ µ6 t and t 7→ 1/t, hence we have AutB (P1 ) = D6 . The trisections T
on Q preserved under this action are given by
(
T :
Y 2 = XZ
W 3 + aX Z W + b(X 3 + Z 3 ) = 0
(39)
In local coordinates we get
w 3 + at 2 w + b(1 + t 6 ) = 0.
(40)
Whenever a = 0 and b 6= 0, this corresponds to the configuration II 6 . For this case we have AutB (P1 ) = D6 .
• AutB (P1 ) = A4 case: If we consider the action of A4 on Q generated by
Θ2 : P3 → P3
[X , Y , Z , W ] 7→ [X , µ3 Y , µ32 Z , W ]
Σ1 : P → P
[X , Y , Z , W ] 7→ [X − 2Y + Z , −2X + Y + Z , 4X + 4Y + Z , −3W ],
3
3
(41)
then it can be checked that the induced group on P1 is also A4 . The trisections T on Q which are preserved under this
action are given by
(
Y 2 = XZ
T :
(42)
W 3 + a(X 2 − Y Z )W + b(8X 3 + 20X Y Z − Z 3 ) = 0
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Automorphism groups of rational elliptic surfaces with section and constant J-map
In local coordinates we get
w 3 + a(1 − t 3 )w + b(8 + 20t 3 − t 6 ) = 0.
(43)
By Tate’s Algorithm, whenever a 6= 0 and b = 0, this corresponds to the configuration III 4 , and whenever a = 0 and
b 6= 0 this corresponds to the configuration II 6 . In all these cases we have AutB (P1 ) = A4 .
• AutB (P1 ) = S4 case: If we consider the action of the Octahedral Group (isomorphic to S4 ) on the quadric cone Q
generated by
∆2 : P3 → P3
[X , Y , Z , W ] 7→ [X , iY , −Z , −iW ]
Σ2 : P → P
[X , Y , Z , W ] 7→ [X + 2iY − Z , iX + iZ , −X + 2iY + Z , −2W ],
3
3
(44)
then it can be checked that the induced group on P1 is also S4 . The trisections T on Q which are preserved under this
action are given by
(
Y 2 = XZ
T :
(45)
W 3 + a(X 2 Y − Y Z 2 ) = 0
In local coordinates we get
w 3 + a(t − t 5 ) = 0.
(46)
If a 6= 0, this corresponds to the configuration II 6 . For this case we have AutB (P1 ) is the Octahedral Group.
5.5.
Autσ (B) Groups Corresponding to Non-cyclic AutB (P1 ) Groups
After showing the existence of non-cyclic AutB (P1 ) groups above, in this section we show what the groups Autσ (B)
corresponding to each case are.
• AutB (P1 ) = D2 : In this case Autσ (B) is a C M(B) = Z/6Z extension of AutB (P1 ) = D2 = Z/2Z × Z/2Z for the
configurations IV 2 II 2 and II 6 which have J ≡ 0. Moreover, C M(B) is in the center of Autσ (B). It can be checked that
the only order 24 groups which have these properties are Z/3Z × D4 , Z/3Z × Q8 , Z/12Z × Z/2Z and Z/3Z × (Z/2Z)3 .
For the configuration IV 2 II 2 , two of the automorphisms α ∈ Autσ (B) whose induced automorphism φ(α) fix the points
corresponding to the II fibers have ord(α) = 4 and four of such α have ord(α) = 12. There are no other α ∈ Autσ (B)
with ord(α) = 12. Among the four candidate groups for Autσ (B) listed above, the only one with the number of order 12
elements equal to 4 is Z/3Z × D4 . Therefore Autσ (B) = Z/3Z × D4 for the configuration IV 2 II 2 if AutB (P1 ) = D2 . For the
configuration II 6 , if AutB (P1 ) = D2 , two of the fixed points of the induced automorphisms correspond to II fibers and the
other fixed points correspond to smooth fibers. Thus, the number of order 12 elements of Autσ (B) is again 4. Therefore,
we have Autσ (B) = Z/3Z × D4 .
For the configuration III 4 , Autσ (B) is a C M(B) = Z/4Z extension of AutB (P1 ) = D2 and C M(B) is in the center of
Autσ (B). When AutB (P1 ) = D2 , the fibers over the fixed points of the induced automorphisms are smooth fibers, hence
if ord(φ(α)) = 2, then ord(α) = 2 or 4 for α ∈ Autσ (B). We will give the presentation of Autσ (B) as follows. Let A
be the generator of C M(B), R and S be two elements of Autσ (B) such that ord(R) = ord(S) = 2, and the induced
automorphisms φ(R) = r and φ(S) = s generate AutB (P1 ) = D2 . Thus, we have
D2 = hr, s | r 2 = s2 = 1, rs = sri
and
φ
0 → hAi ,→ Autσ (B) →
− D2 → 0
is short exact. Note that φ(RS) = rs has order 2, hence ord(RS) is either 2 or 4. Then we can write the presentation
of Autσ (B) as
Autσ (B) = hA, R, S | R 2 = S 2 = A4 = 1, AR = RA, AS = SA, RSRS = Ak i
1790
T. Karayayla
where we should determine whether k = 0 or k = 2 (since ord(RS) is either 2 or 4 and φ(RSRS) = 1). If we consider
the two smooth fibers E1 and E2 over the fixed points of s, then R restricts to an invertible map between E1 and E2 .
Since ord(S) = ord(φ(S)) = 2, S comes from the pull-back process of a surface with configuration I0∗ III 2 via the degree
2 map g2 on P1 . Here the fibers E1 and E2 are the fibers corresponding to the I0∗ and the smooth fiber over the fixed
points of g2 . Hence the action of S on E1 is by multiplication by −1 and its action on E2 is the identity or vice versa.
Then RSRS acts on both E1 and E2 as multiplication by −1, the same way as A2 . Therefore, we have k = 2 and the
presentation of Autσ (B) is
Autσ (B) = hA, R, S | R 2 = S 2 = A4 = 1, AR = RA, AS = SA, RSRS = A2 i.
(47)
• AutB (P1 ) = D3 : This case occurs for all B with the configuration IV 3 and for some B with the configuration II 6 . In
both cases C M(B) = Z/6Z is in the center of Autσ (B). Let r and s be the two generators of AutB (P1 ) = D3 such that
D3 = hr, s | s2 = r 3 = 1, srs = r 2 i.
Since AutB (P1 ) = D3 , from the previous section we know that fixed points of s correspond to smooth fibers for the II 6
case, and to a IV fiber and a smooth fiber for the IV 3 case. In any case there exists S ∈ Autσ (B) with ord(S) = 2 and
φ(S) = s. Also, there exists R ∈ Autσ (B) with ord(R) = 3 and φ(R) = r. Let C M(B) = hAi. Since
φ
0 → hAi ,→ Autσ (B) −
→ D3 → 0
is exact, we can write a presentation for Autσ (B) as:
Autσ (B) = hA, R, S | S 2 = R 3 = A6 = 1, AR = RA, AS = SA, SRS = R 2 Ak i
where k should be determined. Since ord(S) = 2, we get ord(SRS) = ord(R) = ord(R 2 ) = 3, then we have k ≡ 0
mod 2. If we take R 0 = RA4 , then another presentation for Autσ (B) is:
hA, R, S | S 2 = (R 0 )3 = A6 = 1, AS = SA, R 0 A = AR 0 , SR 0 S = (R 0 )2 Ak+2 i.
Therefore, all even k give isomorphic groups. If we take k = 0, the first presentation gives:
Autσ (B) = hA, R, S | S 2 = R 3 = A6 = 1, AS = SA, AR = RA, SRS = R 2 i
(48)
= Z/6Z × D3 .
• AutB (P1 ) = D4 : This case occurs for some B with the configuration III 4 . In this case C M(B) = Z/4Z is in the center
of Autσ (B). Let s ∈ AutB (P1 ) = D4 be one of the order 2 induced automorphisms fixing the points corresponding to two
of the III fibers. Then if S ∈ Autσ (B) is such that φ(S) = s, then ord(S) = 8. Let r ∈ AutB (P1 ) be one of the order 4
induced automorphisms. The fixed points of r correspond to smooth fibers, and there exists R ∈ Autσ (B) with φ(R) = r
and ord(R) = 4. Let C M(B) = hAi. Note that S 2 = A: We have
D4 = hr, s | r 4 = s2 = 1, srs = s3 i
and
φ
0 → hAi ,→ Autσ (B) →
− D4 → 0
is exact.Then we can write a presentation of Autσ (B) as:
Autσ (B) = hA, R, S | A4 = R 4 = S 8 = 1, AR = RA, AS = SA, S 2 = A, SRS = R 3 Ak i
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Automorphism groups of rational elliptic surfaces with section and constant J-map
where we should determine k. This presentation can be equivalently written as:
Autσ (B) = hR, S | R 4 = S 8 = 1, RS 2 = S 2 R, RSRS = S 2k i.
Note that the fixed points of the induced automorphism φ(RS) = rs correspond to smooth fibers, hence we have
ord(RS) = 2 or 4. Then RSRS is either the identity or A2 = S 4 . This argument shows that k = 0 or 2 in the above
presentation. But k = 0 and k = 2 give isomorphic groups since if we take R 0 = S 2 R, then a presentation of the group
can be written as:
hR 0 , S | (R 0 )4 = S 8 = 1, R 0 S 2 = S 2 R 0 , R 0 SR 0 S = S 2k+4 i.
Therefore, by taking k = 0 we get
Autσ (B) = hR, S | R 4 = S 8 = 1, S 2 R = RS 2 , RSRS = 1i.
(49)
• AutB (P1 ) = D6 : This case occurs for some B with the configuration II 6 . In this case we have C M(B) = Z/6Z and
C M(B) is in the center of Autσ (B). Let r ∈ AutB (P1 ) = D6 be one of the order 6 induced automorphisms. There exists
R ∈ Autσ (B) with φ(R) = r and ord(R) = 6. Let s ∈ AutB (P1 ) be an order 2 induced automorphism such that s 6= r 3
and the fixed points of s correspond to smooth fibers. Then there exists S ∈ Autσ (B) such that φ(S) = s and ord(S) = 2.
If C M(B) = hAi, then we have
AutB (P1 ) = D6 = hs, r | s2 = r 6 = 1, srs = r 5 i
and
φ
0 → hAi ,→ Autσ (B) →
− D6 → 0
is exact. Then we can write a presentation of Autσ (B) as
Autσ (B) = hA, R, S | A6 = R 6 = S 2 = 1, AR = RA, AS = SA, SRS = R 5 Ak i
where we should determine k. The relation SRS = R 5 Ak can be written as SRSR = Ak . The induced automorphism
φ(SR) = sr has fixed points which correspond to II fibers and ord(sr) = 2. Then ord(SR) = 4 or 12. Hence
ord(SRSR) = 2 or 6. Since φ(SRSR) = 1, we have SRSR = Ak where k is odd. But all odd k give isomorphic groups
when substituted in the above presentation. To see this, if we take R 0 = RA, then we can write another presentation of
Autσ (B) as
hA, R 0 , S | A6 = (R 0 )6 = S 2 = 1, AR 0 = R 0 A, AS = SA, SR 0 S = (R 0 )5 Ak+2 i.
Therefore, by taking k = 1 we get
Autσ (B) = hA, R, S | A6 = R 6 = S 2 = 1, AS = SA, AR = RA, SRS = R 5 Ai.
(50)
• AutB (P1 ) = A4 : This case occurs for some B with the configuration III 4 and for some B with the configuration II 6 . If
AutB (P1 ) = A4 and the configuration of singular fibers of B is II 6 , then C M(B) = Z/6Z is in the center of Autσ (B). A
presentation of the group AutB (P1 ) = A4 can be given by
hr, s | r 3 = s2 = (rs)3 = 1i.
There exists S ∈ Autσ (B) such that φ(S) = s and ord(S) = 4, and there exists R ∈ Autσ (B) such that φ(R) = r and
ord(R) = 3. If we have C M(B) = hAi, then we have S 2 = A3 . Since
φ
0 → hAi ,→ Autσ (B) →
− A4 → 0
1792
T. Karayayla
is exact, we can write a presentation of Autσ (B) as follows:
Autσ (B) = hA, R, S | A6 = R 3 = 1, S 2 = A3 , AR = RA, AS = SA, (RS)3 = An i
where we should determine n. If we take S 0 = A3 S, then another presentation of Autσ (B) is given by
Autσ (B) = hA, R, S 0 | A6 = R 3 = 1, (S 0 )2 = A3 , AR = RA, AS 0 = S 0 A, (RS 0 )3 = An+3 i.
Therefore, two values of n give isomorphic groups if they are congruent modulo 3. Note that φ(RS) has order 3 and
its fixed points correspond to smooth fibers. Then RS acts on those smooth fibers by a complex multiplication of some
order which divides 6. Therefore (RS)3 acts on those smooth fibers as the identity or complex multiplication by −1. This
shows that (RS)3 = A3 or the identity. Thus, we conclude that n ≡ 0 mod 3, hence taking n = 0 we get:
Autσ (B) = hA, R, S | A6 = R 3 = 1, S 2 = A3 , AR = RA, AS = SA, (RS)3 = 1i.
(51)
If we now consider the case AutB (P1 ) = A4 where the configuration of singular fibers of B is III 4 , in this case we have
C M(B) = Z/4Z and C M(B) is in the center of Autσ (B). If we number the points on P1 corresponding to the four III
fibers as p1 , p2 , p3 and p4 , then AutB (P1 ) = A4 is generated by two elements r and s which permute those four points
by acting on the indices as the permutations (234) and (12)(34). A presentation of AutB (P1 ) = A4 is then given by
A4 = hr, s | r 3 = s2 = (rs)3 = 1i.
There exists R ∈ Autσ (B) such that φ(R) = r with ord(R) = 3, and there exists S ∈ Autσ (B) such that φ(S) = s with
ord(S) = 2. If we let C M(B) = hAi, then the following sequence is exact
φ
0 → hAi ,→ Autσ (B) →
− A4 → 0
and a presentation of Autσ (B) can be written as:
Autσ (B) = hA, R, S | A4 = R 3 = S 2 = 1, AR = RA, AS = SA, (RS)3 = An i
where we should determine n. Note that two values of n give isomorphic groups if they are congruent modulo 2 since
by taking S 0 = SA2 the same group has another presentation as the following:
hA, R, S 0 | A4 = R 3 = (S 0 )2 = 1, AR = RA, AS 0 = S 0 A, (RS 0 )3 = An+2 i.
To determine n mod 2, let Ej , Fj (j = 1, 2,3) be the pairs of smooth fibers which are fibered over the fixed points of
the three order 2 induced automorphisms in A4 such that R(Ej ) = Ej+1 , R(Fj ) = Fj+1 (take the indices modulo 3) and
S(E1 ) = E1 , S(F1 ) = F1 . Then we have S(E2 ) = F2 and S(E3 ) = F3 . Note that all Ej and Fj are isomorphic elliptic curves
with j-invariant 1, hence both R and S restrict to elliptic curve isomorphisms between those elliptic curves since the zero
of one curve is mapped to the zero of the other. Such elliptic curve isomorphisms are given by complex multiplications
once coordinates are chosen for the elliptic curves. The coordinates on Ej and Fj can be chosen so that the restriction
maps R : Ej → Ej+1 and R : Fj → Fj+1 are all given by w 7→ w. The crucial point in this argument is that since
ord(S) = ord(φ(S)) = 2, without loss of generality S acts on E1 as the identity while it acts on F1 as multiplication
by −1. This follows from the fact that S is obtained from the pull-back of an elliptic surface with configuration I0∗ III 2
by the degree 2 map g2 on P1 such that without loss of generality F1 corresponds to the fiber I0∗ and E1 corresponds
to the smooth fiber over the other fixed point of g2 . Thus, we can write the restriction maps of S as S : E1 → E1 ,
w 7→ w, S : F1 → F1 , w 7→ −w, S : E2 → F2 , w 7→ kw, S : F2 → E2 , w 7→ (1/k)w, S : E3 → F3 , w 7→ mw and finally
S : F3 → E3 , w 7→ (1/m)w for some k, m ∈ C such that k 4 = m4 = 1. Then it can be checked that (RS)3 acts on E1 , E2
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Automorphism groups of rational elliptic surfaces with section and constant J-map
and F3 as multiplication by k/m and on F1 , F2 and E3 as multiplication by −m/k. But (RS)3 = An is in C M(B) hence it
acts globally on each smooth fiber of B as the complex multiplication by the same number. Thus, we have k/m = −m/k
hence k/m = ±i. Therefore (RS)3 = An is a complex multiplication of order 4. Then we get n = 1 or 3 and n = 1
mod 2. By taking n = 1, we can write
Autσ (B) = hA, R, S | A4 = R 3 = S 2 = 1, AR = RA, AS = SA, (RS)3 = Ai.
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• AutB (P1 ) = S4 (Octahedral Group) This case occurs for some B with the configuration of singular fibers II 6 . In this
case we have C M(B) = Z/6Z and C M(B) is in the center of Autσ (B). The group AutB (P1 ) which is the octahedral group
is isomorphic to the symmetric group S4 , and a presentation for this group can be written as:
AutB (P1 ) = S4 = hr, s | s4 = r 2 = (sr)3 = 1i.
The fixed points of the order 4 generator s of AutB (P1 ) correspond to two II fibers, hence there exists S ∈ Autσ (B) such
that φ(S) = s and ord(S) = 24. The fixed points of the order 2 generator r of AutB (P1 ) correspond to smooth fibers,
hence there exists R ∈ Autσ (B) such that φ(R) = r and ord(R) = 2. If we let C M(B) = hAi, then we have S 4 = A and
the following sequence is exact
φ
0 → hAi ,→ Autσ (B) →
− S4 → 0.
Therefore, we can write a presentation for Autσ (B) as:
Autσ (B) = hA, R, S | A6 = R 2 = 1, S 4 = A, AR = RA, (SR)3 = An i
where we should determine n. Note that ord(φ(SR)) = ord(sr) = 3, hence ord(SR) = 3 or 6 and ord((SR)3 ) = 1 or 2.
Thus, we should have n = 0 or 3. In fact both n = 0 and n = 3 give isomorphic groups since if we take R 0 = RA3 , then
another presentation of the same group can be written as:
hA, R 0 , S | A6 = (R 0 )2 = 1, S 4 = A, AR 0 = R 0 A, (SR 0 )3 = An+3 i.
Therefore, by taking n = 0 we get:
Autσ (B) = hA, S, R | A6 = R 2 = 1, S 4 = A, AR = RA, (SR)3 = 1i.
6.
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Infinite AutB (P1 ) case
In the previous section we have excluded the surfaces B which have only two singular fibers from the discussion. In this
section we determine the groups Autσ (B) for such surfaces and show that these groups are infinite.
• I0∗ I0∗ : The Weierstrass fibration of any rational elliptic surface B with this configuration is the quotient of the product
surface E × P1 (where E is an elliptic curve with j-invariant equal to the constant J-value of B) by the action (w, z) 7→
(−w, −z). Any α ∈ Autσ (B) lifts to an automorphism of E × P1 which respects this action and which maps fibers over
P1 to fibers, and which preserves the zero section {0} × P1 . Conversely, any such automorphism on E × P1 induces an
automorphism in Autσ (P1 ). If we define the group DC∗ by
DC∗ = {τ ∈ Aut(P1 )|τ = cz or τ = c/z, c ∈ C∗ },
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Autσ (B) = (DC∗ × C M(B))/h(−z, −I)i.
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then it can be shown that
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T. Karayayla
• II ∗ II: The Weierstrass fibration of any rational elliptic surface B with this configuration is the quotient of the product
space E × P1 by the action (w, z) 7→ (µ6 w, µ6 z). Any α ∈ Autσ (B) is induced by an automorphism of E × P1 which
preserves the zero section, maps fibers to fibers and respects this action. It can be checked that
Autσ (B) = ({cz ∈ Aut(P1 )|c ∈ C∗ } × C M(B))/h(µ6 z, µ6 I)i
= (C∗ × Z/6Z)/h(µ6 , 1)i = C∗ .
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• III ∗ III: The Weierstrass fibration of any rational elliptic surface B with this configuration is the quotient of a product
surface E × P1 (where E is an elliptic curve with j-invariant 1) by the action (w, z) 7→ (iw, iz). Any α ∈ Autσ (B) is
induced by an automorphism of E × P1 which preserves the zero section, maps fibers to fibers and respects this action.
It can be checked that
Autσ (B) = ({cz ∈ Aut(P1 )|c ∈ C∗ } × C M(B))/{(iz, iI)}
= (C∗ × Z/4Z)/h(i, 1)i = C∗ .
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• IV ∗ IV : The Weierstrass fibration of any rational elliptic surface B with this configuration is the quotient of the product
surface E × P1 (where E is an elliptic curve with j invariant 0) by the action (w, z) 7→ (µ3 w, µ3 z). Any α ∈ Autσ (B) is
induced by an automorphism of E × P1 which preserves the zero section, maps fibers to fibers and respects this action.
It can be shown that
Autσ (B) = ({cz ∈ Aut(P1 )|c ∈ C∗ } × C M(B)))/h(µ3 z, µ3 I)i
= (C∗ × Z/6Z)/h(µ3 , 2)i = C∗ × Z/2Z.
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References
[1] M. Artin, Algebra. Prentice Hall, Englewood Cliffs NJ, 1991.
[2] V. Bouchard and R. Donagi, On a class of non-simply connected Calabi-Yau 3-folds. Communications in Number
Theory and Physics 2, no. 1 (2008), 1–61.
[3] A. Fauntleroy, On the moduli of curves on rational ruled surfaces. American Journal of Mathematics 109, no. 3 (1987),
417–52.
[4] T. Karayayla, The classification of automorphism groups of rational elliptic surfaces with section. Advances in
Mathematics 230, no 1 (2012), 1–54.
[5] K. Kodaira, On compact complex analytic surfaces II. Annals of Mathematics 77 (1963), 563–626.
[6] R. Miranda, Persson’s list of singular fibers for a rational elliptic surface. Mathematische Zeitschrift 205 (1990)
191–211.
[7] R. Miranda, The basic theory of elliptic surfaces. Universita di Pisa Dipartimento di Matematica, 1989.
[8] R. Miranda and U. Persson, On extremal rational elliptic surfaces. Mathematische Zeitschrift 193 (1986), 537–58.
[9] K. Oguiso and T. Shioda, The Mordell-Weil lattice of a rational elliptic surface. Commentarii Mathematici Universitatis Sancti Pauli 40 (1991), 83–99.
[10] U. Persson, Configurations of Kodaira fibers on rational elliptic surfaces. Mathematische Zeitschrift 205, no.1 (1990),
1–47.
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