Linearisation of finite Abelian subgroups of the
Cremona group of the plane
Jérémy Blanc
November 4, 2018
Abstract
Given a finite Abelian subgroup of the Cremona group of the plane, we
provide a way to decide whether it is birationally conjugate to a group of
automorphisms of a minimal surface.
In particular, we prove that a finite cyclic group of birational transforma-
tions of the plane is linearisable if and only if none of its non-trivial elements
fix a curve of positive genus. For finite Abelian groups, there exists only one
surprising exception, a group isomorphic to Z/2Z×Z/4Z, whose non-trivial
elements do not fix a curve of positive genus but which is not conjugate to
a group of automorphisms of a minimal rational surface.
We also give some descriptions of automorphisms (not necessarily of finite
order) of del Pezzo surfaces and conic bundles.
1 Introduction
1.1 The main questions and results
In this paper, every surface will be complex, rational, algebraic and smooth, and
except for C2, will also be projective. By an automorphism of a surface we mean
a biregular algebraic morphism from the surface to itself. The group of automor-
phisms (respectively of birational transformations) of a surface S will be denoted
by Aut(S) (respectively by Bir(S)).
The group Bir(P2) is classically called the Cremona group. Taking some sur-
face S, any birational map S 99K P2 conjugates Bir(S) to Bir(P2); any subgroup
of Bir(S) may therefore be viewed as a subgroup of the Cremona group, up to
conjugacy.
The minimal surfaces are P2, P1×P1 and the Hirzebruch surfaces Fn for n ≥ 2;
their groups of automorphisms are a classical object of study, and their structures
are well known (see for example [Bea1]). These groups are in fact the maximal
connected algebraic subgroups of the Cremona group (see [Mu-Um], [Um]).
Given some group acting birationally on a surface, we would like to determine
some geometric properties that allow us to decide whether the group is conjugate to
a group of automorphisms of a minimal surface, or equivalently to decide whether
http://arxiv.org/abs/0704.0537v2
it belongs to a maximal connected algebraic subgroup of the Cremona group. This
conjugation looks like a linearisation, as we will see below, and explains our title.
We observe that the set of points of a minimal surface which are fixed by a
non-trivial automorphism is the union of a finite number of points and rational
curves. Given a group G of birational transformations of a surface, the following
properties are thus related (note that for us the genus is the geometric genus, so
that a curve has positive genus if and only if it is not rational); property (F ) is
our candidate for the geometric property for which we require:
(F ) No non-trivial element of G fixes (pointwise) a curve of positive genus.
(M) The group G is birationally conjugate to a group of automorphisms of
a minimal surface.
The fact that a curve of positive genus is not collapsed by a birational trans-
formation of surfaces implies that property (F ) is a conjugacy invariant; it is
clear that the same is true of property (M). The above discussion implies that
(M) ⇒ (F ); we would like to prove the converse.
The implication (F ) ⇒ (M) is true for finite cyclic groups of prime order
(see [Be-Bl]). The present article describes precisely the case of finite Abelian
groups. We prove that (F ) ⇒ (M) is true for finite cyclic groups of any order,
and that we may restrict the minimal surfaces to P2 or P1 × P1. In the case of
finite Abelian groups, there exists, up to conjugation, only one counterexample
to the implication, which is represented by a group isomorphic to Z/2Z × Z/4Z
acting biregularly on a special conic bundle. Precisely, we will prove the following
results, announced without proof as Theorems 4.4 and 4.5 in [Bla3]:
Theorem 1. Let G be a finite cyclic subgroup of order n of the Cremona group.
The following conditions are equivalent:
• If g ∈ G, g 6= 1, then g does not fix a curve of positive genus.
• G is birationally conjugate to a subgroup of Aut(P2).
• G is birationally conjugate to a subgroup of Aut(P1 × P1).
• G is birationally conjugate to the group of automorphisms of P2 generated
by (x : y : z) 7→ (x : y : e2iπ/nz).
Theorem 2. Let G be a finite Abelian subgroup of the Cremona group. The
following conditions are equivalent:
• If g ∈ G, g 6= 1, then g does not fix a curve of positive genus.
• G is birationally conjugate to a subgroup of Aut(P2), or to a subgroup of
Aut(P1 × P1) or to the group Cs24 isomorphic to Z/2Z × Z/4Z, generated
by the two elements
(x : y : z) 99K (yz : xy : −xz),
(x : y : z) 99K (yz(y − z) : xz(y + z) : xy(y + z)).
Moreover, this last group is conjugate neither to a subgroup of Aut(P2), nor to a
subgroup of Aut(P1 × P1).
Then, we discuss the case in which the group is infinite, respectively non-
Abelian (Section 11) and provide many examples of groups satisfying (F ) but not
Note that many finite groups which contain elements that fix a non-rational
curve are known, see for example [Wim] or more recently [Bla2] and [Do-Iz]. This
can also occur if the group is infinite, see [BPV] and [Bla5]. In fact, the set of
non-rational curves fixed by the elements of a group is a conjugacy invariant very
useful in describing conjugacy classes (see [Ba-Be], [dFe], [Bla4]).
1.2 How to decide
Given a finite Abelian group of birational transformations of a (rational) surface,
we thus have a good way to determine whether the group is birationally conjugate
to a group of automorphisms of a minimal surface (in fact to P2 or P1 × P1). If
some non-trivial element fixes a curve of positive genus (i.e. if condition (F ) is not
satisfied), this is false. Otherwise, if the group is not isomorphic to Z/2Z×Z/4Z,
it is birationally conjugate to a subgroup of Aut(P2) or of Aut(P1 × P1). There
are exactly four conjugacy classes of groups isomorphic to Z/2Z×Z/4Z satisfying
condition (F ) (see Theorem 5); three are conjugate to a subgroup of Aut(P2) or
Aut(P1 × P1), and the fourth (the group Cs24 of Theorem 2, described in detail
in Section 7) is not.
1.3 Linearisation of birational actions
Our question is related to that of linearisation of birational actions on C2. This
latter question has been studied intensively for holomorphic or polynomial actions,
see for example [De-Ku], [Kra] and [vdE]. Taking some group acting birationally
on C2, we would like to know if we may birationally conjugate this action to have
a linear action. Note that working on P2 or C2 is the same for this question.
Theorem 1 implies that for finite cyclic groups, being linearisable is equivalent
to fulfilling condition (F ). This is not true for finite Abelian groups in general,
since some groups acting biregularly on P1 × P1 are not birationally conjugate to
groups of automorphisms of P2. Note that Theorem 1 implies the following result
on linearisation, also announced in [Bla3] (as Theorem 4.2):
Theorem 3. Any birational map which is a root of a non-trivial linear automor-
phism of finite order of the plane is conjugate to a linear automorphism of the
plane.
1.4 The approach and other results
Our approach – followed in all the modern articles on the subject – is to view the
finite subgroups of the Cremona group as groups of (biregular) automorphisms of
smooth projective rational surfaces and then to assume that the action is minimal
(i.e. that it is not possible to blow-down some curves and obtain once again a
biregular action on a smooth surface). Manin and Iskovskikh ([Man] and [Isk2])
proved that the only possible cases are action on del Pezzo surfaces or conic bun-
dles. We will clarify this classification, for finite Abelian groups fillfulling (F), by
proving the following result:
Theorem 4. Let S be some smooth projective rational surface and let G ⊂ Aut(S)
be a finite Abelian group of automorphisms of S such that
• the pair (G,S) is minimal;
• if g ∈ G, g 6= 1, then g does not fix a curve of positive genus.
Then, one of the following occurs:
1. The surface S is minimal, i.e. S ∼= P2, or S ∼= Fn for some integer n 6= 1.
2. The surface S is a del Pezzo surface of degree 5 and G ∼= Z/5Z.
3. The surface S is a del Pezzo surface of degree 6 and G ∼= Z/6Z.
4. The pair (G,S) is isomorphic to the pair (Cs24, Ŝ4) defined in Section 7.
We will then prove that all the pairs in cases 1, 2 and 3 are birationally equiv-
alent to a group of automorphisms of P1 × P1 or P2, and that this is not true
for case 4. In fact, we are able to provide the precise description of all conjugacy
classes of finite Abelian subgroups of Bir(P2) satisfying (F ):
Theorem 5. Let G be a finite Abelian subgroup of the Cremona group such that
no non-trivial element of G fixes a curve of positive genus. Then, G is birationally
conjugate to one and only one of the following:
[1] G ∼= Z/nZ× Z/mZ g.b. (x, y) 7→ (ζnx, y) and (x, y) 7→ (x, ζmy)
[2] G ∼= Z/2Z× Z/2nZ g.b. (x, y) 7→ (x−1, y) and (x, y) 7→ (−x, ζ2ny)
[3] G ∼= (Z/2Z)2 × Z/2nZ g.b. (x, y) 7→ (±x±1, y) and (x, y) 7→ (x, ζ2ny)
[4] G ∼= (Z/2Z)3 g.b. (x, y) 7→ (±x,±y) and (x, y) 7→ (x−1, y−1)
[5] G ∼= (Z/2Z)4 g.b. (x, y) 7→ (±x±1,±y±1)
[6] G ∼= Z/2Z× Z/4Z g.b. (x, y) 7→ (x−1, y−1) and (x, y) 7→ (−y, x)
[7] G ∼= (Z/2Z)3 g.b. (x, y) 7→ (−x,−y), (x, y) 7→ (x−1, y−1),
and (x, y) 7→ (y, x)
[8] G ∼= (Z/2Z)× (Z/4Z) g.b. (x : y : z) 99K (yz(y − z) : xz(y + z) : xy(y + z))
and (x : y : z) 99K (yz : xy : −xz)
[9] G ∼= (Z/3Z)2 g.b. (x : y : z) 7→ (x : ζ3y : (ζ3)
and (x : y : z) 7→ (y : z : x)
(where n,m are positive integers, n divides m and ζn = e
2iπ/n).
Furthermore, the groups in cases [1] through [7] are birationally conjugate to sub-
groups of Aut(P1×P1), but the others are not. The groups in cases [1] and [9] are
birationally conjugate to subgroups of Aut(P2), but the others are not.
To prove these results, we will need a number of geometric results on automor-
phisms of rational surfaces, and in particular on automorphisms of conic bundles
and del Pezzo surfaces (Sections 3 to 9). We give for example the classification of
all the twisting elements (that exchange the two components of a singular fibre)
acting on conic bundles in Proposition 6.5 (for the elements of finite order) and
Proposition 6.8 (for those of infinite order); these are the most important elements
in this context (see Lemma 3.8). We also prove that actions of (possibly infinite)
Abelian groups on del Pezzo surfaces satifying (F ) are minimal only if the degree
is at least 5 (Section 9) and describe these cases precisely (Sections 4, 5 and 9).
We also show that a finite Abelian group acting on a projective smooth surface
S such that (KS)
2 ≥ 5 is birationally conjugate to a group of automorphisms of
P1 × P1 or P2 (Corollary 9.10) and in particular satisfies (F ).
1.5 Comparison with other work
Many authors have considered the finite subgroups of Bir(P2). Among them,
S. Kantor [Kan] gave a classification of the finite subgroups, which was incomplete
and included some mistakes; A. Wiman [Wim] and then I.V. Dolgachev and V.A.
Iskovskikh [Do-Iz] successively improved Kantor’s results. The long paper [Do-Iz]
expounds the general theory of finite subgroups of Bir(P2) according to the modern
techniques of algebraic geometry, and will be for years to come the reference on
the subject. Our viewpoint and aim differ from those of [Do-Iz]: we are only
interested in Abelian groups in relation with the above conditions (F) and (M);
this gives a restricted setting in which the theoretical approach is simplified and the
results obtained are more accurate. In the study of del Pezzo surfaces, using the
classification [Do-Iz] of subgroups of automorphisms would require the examination
of many cases; for the sake of readibility we prefered a direct proof. The two
main theorems of [Do-Iz] on automorphism of conic bundles (Proposition 5.3 and
Theorem 5.7(2)) do not exclude groups satisfying property (F ) and do not give
explicit forms for the generators of the groups or the surfaces.
1.6 Aknowledgements
This article is part of my PhD thesis [Bla2]; I am grateful to my advisor T. Vust
for his invaluable help during these years, to I. Dolgachev for helpful discussions,
and thank J.-P. Serre and the referees for their useful remarks on this paper.
2 Automorphisms of P2 or P1 × P1
Note that a linear automorphism of C2 may be extended to an automorphism of
either P2 or P1 × P1. Moreover, the automorphisms of finite order of these three
surfaces are birationally conjugate. For finite Abelian groups, the situation is quite
different. We give here the birational equivalence of these groups.
Notation 2.1. The element [a : b : c] denotes the diagonal automorphism (x : y :
z) 7→ (ax : by : cz) of P2, and ζm = e
2iπ/m.
Proposition 2.2 (Finite Abelian subgroups of Aut(P2)). Every finite Abelian
subgroup of Aut(P2) = PGL(3,C) is conjugate, in the Cremona group Bir(P2), to
one and only one of the following:
1. A diagonal group, isomorphic to Z/nZ×Z/mZ, where n divides m, generated
by [1 : ζn : 1] and [ζm : 1 : 1]. (The case n = 1 gives the cyclic groups).
2. The special group V9, isomorphic to Z/3Z×Z/3Z, generated by [1 : ζ3 : (ζ3)
and (x : y : z) 7→ (y : z : x).
Thus, except for the group V9, two isomorphic finite Abelian subgroups of PGL(3,C)
are conjugate in Bir(P2).
Proof. First of all, a simple calculation shows that every finite Abelian subgroup
of PGL(3,C) is either diagonalisable or conjugate to the group V9. Furthermore,
since this last group does not fix any point, it is not diagonalisable, even in Bir(P2)
[Ko-Sz, Proposition A.2].
Let T denote the torus of PGL(3,C) constituted by diagonal automorphisms
of P2. Let G be a finite subgroup of T ; as an abstract group it is isomorphic to
Z/nZ× Z/mZ, where n divides m. Now we can conjugate G by a birational map
of the form h : (x, y) 99K (xayb, xcyd) so that it contains [ζm : 1 : 1] (see [Be-Bl]
and [Bla1]). Since h normalizes the torus T , the group G remains diagonal and
contains the n-torsion of T , hence it contains [1 : ζn : 1].
Corollary 2.3. Every finite Abelian group of linear automorphisms of C2 is bi-
rationally conjugate to a diagonal group, isomorphic to Z/nZ × Z/mZ, where n
divides m, generated by (x, y) 7→ (ζnx, y) and (x, y) 7→ (x, ζmy).
Proof. This follows from the fact that the group GL(2,C) of linear automorphisms
of C2 extends to a group of automorphisms of P2 that leaves the line at infinity
invariant and fixes one point.
Example 2.4. Note that Aut(P1 × P1) contains the group (C∗)2 ⋊ Z/2Z, where
(C∗)2 is the group of automorphisms of the form (x, y) 7→ (αx, βy), α, β ∈ C∗, and
Z/2Z is generated by the automorphism (x, y) 7→ (y, x).
The birational map (x, y) 99K (x : y : 1) from P1×P1 to P2 conjugates (C∗)2⋊
Z/2Z to the group of automorphisms of P2 generated by (x : y : z) 7→ (αx : βy : z),
α, β ∈ C∗ and (x : y : z) 7→ (y : x : z).
Proposition 2.5 (Finite Abelian subgroups of Aut(P1 × P1)). Up to birational
conjugation, every finite Abelian subgroup of Aut(P1×P1) is conjugate to one and
only one of the following:
[1] G ∼= Z/nZ× Z/mZ g.b. (x, y) 7→ (ζnx, y) and (x, y) 7→ (x, ζmy)
[2] G ∼= Z/2Z× Z/2nZ g.b. (x, y) 7→ (x−1, y) and (x, y) 7→ (−x, ζ2ny)
[3] G ∼= (Z/2Z)2 × Z/2nZ g.b. (x, y) 7→ (±x±1, y) and (x, y) 7→ (x, ζ2ny)
[4] G ∼= (Z/2Z)3 g.b. (x, y) 7→ (±x,±y) and (x, y) 7→ (x−1, y−1)
[5] G ∼= (Z/2Z)4 g.b. (x, y) 7→ (±x±1,±y±1)
[6] G ∼= Z/2Z× Z/4Z g.b. (x, y) 7→ (x−1, y−1) and (x, y) 7→ (−y, x)
[7] G ∼= (Z/2Z)3 g.b. (x, y) 7→ (−x,−y), (x, y) 7→ (x−1, y−1),
and (x, y) 7→ (y, x)
(where n,m are positive integers, n divides m and ζn = e
2iπ/n).
Furthermore, the groups in [1] are conjugate to subgroups of Aut(P2), but the others
are not.
Proof. Recall that Aut(P1 × P1) = (PGL(2,C) × PGL(2,C)) ⋊ Z/2Z. Let G be
some finite Abelian subgroup of Aut(P1 × P1); we now prove that G is conjugate
to one of the groups in cases [1] through [7].
First of all, if G is a subgroup of the group (C∗)2⋊Z/2Z given in Example 2.4,
then it is conjugate to a subgroup of Aut(P2) and hence to a group in case [1].
Assume that G ⊂ PGL(2,C)×PGL(2,C) and denote by π1 and π2 the projec-
tions πi : PGL(2,C)×PGL(2,C) → PGL(2,C) on the i-th factor. Since π1(G) and
π2(G) are finite Abelian subgroups of PGL(2,C) each is conjugate to a diagonal
cyclic group or to the group x 99K ±x±1, isomorphic to (Z/2Z)2. We enumerate
the possible cases.
If both groups π1(G) and π2(G) are cyclic, the group G is conjugate to a
subgroup of the diagonal torus (C∗)2 of automorphisms of the form (x, y) 7→
(αx, βy), α, β ∈ C∗.
If exactly one of the two groups π1(G) and π2(G) is cyclic we may assume, up
to conjugation in Aut(P1 × P1), that π2(G) is cyclic, generated by y 7→ ζmy, for
some integer m ≥ 1, and that π1(G) is the group x 99K ±x
±1. We use the exact
sequence 1 → G ∩ kerπ2 → G → π2(G) → 1 and find, up to conjugation, two
possibilities for G:
(a) G is generated by (x, y) 7→ (x−1, y) and (x, y) 7→ (−x, ζmy).
(b) G is generated by (x, y) 7→ (±x±1, y) and (x, y) 7→ (x, ζmy).
If m is even, we obtain respectively [2] and [3] for n = m/2. If m is odd, the two
groups are equal; conjugating by ϕ : (x, y) 99K (x, y(x + x−1)) (which conjugates
(x, y) 7→ (−x, y) to (x, y) 7→ (−x,−y)) we obtain the group [2] for n = m.
If both groups π1(G) and π2(G) are isomorphic to (Z/2Z)
2, then up to conju-
gation, we obtain three groups, namely
(a) G is generated by (x, y) 7→ (−x,−y) and (x, y) 7→ (x−1, y−1).
(b) G is generated by (x, y) 7→ (±x,±y) and (x, y) 7→ (x−1, y−1).
(c) G is given by (x, y) 7→ (±x±1,±y±1).
The group [2] with n = 1 is conjugate to (a) by (x, y) 99K (x, x y+x
y+x−1
). The groups
(b) and (c) are respectively equal to [4] and [5].
We now suppose that the group G is not contained in PGL(2,C)×PGL(2,C).
Any element ϕ ∈ Aut(P1×P1) not contained in PGL(2,C)×PGL(2,C) is conjugate
to ϕ : (x, y) 7→ (α(y), x), where α ∈ Aut(P1), and if ϕ is of finite order, α may be
chosen to be y 7→ λy with λ ∈ C∗ a root of unity.
Thus, up to conjugation, G is generated by the group H = G ∩ (PGL(2,C)×
PGL(2,C)) and one element (x, y) 7→ (λy, x), for some λ ∈ C∗ of finite order. Since
the group G is Abelian, every element of H is of the form (x, y) 7→ (β(x), β(y)),
for some β ∈ PGL(2,C) satisfying β(λx) = λβ(x). Three possibilities occur,
depending on the value of λ which may be 1, −1 or something else.
If λ = 1, we conjugate the group by some element (x, y) 7→ (γ(x), γ(y)) so that
H is either diagonal or equal to the group generated by (x, y) 7→ (−x,−y) and
(x, y) 7→ (x−1, y−1). In the first situation, the group is contained in (C∗)2 ⋊Z/2Z
(which gives [1]); the second situation gives [7].
If λ = −1, the group H contains the square of (x, y) 7→ (−y, x), which is
(x, y) 7→ (−x,−y) and is either cyclic or generated by (x, y) 7→ (−x,−y) and
(x, y) 7→ (x−1, y−1). If H is cyclic, it is diagonal, since it contains (x, y) 7→
(−x,−y), so G is contained in (C∗)2 ⋊ Z/2Z. The second possibility gives [6].
If λ 6= ±1, the group H is diagonal and then G is contained in (C∗)2 ⋊ Z/2Z.
We now prove that distinct groups of the list are not birationally conjugate.
First of all, each group of case [1] fixes at least one point of P1 × P1. Since the
other groups of the list don’t fix any point, they are not conjugate to [1] [Ko-Sz,
Proposition A.2].
Consider the other groups. The set of isomorphic groups are those of cases [3]
(with n = 1), [4] and [7] (isomorphic to (Z/2Z)3), and of cases [2] (with n = 2)
and [6] (isomorphic to Z/2Z× Z/4Z).
The groups of cases [2] to [5] leave two pencils of rational curves invariant (the
fibres of the two projections P1 × P1 → P1) which intersect freely in exactly one
point. We prove that this is not the case for [6] and [7]; this shows that these
two groups are not birationally conjugate to any of the previous groups. Take
G ⊂ Aut(P1×P1) to be either [6] or [7]. We have then Pic(P1×P1)G = Zd, where
d = − 1
KP1×P1 is the diagonal of P
1×P1. Suppose that there exist two G-invariant
pencils Λ1 = n1d and Λ2 = n2d of rational curves, for some positive integers n1, n2
(we identify here a pencil with the class of its elements in Pic(P1 × P1)G). The
intersection Λ1 · Λ2 = 2n1n2 is an even integer. Note that the fixed part of the
intersection is also even, since G is of order 8 and acts without fixed points on
P1 × P1. The free part of the intersection is then also an even integer and hence
is not 1.
Let us now prove that [4] is not birationally conjugate to [3] (with n = 1).
This follows from the fact that [4] contains three subgroups that are fixed-point
free (the groups generated by (x, y) 7→ (x−1, y−1) and one of the three involutions
of the group (x, y) 7→ (±x,±y)), whereas [3] (with n = 1) contains only one such
subgroup, which is (x, y) 7→ (±x±1, y).
We now prove the last assertion. The finite Abelian groups of automorphisms
of P2 are conjugate either to [1] or to the group V9, isomorphic to (Z/3Z)
2 (see
Proposition 2.2). As no group of the list [2] through [7] is isomorphic to (Z/3Z)2,
we are done.
Summary of this section. We have found that the groups common to the
three surfaces C2,P2 and P1 × P1 are the ”diagonal” ones (generated by (x, y) 7→
(ζnx, y) and (x, y) 7→ (x, ζmy)). On P
2 there is only one more group, which is the
special group V9, and on P
1 × P1 there are 2 families ([2] and [3]) and 4 special
groups ([4], [5], [6] and [7]).
3 Some facts about automorphisms of conic bun-
We first consider conic bundles without mentioning any group action on them. We
recall some classical definitions:
Definition 3.1. Let S be a rational surface and π : S → P1 be a morphism. We
say that the pair (S, π) is a conic bundle if a general fibre of π is isomorphic to P1,
with a finite number of exceptions: these singular fibres are the union of smooth
rational curves F1 and F2 such that (F1)
2 = (F2)
2 = −1 and F1 · F2 = 1.
Let (S, π) and (S̃, π̃) be two conic bundles. We say that ϕ : S 99K S̃ is a
birational map of conic bundles if ϕ is a birational map which sends a general fibre
of π on a general fibre of π̃.
We say that a conic bundle (S, π) is minimal if any birational morphism of
conic bundles (S, π) → (S̃, π̃) is an isomorphism.
We remind the reader of the following well-known result:
Lemma 3.2. Let (S, π) be a conic bundle. The following conditions are equivalent:
• (S, π) is minimal.
• The fibration π is smooth, i.e. no fibre of π is singular.
• S is a Hirzebruch surface Fm, for some integer m ≥ 0. �
Blowing-down one irreducible component in any singular fibre of a conic bundle
(S, π), we obtain a birational morphism of conic bundles S → Fm for some integer
m ≥ 0. Note that m depends on the choice of the blown-down components. The
following lemma gives some information on the possibilities. Note first that since
the sections of Fm have self-intersection≥ −m, the self-intersections of the sections
of π are also bounded from below.
Lemma 3.3. Let (S, π) be a conic bundle on a surface S 6∼= P1 × P1. Let −n be
the minimal self-intersection of sections of π and let r be the number of singular
fibres of π. Then n ≥ 1 and:
1. There exists a birational morphism of conic bundles p− : S → Fn such that:
(a) p− is the blow-up of r points of Fn, none of which lies on the exceptional
section En.
(b) The strict pull-back Ẽn of En by p− is a section of π with self-intersection
2. If there exist two different sections of π with self-intersection −n, then r ≥
2n. In this case, there exist birational morphisms of conic bundles p0 : S →
F0 = P
1 × P1 and p1 : S → F1.
Proof. We denote by s a section of π of minimal self-intersection −n, for some
integer n (this integer is in fact positive, as will appear in the proof). Note that
this curve intersects exactly one irreducible component of each singular fibre.
If r = 0, the lemma is trivially true: take p− to be the identity map. We now
suppose that r ≥ 1, and denote by F1, ..., Fr the irreducible components of the
singular fibres which do not intersect s. Blowing these down, we get a birational
morphism of conic bundles p− : S → Fm, for some integerm ≥ 0. The image of the
section s by p− is a section of the conic bundle of Fm of minimal self-intersection,
so we get m = n, and n ≥ 0. If we had n = 0, then taking some section s̃ of
P1×P1 of self-intersection 0 passing through at least one blown-up point, its strict
pull-back by p− would be a section of negative self-intersection, which contradicts
the minimality of s2 = −n = 0. We find finally that m = n > 0, and that p−(s)
is the unique section Fn of self-intersection −n. This proves the first assertion.
We now prove the second assertion. Suppose that some section t 6= s has self-
intersection −n. The Picard group of S is generated by s = p∗−(En), the divisor f
of a fibre of π and F1, ..., Fr. Write t as t = s+ bf −
i=1 aiFi, for some integers
b, a1, ..., ar, with a1, ..., ar ≥ 0. We have t
2 = −n and t · (t+KS) = −2 (adjunction
formula), where KS = p
−(KFn) +
i=1 Fi = −(n + 2)f − 2s +
i=1 Fi. These
relations give:
s2 = t2 = s2 −
i=1 a
i + 2b,
n− 2 = t ·KS = −(n+ 2) + 2n− 2b+
i=1 ai,
whence
i=1 ai =
i=1 a
i = 2b, so each ai is equal to 0 or 1 and consequently
2b ≤ r. Since s · t = b− n ≥ 0, we find that r ≥ 2n, as announced.
Finally, by contracting f − F1, f − F2, ..., f − Fn, Fn+1, Fn+2, ..., Fr, we obtain
a birational morphism p0 of conic bundles which sends s on a section of self-
intersection 0 and whose image is thus F0. Similarly, the morphism p1 : S → F1
is given by the contraction of f − F1, f − F2, ..., f − Fn−1, Fn, Fn+1, ..., Fr.
We now add some group actions on the conic bundles, and give natural defi-
nitions (note that we will restrict ourselves to finite or Abelian groups only when
this is needed and will then say so):
Definition 3.4. Let (S, π) be some conic bundle.
• We denote by Aut(S, π) ⊂ Aut(S) the group of automorphisms of the conic
bundle, i.e. automorphisms of S that send a general fibre of π on another
general fibre.
Let G ⊂ Aut(S, π) be some group of automorphisms of the conic bundle (S, π).
• We say that a birational map of conic bundles ϕ : S 99K S̃ is G-equivariant
if the G-action on S̃ induced by ϕ is biregular (it is clear that it preserves
the conic bundle structure).
• We say that the triple (G,S, π) is minimal if any G-equivariant birational
morphism of conic bundles ϕ : S → S̃ is an isomorphism.
Remark 3.5. We insist on the fact that since a conic bundle is for us a pair (S, π),
an automorphism of S is not necessarily an automorphism of the conic bundle (i.e.
Aut(S) 6= Aut(S, π) in general). One should be aware that in the literature, conic
bundle sometimes means ”a variety admitting a conic bundle structure”.
Remark 3.6. If G ⊂ Aut(S, π) is such that the pair (G,S) is minimal, so is the
triple (G,S, π). The converse is not true in general (see Remark 4.7).
Note that any automorphism of the conic bundle acts on the set of singular
fibres and on its irreducible components. The permutation of the two components
of a singular fibre is very important (Lemma 3.8). For this reason, we introduce
some terminology:
Definition 3.7. Let g ∈ Aut(S, π) be an automorphism of the conic bundle (S, π).
Let F = {F1, F2} be a singular fibre. We say that g twists the singular fibre F if
g(F1) = F2 (and consequently g(F2) = F1).
If g twists at least one singular fibre of π, we will say that g twists the conic
bundle (S, π), or simply (if the conic bundle is implicit) that g is a twisting element.
Here is a simple but very important observation:
Lemma 3.8. Let G ⊂ Aut(S, π) be a group of automorphisms of a conic bundle.
The following conditions are equivalent:
1. The triple (G,S, π) is minimal.
2. Any singular fibre of π is twisted by some element of G. �
Remark 3.9. An automorphism of a conic bundle with a non-trivial action on
the basis of the fibration may twist at most two singular fibres. However, an
automorphism with a trivial action on the basis of the fibration may twist a large
number of fibres. We will give in Propositions 6.5 and 6.8 a precise description of
all twisting elements.
The following lemma is a direct consequence of Lemma 3.3; it provides infor-
mation on the structure of the underlying variety of a conic bundle admitting a
twisting automorphism.
Lemma 3.10. Suppose that some automorphism of the conic bundle (S, π) twists
at least one singular fibre. Then, the following occur.
1. There exist two birational morphisms of conic bundles p0 : S → F0 and
p1 : S → F1 (which are not g-equivariant).
2. Let −n be the minimal self-intersection of sections of π and let r be the
number of singular fibres of π. Then, r ≥ 2n ≥ 2.
Proof. Note that any section of π touches exactly one component of each singular
fibre. Since g twists some singular fibre, its action on the set of sections of S is
fixed-point-free. The number of sections of minimal self-intersection is then greater
than 1 and we apply Lemma 3.3 to get the result.
Remark 3.11. A result of the same kind can be found in [Isk1], Theorem 1.1.
Lemma 3.12. Let G ⊂ Aut(S, π) be a group of automorphisms of the conic bundle
(S, π), such that:
• π has at most 3 singular fibres (or equivalently (KS)
2 ≥ 5);
• the triple (G,S, π) is minimal.
Then, S is either a Hirzeburch surface or a del Pezzo surface of degree 5 or 6,
depending on whether the number of singular fibres is 0, 3 or 2 respectively.
Proof. Let −n be the minimal self-intersection of sections of π and let r ≤ 3 be
the number of singular fibres of π. If r = 0, we are done, so we may suppose that
r > 0. Since (G,S, π) is minimal, every singular fibre is twisted by some element
of G (Lemma 3.8). From Lemma 3.10, we get r ≥ 2n ≥ 2, whence r = 2 or 3 and
n = 1, and we obtain the existence of some birational morphism of conic bundles
(not G-equivariant) p1 : S → F1. So the surface S is obtained by the blow-up
of 2 or 3 points of F1, not on the exceptional section (Lemma 3.3), and thus by
blowing-up 3 or 4 points of P2, no 3 of which are collinear (otherwise we would
have a section of self-intersection ≤ −2). The surface is then a del Pezzo surface
of degree 6 or 5.
Remark 3.13. We conclude this section by mentioning an important exact se-
quence. Let G ⊂ Aut(S, π) be some group of automorphisms of a conic bundle
(S, π). We have a natural homomorphism π : G → Aut(P1) = PGL(2,C) that
satisfies π(g)π = πg, for every g ∈ G. We observe that the group G′ = kerπ of
automorphisms that leave every fibre invariant embeds in the group PGL(2,C(x))
of automorphisms of the generic fibre P1(C(x)). Then we get the exact sequence
1 → G′ → G
→ π(G) → 1. (1)
This restricts the structure of G; for example if G is Abelian and finite, so are
G′ and π(G), and we know that the finite Abelian subgroups of PGL(2,C) and
PGL(2,C(x)) are either cyclic or isomorphic to (Z/2Z)2.
We also see that the group G is birationally conjugate to a subgroup of the
group of birational transformations of P1 × P1 of the form (written in affine coor-
dinates):
(x, y) 99K
ax+ b
cx+ d
α(x)y + β(x)
γ(x)y + δ(x)
where a, b, c, d ∈ C, α, β, γ, δ ∈ C(x), and (ad− bc)(αδ − βγ) 6= 0.
This group, called the de Jonquières group, is the group of birational transfor-
mations of P1 ×P1 that preserve the fibration induced by the first projection, and
is isomorphic to PGL(2,C(x))⋊ PGL(2,C).
The subgroups of this group can be studied algebraically (as in [Bea2] and
[Bla4]) but we will not adopt this point of view here.
4 The del Pezzo surface of degree 6
There is a single isomorphism class of del Pezzo surfaces of degree 6, since all
sets of three non-collinear points of P2 are equivalent under the action of linear
automorphisms. Consider the surface S6 of degree 6 defined by the blow-up of
the points A1 = (1 : 0 : 0), A2 = (0 : 1 : 0) and A3 = (0 : 0 : 1). We may view
it in P2 × P2, defined as {
(x : y : z), (u : v : w)
| ux = vy = wz}, where the
blow-down p : S6 → P
2 is the restriction of the projection on one copy of P2,
explicitly p :
(x : y : z), (u : v : w)
7→ (x : y : z). There are exactly 6 exceptional
divisors, which are the pull-backs of the Ai’s by the two projection morphisms.
We write Ei = p
−1(Ai) and denote by Dij the strict pull-back by p of the line of
P2 passing through Ai and Aj .
The group of automorphisms of S6 is well known (see for example [Wim],
[Do-Iz]). It is isomorphic to (C∗)2 ⋊ (Sym3 × Z/2Z), where (C
∗)2 ⋊ Sym3 is the
lift on S6 of the group of automorphisms of P
2 that leave the set {A1, A2, A3}
invariant, and Z/2Z is generated by the permutation of the two factors (it is the
lift of the standard quadratic transformation (x : y : z) 99K (yz : xz : xy) of P2);
the action of Z/2Z on (C∗)2 sends an element on its inverse.
There are three conic bundle structures on the surface S6. Let π1 : S6 → P
be the morphism defined by
(x : y : z), (u : v : w)
(y : z) if (x : y : z) 6= (1 : 0 : 0),
(w : v) if (u : v : w) 6= (1 : 0 : 0).
Note that p sends the fibres of π1 on lines of P
2 passing through A1. There are
exactly two singular fibres of this fibration, namely
π−11 (1 : 0) = {E2, D12} and π
1 (0 : 1) = {E3, D13};
and E1, D23 are sections of π1.
Lemma 4.1. The group Aut(S6, π1) of automorphisms of the conic bundle (S6, π1)
acts on the hexagon {E1, E2, E3, D12, D13, D23} and leaves the set {E1, D23} in-
variant.
1. The action on the hexagon gives rise to the exact sequence
1 → (C∗)2 → Aut(S6, π1) → (Z/2Z)
2 → 1.
2. This exact sequence is split and Aut(S6, π1) = (C
∗)2 ⋊ (Z/2Z)2, where
(a) (C∗)2 is the group of automorphisms of the form(
(x : y : z), (u : v : w)
(x : αy : βz), (αβu : βv : αw)
, α, β ∈ C∗.
(b) The group (Z/2Z)2 is generated by the automorphisms
(x : y : z), (u : v : w)
(x : z : y), (u : w : v)
whose action on the set of exceptional divisors is (E2 E3)(D12 D13);
(x : y : z), (u : v : w)
(u : v : w), (x : y : z)
whose action is (E1 D23)(E2 D13)(E3 D12).
(c) The action of (Z/2Z)2 on (C∗)2 is generated by permutation of the
coordinates and inversion.
Proof. Since Aut(S6) acts on the hexagon, so does Aut(S6, π1) ⊂ Aut(S6). Since
the group Aut(S6, π1) sends a section on a section, the set {E1, D23} is invariant.
The group (C∗)2 leaves the conic bundle invariant, and is the kernel of the
action of Aut(S6, π1) on the hexagon. As the set {E1, D23} is invariant, the
image is contained in the group (Z/2Z)2 generated by (E2 E3)(D12 D13) and
(E1 D23)(E2 D13)(E3 D12). The rest of the lemma follows directly.
By permuting coordinates, we have two other conic bundle structures on the
surface S6, given by the following morphisms π2, π3 : S6 → P
(x : y : z), (u : v : w)
(x : z) if (x : y : z) 6= (0 : 1 : 0),
(w : u) if (u : v : w) 6= (0 : 1 : 0).
(x : y : z), (u : v : w)
(x : y) if (x : y : z) 6= (0 : 0 : 1),
(v : u) if (u : v : w) 6= (0 : 0 : 1).
The description of the exceptional divisors on S6 shows that π1, π2 and π3 are
the only conic bundle structures on S6.
Lemma 4.2. For i = 1, 2, 3, the pair (Aut(S6, πi), S6) is not minimal. More
precisely the morphism πj×πk : S6 → P
1×P1 conjugates Aut(S6, πi) to a subgroup
of Aut(P1 × P1), where {i, j, k} = {1, 2, 3}.
Proof. The union of the sections E1 and D23 is invariant by the action of the
whole group Aut(S6, π1). Since these two exceptional divisors don’t intersect, we
can contract both and get a birational Aut(S6, π1)-equivariant morphism from S6
to P1×P1: the pair (Aut(S6, π1), S6) is thus not minimal; explicitly, the birational
morphism is given by q 7→ (π2(q), π3(q)), as stated in the lemma. We obtain the
other cases by permuting coordinates.
Remark 4.3. The subgroup of Aut(P1×P1) obtained in this manner doesn’t leave
any of the two fibrations of P1 × P1 invariant.
Corollary 4.4. If (G,S6) is a minimal pair (where G ⊂ Aut(S6)), then G does
not preserve any conic bundle structure. �
We conclude this section with a fundamental example; we will use several times
the following automorphism κα,β of (S6, π1):
Example 4.5. For any α, β ∈ C∗, we define κα,β to be the following automorphism
of (S6, π1):
κα,β :
(x : y : z), (u : v : w)
(u : αw : βv), (x : α−1z : β−1y)
Note that κα,β twists the two singular fibres of π1 (see Lemma 4.6 below); its
action on the basis of the fibration is (x1 : x2) 7→ (αx1 : βx2) and
κ2α,β(
(x : y : z), (u : v : w)
(x : αβ−1y : α−1βz), (u : α−1βv : αβ−1w)
So κα,β is an involution if and only if its action on the basis of the fibration is
trivial.
Lemma 4.6. Let g ∈ Aut(S6, π1) be an automorphism of the conic bundle (S6, π1).
The following conditions are equivalent:
• the triple (< g >, S6, π1) is minimal;
• g twists the two singular fibres of π1;
• the action of g on the exceptional divisors of S6 is (E1 D23)(E2 D12)(E3 D13);
• g = κα,β for some α, β ∈ C
Proof. According to Lemma 4.1 the action of Aut(S6, π1) on the exceptional curves
is isomorphic to (Z/2Z)2 and hence the possible actions of g 6= 1 are these:
1. id, 2. (E2 E3)(D12 D13),
3. (E1 D23)(E2 D13)(E3 D12), 4. (E1 D23)(E2 D12)(E3 D13).
In the first three cases, the triple (< g >, S6, π1) is not minimal. Indeed, the
blow-down of {E2, E3} or {E2, D13} gives a g-equivariant birational morphism of
conic bundles.
Hence, if (< g >, S6, π1) is minimal, its action on the exceptional curves is the
fourth one above, as stated in the lemma, and it then twists the two singular fibres
of π1. Conversely if g twists the two singular fibres of π1, the triple (< g >, S6, π1)
is minimal (by Lemma 3.8).
It remains to see that the last assertion is equivalent to the others. This follows
from Lemma 4.1; indeed this lemma implies that (C∗)2κ1,1 is the set of elements
of Aut(S6, π1) inducing the permutation (E1 D23)(E2 D12)(E3 D13).
Remark 4.7. The pair (Aut(S6, π1), S6) is not minimal (Lemma 4.2). Consequently
< κα,β > is an example of a group whose action on the surface is not minimal,
but whose action on a conic bundle is minimal.
5 The del Pezzo surface of degree 5
As for the del Pezzo surface of degree 6, there is a single isomorphism class of del
Pezzo surfaces of degree 5. Consider the del Pezzo surface S5 of degree 5 defined
by the blow-up p : S5 → P
2 of the points A1 = (1 : 0 : 0), A2 = (0 : 1 : 0),
A3 = (0 : 0 : 1) and A4 = (1 : 1 : 1). There are 10 exceptional divisors on S5,
namely the divisor Ei = p
−1(Ai), for i = 1, ..., 4, and the strict pull-back Dij of
the line of P2 passing through Ai and Aj , for 1 ≤ i < j ≤ 4. There are 5 sets of 4
skew exceptional divisors on S5, namely
F1 = {E1, D23, D24, D34}, F2 = {E2, D13, D14, D34}, F3 = {E3, D12, D14, D24},
F4 = {E4, D12, D13, D23}, F5 = {E1, E2, E3, E4}.
Proposition 5.1. The action of Aut(S5) on the five sets F1, ..., F5 of four skew
exceptional divisors of S5 gives rise to an isomomorphism
ρ : Aut(S5) → Sym5.
Furthermore, the actions of Symn, Altm ⊂ Aut(S5) on S5 given by the canonical
embedding of these groups into Sym5 are fixed-point free if and only if n = 3, 4, 5,
respectively m = 4, 5.
Proof. Since any automorphism in the kernel of ρ leaves E1, E2, E3 and E4 invari-
ant and hence is the lift of an automorphism of P2 that fixes the 4 points, the
homomorphism ρ is injective.
We now prove that ρ is also surjective. Firstly, the lift of the group of au-
tomorphisms of P2 that leave the set {A1, A2, A3, A4} invariant is sent by ρ on
Sym4 = Sym{F1,F2,F3,F4}. Secondly, the lift of the standard quadratic transforma-
tion (x : y : z) 99K (yz : xz : xy) is an automorphism of S5, as its lift on S6 is an
automorphism, and as it fixes the point A4; its image by ρ is (F4 F5).
It remains to prove the last assertion. First of all, it is clear that the actions
of the cyclic groups Alt3 and Sym2 fix some points. The group Sym3 ⊂ Aut(P
of permutations of A1, A2 and A3 fixes exactly one point, namely (1 : 1 : 1). The
blow-up of this point gives a fixed-point free action on F1, and thus its lift on S5
is also fixed-point free. The group Alt4 ⊂ Aut(P
2) contains the element (x : y :
z) 7→ (z : x : y) (which corresponds to (1 2 3)) that fixes exactly three points, i.e.
(1 : a : a2) for a3 = 1. It also contains the element (x : y : z) 7→ (z − y : z − x : z)
(which corresponds to (1 2)(3 4)) that does not fix (1 : a : a2) for a3 = 1. Thus,
the action of Alt4 on P
2 is fixed-point free and the same is true on S5.
Remark 5.2. The structure of Aut(S5) is classical and can be found for example
in [Wim] and [Do-Iz].
Lemma 5.3. Let π : S5 → P
1 be some morphism inducing a conic bundle (S5, π).
There are exactly four exceptional curves of S5 which are sections of π; the blow-
down of these curves gives rise to a birational morphism p : S5 → P
2 which
conjugates the group Aut(S5, π) ∼= Sym4 to the subgroup of Aut(P
2) that leaves
invariant the four points blown-up by p. In particular, the pair (Aut(S5, π), S5) is
not minimal.
Proof. Blowing-down one component in any singular fibre, we obtain a birational
morphism of conic bundles (not Aut(S5, π)-equivariant) from S5 to some Hirze-
bruch surface Fn. Since S5 does not contain any curves of self-intersection ≤ −2, n
is equal to 0 or 1. Changing the component blown-down in a singular fibre performs
an elementary link Fn 99K Fn±1; we may then assume that n = 1, and that F1 is
the blow-up of A1 ∈ P
2. Consequently, the fibres of the conic bundles correspond
to the lines passing through A1. Denoting by A2, A3, A4 the other points blown-up
by the constructed birational morphism S5 → P
2 and using the same notation as
before, the three singular fibres are {Ei, D1i} for i = 2, ..., 4, and the other excep-
tional curves are four skew sections of the conic bundle, namely the elements of
F1 = {E1, D23, D24, D34}. The blow-down of F1 gives an Aut(S5, π)-equivariant
birational morphism (that is not a morphism of conic bundles) p : S5 → P
2 and
conjugates Aut(S5, π) to a subgroup of the group Sym4 ⊂ Aut(P
2) of automor-
phisms that leaves the four points blown-up by p invariant. The fibres of π are
sent on the conics passing through the four points, so the lift of the whole group
Sym4 belongs to Aut(S5, π).
Corollary 5.4. Let G be some group of automorphisms of a conic bundle (S, π)
such that the pair (G,S) is minimal and (KS)
2 ≥ 5 (or equivalently such that the
number of singular fibres of π is at most 3). Then, the fibration is smooth, i.e. S
is a Hirzebruch surface.
Proof. Since (G,S) is minimal, so is the triple (G,S, π). By Lemma 3.12, the
surface S is either a Hirzebruch surface, or a del Pezzo surface of degree 5 or 6.
Corollary 4.4 shows that the del Pezzo surface of degree 6 is not possible and
Lemma 5.3 eliminates the possibility of the del Pezzo surface of degree 5.
6 Description of twisting elements
In this section, we describe the twisting automorphisms of conic bundles, which
are the most important automorphisms (see Lemma 3.8).
Lemma 6.1 (Involutions twisting a conic bundle). Let g ∈ Aut(S, π) be a twist-
ing automorphism of the conic bundle (S, π). Then, the following properties are
equivalent:
1. g is an involution;
2. π(g) = 1, i.e. g has a trivial action on the basis of the fibration;
3. the set of points of S fixed by g is an irreducible hyperelliptic curve of genus
(k − 1) – a double covering of P1 by means of π, ramified over 2k points –
plus perhaps a finite number of isolated points, which are the singular points
of the singular fibres not twisted by g.
Furthermore, if the three conditions above are satisfied, the number of singular
fibres of π twisted by g is 2k ≥ 2.
Proof. 1 ⇒ 2: By contracting some exceptional curves, we may assume that the
triple (< g >, S, π) is minimal. Suppose that g is an involution and π(g) 6= 1.
Then g may twist only two singular fibres, which are the fibres of the two points
of P1 fixed by π(g). Hence, the number of singular fibres is ≤ 2. Lemma 3.12
tells us that S is a del Pezzo surface of degree 6 and then Lemma 4.6 shows that
g = κα,β (Example 4.5) for some α, β ∈ C
∗. But such an element is an involution
if and only if it acts trivially on the basis of the fibration.
(1 and 2) ⇒ 3: Suppose first that (< g >, S, π) is minimal. This implies that g
twists every singular fibre of π. Therefore, since π(g) = 1 and g2 = 1, on a singular
fibre there is one point fixed by g (the singular point of the fibre) and on a general
fibre there are two fixed points. The set of points of S fixed by g is thus a smooth
irreducible curve. The projection π gives it as a double covering of P1 ramified over
the points whose fibres are singular and twisted by g. By the Riemann-Hurwitz
formula, this number is even, equal to 2k and the genus of the curve is k − 1.
The situation when (< g >, S, π) is not minimal is obtained from this one, by
blowing-up some fixed points. This adds in each new singular fibre (not twisted
by the involution) an isolated point, which is the singular point of the singular
fibre. We then get the third assertion and the final remark.
3 ⇒ 2: This implication is clear.
2 ⇒ 1: If π(g) = 1, then, g2 leaves every component of every singular fibre of
π invariant. Let p1 : S → F1 be the birational morphism of conic bundles given by
Lemma 3.10; it is a g2-equivariant birational morphism which conjugates g2 to an
automorphism of F1 that necessarily fixes the exceptional section. The pull-back
by p1 of this section is a section C of π, fixed by g
2. Since C touches exactly
one component of each singular fibre (in particular those that are twisted by g),
g sends C on another section D also fixed by g2. The union of the sections D and
C intersects a general fibre in two points, which are exchanged by the action of g.
This implies that g has order 2.
We now give some further simple results on twisting involutions.
Corollary 6.2. Let (S, π) be some conic bundle. No involution twisting (S, π) has
a root in Aut(S, π) which acts trivially on the basis of the fibration.
Proof. Such a root must twist a singular fibre and so (Lemma 6.1) is an involution.
Remark 6.3. There may exist some roots in Aut(S, π) of twisting involutions which
act non trivially on the basis of the fibration.
Take for example four general points A1, ..., A4 of the plane and denote by g ∈
Aut(P2) the element of order 4 that permutes these points cyclically. The blow-up
of these points conjugates g to an automorphism of the del Pezzo surface S5 of
degree 5 (see Section 5). The pencil of conics of P2 passing through the four points
induces a conic bundle structure on S5, with three singular fibres which are the
lift of the pairs of two lines passing through the points. The lift on S5 of g is an
automorphism of the conic bundle whose square is a twisting involution.
Corollary 6.4. Let (S, π) be some conic bundle and let g ∈ Aut(S, π). The
following conditions are equivalent.
1. g twists more than 2 singular fibres of π.
2. g fixes a curve of positive genus.
And these conditions imply that g is an involution which acts trivially on the basis
of the fibration and twists at least 4 singular fibres.
Proof. The first condition implies that g acts trivially on the basis of the fibration,
and thus (by Lemma 6.1) that g is an involution which fixes a curve of positive
genus.
Suppose that g fixes a curve of positive genus. Then, g acts trivially on the
basis of the fibration, and fixes 2 points on a general fibre. Consequently, the curve
fixed by g is a smooth hyperelliptic curve; we get the remaining assertions from
Lemma 6.1.
As we mentioned above, the automorphisms that twist some singular fibre are
fundamental (Lemma 3.8). We now describe these elements and prove that the
only possibilities are twisting involutions, roots of twisting involutions (of even or
odd order) and elements of the form κα,β (see Example 4.5):
Proposition 6.5 (Classification of twisting elements of finite order). Let g ∈
Aut(S, π) be a twisting automorphism of finite order of a conic bundle (S, π).
Let n be the order of its action on the basis.
Then gn is an involution that acts trivially on the basis of the fibration and
twists an even number 2k of singular fibres; furthermore, exactly one of the fol-
lowing situations occurs:
1. n = 1.
2. n > 1 and k = 0; in this case n is even and there exists a g-equivariant bi-
rational morphism of conic bundles η : S → S6 (where S6 is the del Pezzo
surface of degree 6) such that ηgη−1 = κα,β for some α, β ∈ C
∗ (see Exam-
ple 4.5).
3. n > 1 is odd and k > 0; here g twists 1 or 2 fibres, which are the fibres twisted
by gn that are invariant by g.
4. n is even and k > 0; here g twists r = 1 or 2 singular fibres; none of them
are twisted by gn; moreover the action of g on the set of 2k fibres twisted by
gn is fixed-point free; furthermore, n divides 2k, and 2k/n ≡ r (mod 2).
Proof. Lemma 6.1 describes the situation when n = 1. We now assume that n > 1;
by blowing-down some components of singular fibres we may also suppose that the
triple (G,S, π) is minimal.
Denote by a1, a2 ∈ P
1 the two points fixed by π(g) ∈ Aut(P1). For i 6≡ 0
(mod n) the element π(gi) fixes only two points of P1, namely a1 and a2 (since
π(g) has order n); the only possible fibres twisted by gi are thus π−1(a1), π
−1(a2).
Suppose that gn does not twist any singular fibre. By minimality there are
at most 2 singular fibres (π−1(a1) and/or π
−1(a2)) of π and g twists each one.
Lemma 3.12 tells us that S is a del Pezzo surface of degree 6 and Lemma 4.6
shows that
g = κα,β :
(x : y : z), (u : v : w)
(u : αw : βv) , (x : α−1z : β−1y)
for some α, β ∈ C∗. We compute the square of g and find
(x : y : z), (u : v : w)
(x : αβ−1y : α−1βz) , (u : α−1βv : αβ−1w)
Consequently, the order of g is 2n. The fact that gi twists π−1(a1) and π
−1(a2)
when i is odd implies that n is even. Case 2 is complete.
If gn twists at least one singular fibre, it twists an even number of singular
fibres (Lemma 6.1) which we denote by 2k, and gn is an involution. If n is odd,
each fibre twisted by gn is twisted by g, and conversely; this yields case 3. It
remains to consider the more difficult case when n is even.
Firstly we observe that there are r + 2k singular fibres with r ∈ {1, 2}, cor-
responding to the points a1 and/or a2, c1, ..., c2k of P
1, the first r of them be-
ing twisted by g and the 2k others by gn. Under the permutation π(g), the
set {c1, ..., c2k} decomposes into disjoint cycles of length n (this action is fixed-
point-free); this shows that n divides 2k. We write t = 2k/n ∈ N and set
{c1, ..., c2k} = ∪
i=1Ci, where each Ci ⊂ P
1 is an orbit of π(g) of size n. To
deduce the congruence r ≡ t (mod 2), we study the action of g on Pic(S).
For i ∈ {1, ..., t}, choose Fi to be a component in the fibre of the singular fibre
of some point of Ci, and for i ∈ {1, r} choose Li to be a component in the fibre of
ai. Let us write
i=1(Fi + g(Fi) + ...+ g
n−1(Fi)) +
i=1 Li ∈ Pic(S).
Denoting by f ⊂ S a general fibre of π, we find the equalities g(Li) = f − Li
and gn(Fi) = f − Fi in Pic(S), which yield (once again in Pic(S)):
g(R) = R+ (r + t)f − 2(
i=1 Li +
i=1 Fi).
The contraction of the divisor R gives rise to a birational morphism of conic
bundles (not g-equivariant) ν : S → Fm for some integer m ≥ 0. Denote by s ⊂ S
the pull-back by ν of a general section of Fm of self-intersection m (which does
not pass through any of the base-points of ν−1). The canonical divisor KS of S is
then equal in Pic(S) to the divisor −2s+ (m − 2)f + R. We compute g(2s) and
2(g(s)− s) = g(2s)− 2s in Pic(S):
g(2s) = g(−KS + (m− 2)f +R) = −KS + (m− 2)f + g(R);
g(2s)− 2s = g(R)−R = (r + t)f − 2(
i=1 Li +
i=1 Fi).
This shows that (r + t)f ∈ 2Pic(S), which implies that r ≡ t (mod 2). Case 4 is
complete.
Corollary 6.6. If g ∈ Aut(S, π) is a root of a twisting involution h that fixes a
rational curve (i.e. that twists 2 singular fibres) and if g twists at least one fibre
not twisted by h, then g2 = h, g twists exactly one singular fibre, and it exchanges
the two fibres twisted by h.
Proof. We apply Proposition 6.5 and obtain case 4 with k = 1.
Corollary 6.6 and the following result will be useful in the sequel.
Lemma 6.7. Let g ∈ Aut(S, π) be a non-trivial automorphism of finite order
that leaves every component of every singular fibre of π invariant (i.e. that acts
trivially on Pic(S)) and let h ∈ Aut(S, π) be an element that commutes with g.
Then, either no singular fibre of π is twisted by h or each singular fibre of π which
is invariant by h is twisted by h.
Proof. If no twisting element belongs to Aut(S, π), we are done. Otherwise, the
birational morphism of conic bundles p0 : S → P
1 × P1 given by Lemma 3.10
conjugates g to an element of finite order of Aut(P1 × P1, π1) whose set of fixed
points is the union of two rational curves. The set of points of S fixed by g is thus
the union of two sections and a finite number of points (which are the singular
points of the singular fibres of π). Any element h ∈ Aut(S, π) that commutes
with g leaves the set of these two sections invariant. More precisely, the action on
one invariant singular fibre F implies the action on the two sections: h exchanges
the two sections if and only if it twists F . Since the situation is the same at any
other singular fibre, we obtain the result.
We conclude this section with some results on automorphisms of infinite order
of conic bundles, which will not help us directly here but seem interesting to
observe.
Proposition 6.8 (Classification of twisting elements of infinite order). Let (S, π)
be a conic bundle and g ∈ Aut(S, π) be a twisting automorphism of infinite order.
Then g twists exactly two fibres of π and there exists some g-equivariant bira-
tional morphism of conic bundles η : S → S6, where S6 is the del Pezzo surface of
degree 6 and ηgη−1 = κα,β for some α, β ∈ C
Proof. Assume that the triple (< g >, S, π) is minimal. Lemma 6.1 shows that
no twisting element of infinite order acts trivially on the basis of the fibration.
Consequently, gk acts trivially on the basis if and only if k = 0, whence gk twists a
fibre F if and only if k is odd and g twists F . There thus exist at most 2 singular
fibres of π, and Lemma 3.12 tells us that S is a del Pezzo surface of degree 6.
Lemma 4.6 shows that g = κα,β for some α, β ∈ C
Corollary 6.9. Let g ∈ Aut(S, π) be an element of infinite order; then a birational
morphism conjugates g to an automorphism of a Hirzebruch surface.
Proof. Assume that the triple (< g >, S, π) is minimal. If the fibration is smooth,
we are done. Otherwise, a birational morphism conjugates g to an automorphism
κα,β ∈ Aut(S6) of a conic bundle on the del Pezzo surface of degree 6 (Lemma 6.8).
We conclude by using Lemma 4.2.
7 The example Cs24
We now give the most important example of this paper. This is the only finite
Abelian subgroup of the Cremona group which is not conjugate to a group of
automorphisms of P2 or P1 × P1 but whose non-trivial elements do not fix any
curve of positive genus (Theorem 2).
Let S6 ⊂ P
2 × P2 be the del Pezzo surface of degree 6 (see Section 4) defined
S6 = {
(x : y : z), (u : v : w)
| ux = yv = zw};
we keep the notation of Section 4. We denote by η : Ŝ4 → S6 the blow-up of
A4, A5 ∈ S6 defined by
(0 : 1 : 1) , (1 : 0 : 0)
∈ D23,
(1 : 0 : 0) , (0 : 1 : −1)
∈ E1.
We again denote by E1, E2, E3, D12, D13, D23 the total pull-backs by η of these
divisors of S6. We denote by Ẽ1 and D̃23 the strict pull-backs of E1 and D23 by
η. (Note that for the other exceptional divisors, the strict and total pull-backs are
the same.) Let us illustrate the situations on the surfaces S6 and Ŝ4 respectively:
E2 D15 E4 E3
D12 E5 D14 D13
Let π1 denote the morphism S6 → P
1 defined in Section 4. The morphism
π = π1 ◦ η gives the surface Ŝ4 a conic bundle structure (Ŝ4, π). It has 4 singular
fibres, which are the fibres of (−1 : 1), (0 : 1), (1 : 1) and (1 : 0). We denote by f
the divisor of Ŝ4 corresponding to a fibre of π and set E4 = η
−1(A4), E5 = η
−1(A5).
Note that E4 is one of the components of the singular fibre of (1 : 1); we denote by
D14 = f−E4 the other component, which is the strict pull-back by η of π
1 (1 : 1).
Similarly, we denote by D15 the divisor f−E5, so that the singular fibre of (−1 : 1)
is {E5, D15}.
Lemma 7.1. On the surface Ŝ4 there are exactly 10 irreducible rational smooth
curves of negative self-intersection. Explicitly, the 8 curves
E2, E3, E4, E5, D12, D13, D14, D15
have self-intersection −1 and the two curves
Ẽ1 = E1 − E5 and D̃23 = D23 − E4
have self-intersection −2.
Proof. The difficult part is to show that every rational irreducible smooth curve
of negative self-intersection is one of the ten given above. Let C be such a curve.
Denote by L the pull-back of a general line of P2 by the blow-up pr1 ◦ η : Ŝ4 →
P2 of the five points. If C is collapsed by pr1 ◦ η, then C is one of the curves
Ẽ1, E2, E3, E4, E5. Otherwise, C = mL −
i=1 aiEi, where m, a1, ..., a5 are non-
negative integers, and m > 0. Since C is rational we have C · (C + K
) = −2,
and by hypothesis C2 = −r for some positive integer r. The relations C2 = −r
and C ·K
= r − 2 imply (since K
= −3L+
i=1 Ei) the equations
i=1 a
i = m
2 + r,
i=1 ai = 3m+ r − 2.
If m = r = 1, we find that C is the pull-back of a line passing through two of the
points, so C = D1i for some i ∈ {2, ..., 5}. If m = 2 and r = 1, C is the pull-back
of a conic passing through each blown-up point. The configuration of the points
eliminates this possibility. If m = 1 and r = 2, we obtain a line passing through
three blown-up points, so C = D̃23.
We now prove that if there is no integral solution to (2) for m, r ≥ 2. Since
i=1 ai)
2 ≤ 5(
i=1 a
i ) (by the Cauchy-Schwarz inequality with the vectors
(1, ..., 1) and (a1, ..., a5)), we obtain (3m+ (r − 2))
2 ≤ 5(m2 + r), and this gives
4m2 − 10 + (r − 2) · (6m+ r − 7) ≤ 0.
But this is not possible if m, r ≥ 2, since in this case 4m2 > 10 and 6m+r > 7.
Note that (K
)2 = 4, which is why we denote this surface by Ŝ4; the hat is
here because the surface is not a del Pezzo surface, since it contains irreducible
divisors of self-intersection −2.
Corollary 7.2. There is only one conic bundle structure on Ŝ4, which is the one
induced by π = π1 ◦ η.
Proof. Since (K
)2 = 4, the number of singular fibres of any conic bundle is 4,
and thus it consists of eight (−1)-curves C1, ..., C8. The divisor of a fibre of the
conic bundle is thus equal to 1
i=1 Ci. Since there are exactly eight (−1)-curves
on Ŝ4, there is no choice.
The group of automorphisms of Ŝ4 that leave every curve of negative self-
intersection invariant is isomorphic to C∗ and corresponds to automorphisms of P2
of the form (x : y : z) 7→ (αx : y : z), for α ∈ C∗. Indeed, such automorphisms are
the lifts of automorphisms of S6 leaving invariant every exceptional curve (which
are of the form
(x : y : z), (u : v : w)
(x : αy : βz), (u : α−1v : β−1w)
, for
α, β ∈ C∗) and which fix both points A4 and A5.
Definition 7.3. Let h1 and h2 be the following birational transformations of P
h1 : (x : y : z) 99K (yz : xy : −xz)
h2 : (x : y : z) 99K (yz(y − z) : xz(y + z) : xy(y + z))
and denote respectively by g1, g2 the lift of these elements on Ŝ4 and by Cs24 the
group generated by g1 and g2.
The following lemma shows that Cs24 ⊂ Aut(Ŝ4, π) and describes some of the
properties of this group.
Lemma 7.4. Let h1, h2, g1, g2,Cs24 be as in Definition 7.3. Then:
1. The group Cs24 is a group of automorphisms of Ŝ4 that preserve the conic
bundle (Ŝ4, π), i.e. Cs24 ⊂ Aut(Ŝ4, π).
2. The action of g1 and g2 on the set of irreducible rational curves of negative
self-intersection is respectively:
(Ẽ1 D̃23)(E2 D12)(E3 D13)(E4 E5)(D14 D15),
(Ẽ1 D̃23)(E2 D13)(E3 D12)(E4 D14)(E5 D15).
In particular, both g1 and g2 twist the conic bundle (Ŝ4, π).
3. Both g1 and g2 are elements of order 4 and
2 = (h2)
2 = (x : y : z) 7→ (−x : y : z).
Thus (g1)
2 = (g2)
2 ∈ kerπ is an automorphism of Ŝ4 which leaves every
divisor of negative self-intersection invariant.
4. The group Cs24 is isomorphic to Z/2Z × Z/4Z and the action on the basis
of the fibration π yields the exact sequence
1 →< (h1)
2 >∼= Z/2Z → Cs24
→< π(h1), π(h2) >∼= (Z/2Z)
2 → 1.
5. The group Cs24 contains no involution that twists the conic bundle (Ŝ4, π).
In particular, no element of Cs24 fixes a curve of positive genus.
6. The pair (Cs24, Ŝ4) and the triple (Cs24, Ŝ4, π) are both minimal.
Proof. Observe first that h1 and h2 preserve the pencil of lines of P
2 passing
through the point A1 = (1 : 0 : 0), so g1, g2 are birational transformations of Ŝ4
that send a general fibre of π on another fibre. Then, we compute (h1)
2 = (h2)
(x : y : z) 7→ (−x : y : z). This implies that both h1 and h2 are birational maps of
order 4.
Note that the lift of h1 on the surface S6 is the automorphism
κ1,−1 :
(x : y : z), (u : v : w)
(u : w : −v), (x : z : −y)
(see Example 4.5). Since this automorphism permutes A4 and A5, its lift on Ŝ4
is biregular. The action on the divisors with negative self-intersection is deduced
from that of κ1,−1 (see Lemma 4.6).
Compute the involution
h3 = h1h2 = (x : y : z) 99K (x(y + z) : z(y − z) : −y(y − z)).
Its linear system is
{ax(y + z) + (by + cz)(y − z) = 0 | (a : b : c) ∈ P2},
which is the linear sytem of conics passing through (0 : 1 : 1) and A1 = (1 : 0 : 0),
with tangent y + z = 0 at this point (i.e. passing through A5). Blowing-up these
three points (two on P2 and one in the blow-up of A1), we get an automorphism
g3 of some rational surface. As the points A2 = (0 : 1 : 0) and A3 = (0 : 0 : 1)
are permuted by h3, we can also blow them up and again get an automorphism.
The isomorphism class of the surface obtained is independent of the order of the
blown-up points. We may first blow-up A1, A2, A3 and get S6. Then, we blow-up
the two other base-points of h3, which are in fact A4 (the point (0 : 1 : −1)) and
A5 (the point infinitely near to A1 corresponding to the tangent y + z = 0). This
shows that g3, and therefore g2, belong to Aut(Ŝ4, π).
Since h3 permutes the points A2 and A3, g3 = g1g2 permutes the divisors E2
and E3. It also permutes D12 and D13, since h3 leaves the pencil of lines passing
through A1 invariant. It therefore leaves Ẽ1 and D̃23 invariant, since E2 and E3
touch D̃23 but not E1. The remaining exceptional divisors are E4, E5, D14, D15.
Either g1g2 leaves all four invariant, or it acts as (E4 D15)(E5 D14) (using the
intersection with Ẽ1 and D̃23). Since A4 and A5 are base-points of h1h2, E4 and
E5 are not invariant. Thus, g1g2 acts on the irreducible rational curves of negative
self-intersection as (E2 E3)(D12 D13)(E4 D15)(E5 D14). We obtain the action of
g2 by composing that of g1g2 with that of g1 and thus have proved assertions 1
through 3.
Assertion 4 follows from assertion 3 and the fact that g1 and g2 commute.
Let us prove that Cs24 contains no involution that twists the conic bundle
(Ŝ4, π). Recall that such elements are involutions acting trivially on the basis
of the fibration (see Lemma 6.1). Note that the 2-torsion of Cs24 is equal to
{1, g21, g1g2, g1g
2 }. The elements g1g2 and g1g
2 do not act trivially on the basis
of the fibration, and the element (g1)
2 does not twist any singular fibre since it
leaves every curve of negative self-intersection invariant. This proves assertion 5.
It remains to prove the last assertion. Observe that the orbits of the action of
Cs24 on the exceptional divisors of Ŝ4 are {E2, E3, D12, D13} and {E4, E5, D14, D15}.
Since these orbits cannot be contracted, the pair (Cs24, Ŝ4) is minimal, and so is
the triple (Cs24, Ŝ4, π).
Remark 7.5. The pair (Cs24, Ŝ4) was introduced in [Bla2] and was called Cs.24
because it is a group acting on a conic bundle, which is special, and isomorphic
to Z/2Z× Z/4Z.
8 Finite Abelian groups of automorphisms of conic
bundles - birational representative elements
In this section we use the tools prepared in the previous sections to describe the
finite Abelian groups of automorphisms of conic bundles such that no non-trivial
element fixes a curve of positive genus.
We first treat the case in which no involution twisting the conic bundle belongs
to the group:
Proposition 8.1. Let G ⊂ Aut(S, π) be a finite Abelian group of automorphisms
of the conic bundle (S, π) such that:
• no involution that twists the conic bundle (S, π) belongs to G;
• the triple (G,S, π) is minimal.
Then, one of the following occurs:
• The fibration is smooth, i.e. S is a Hirzebruch surface.
• S is the del Pezzo surface of degree 6.
• The triple (G,S, π) is isomorphic to the triple (Cs24, Ŝ4, π) of Section 7.
Proof. We assume that the fibration is not smooth. Recall that since the triple
(G,S, π) is minimal, any singular fibre of π is twisted by an element of G (by
Lemma 3.8). Since no twisting involution belongs to G, any element g ∈ G that
twists a fibre corresponds to case 2 of Proposition 6.5. In particular, g is the lift
on S of an automorphism of the form κα,β of the del Pezzo surface of degree 6 and
it twists 2 singular fibres, which correspond to the fibres of the two fixed points
of π(g) ∈ PGL(2,C). Furthermore, g is the root of an involution that leaves every
component of every singular fibre of π invariant.
If the number of singular fibres is exactly two, then S is the del Pezzo surface
of degree 6, and we are done.
Now suppose that the number of singular fibres is larger than two. This implies
that π(G) is not a cyclic group (otherwise the non-trivial elements of π(G) would
have the same two fixed points: there would then be at most two singular fibres);
therefore, π(G) is isomorphic to (Z/2Z)2. By a judicious choice of coordinates we
may suppose that
π(G) =
Since a singular fibre corresponds to a fixed point of one of the three elements
of order 2 of π(G), only the fibres of (0 : 1), (1 : 0), (1 : 1), (−1 : 1), (i : 1), (−i : 1)
can be singular. Since the group π(G) acts transitively on the sets {(1 : 0), (0 : 1)},
{(1 : ±1)} and {(1 : ±i)}, there are 4 or 6 singular fibres.
We denote by g1 an element of G which twists the two singular fibres of (1 : 0)
and (0 : 1). Let η : S → S6 denote the birational g1-equivariant morphism given
by Proposition 6.5, which conjugates g1 to the automorphism
−1 = κα,β :
(x : y : z), (u : v : w)
(u : αw : βv), (x : α−1z : β−1y)
of the del Pezzo surface S6 of degree 6, for some α, β ∈ C
∗. In fact, since π(g1)
has order 2, we have β = −α, so ηg1η
−1 = κα,−α. The points blown-up by η are
fixed by
η(g1)
2η−1 = (κα,−α)
(x : y : z), (u : v : w)
(x : −y : −z), (u : −v : −w)
and therefore belong to the curves
E1 = {
(1 : 0 : 0), (0 : a : b)
| (a : b) ∈ P1}
and D23 = {
(0 : a : b), (1 : 0 : 0)
| (a : b) ∈ P1}.
Since these points consist of orbits of ηg1η
−1, half of them lie in E1 and the
other half in D23. In fact, up to a change of coordinates,
(x, y, z), (u, v, w)
(u, v, w), (x, y, z)
, the points that may be blown-up by η are
(0 : 1 : 1) , (1 : 0 : 0)
∈ D23,
κα,−α(A4) = A5 =
(1 : 0 : 0) , (0 : 1 : −1)
∈ E1,
(0 : 1 : i) , (1 : 0 : 0)
∈ D23,
κα,−α(A6) = A7 =
(1 : 0 : 0) , (0 : 1 : i)
∈ E1.
The strict pull-backs Ẽ1 and D̃23 by η of E1 and D23 respectively thus have self-
intersection −2 or −3 in S, depending on the number of points blown-up. By
convention we again denote by E1, E2, E3, D12, D13, D23 the total pull-backs by
η of these divisors. (Note that for E2, E3, D12, D13, the strict and the total pull-
backs are the same.) We set E4 = η
−1(A4),..., E7 = η
−1(A7) and denote by f the
divisor class of the fibre of the conic bundle.
(a) Suppose that η is the blow-up of A4 and A5, which implies that S is the
surface Ŝ4 of Section 7. The Picard group of S is then generated by E1, E2, ..., E5
and f .
Since we assumed that (G,S, π) is minimal, the singular fibres of (1 : 1) and
(−1 : 1) must be twisted. One element g2 twists these two singular fibres and
acts with order 2 on the basis of the fibration, with action (x1 : x2) 7→ (x2 : x1).
Since g1 and g2 twist some singular fibre, both must invert the two curves of self-
intersection −2, namely Ẽ1 and D̃23. The action of g1 and g2 on the irreducible
rational curves of negative self-intersection is then respectively
(Ẽ1 D̃23)(E2 D12)(E3 D13)(E4 E5)(D14 D15),
(Ẽ1 D̃23)(E2 D13)(E3 D12)(E4 D14)(E5 D15).
The elements g1 and g2 thus have the same action on Pic(S) = Pic(Ŝ4) as the
two automorphisms with the same name in Definition 7.3 and Lemma 7.4, which
generate Cs24. Note that the group H of automorphisms of S that leave every
curve of negative self-intersection invariant is isomorphic to C∗ and corresponds
to automorphisms of P2 of the form (x : y : z) 7→ (αx : y : z), for any α ∈ C∗.
Then, g1 and g2 are equal to the lift of the the following birational maps of P
h1 : (x : y : z) 99K (µyz : xy : −xz),
h2 : (x : y : z) 99K (νyz(y − z) : xz(y + z) : xy(y + z)),
for some µ, ν ∈ C∗.
As h1h2(x : y : z) = (µx(y + z) : νz(y − z) : −νy(y − z)) and h2h1(x : y :
z) = (νx(y + z) : µz(y − z) : −µy(y − z)) must be the same by hypothesis, we get
µ2 = ν2.
We observe that π(g1) and π(g2) generate π(G) ∼= (Z/2Z)
2; on the other hand,
by hypothesis an element of G′ does not twist a singular fibre and hence belongs
to H . As the only elements of H which commute with g1 are id and (g1)
2 (which
is the lift of (h1)
2 : (x : y : z) 7→ (−x : y : z)), we see that g1 and g2 generate the
whole group G.
Conjugating h1 and h2 by (x : y : z) 7→ (αx : y : z), where α ∈ C
∗, α2 = µ, we
may suppose that µ = 1. So ν = ±1 and we get in both cases the same group,
because (h1)
2(x : y : z) = (−x : y : z). The triple (G,S, π) is hence isomorphic to
the triple (Cs24, Ŝ4, π) of Section 7.
(b) Suppose that η is the blow-up of A6 and A7. We get a case isomorphic
to the previous one, using the automorphism
(x : y : z), (u : v : w)
(x : y :
iz), (u : v : −iw)
of S6.
(c) Suppose that η is the blow-up of A4, A5, A6 and A7. The Picard group of
S is then generated by E1, E2, ..., E6, E7 and f . Since (G,S, π) is minimal, there
must be two elements g2, g3 ∈ G that twist respectively the fibres of (±1 : 1) and
those of (±i : 1). As in the previous example, the three actions of these elements
on the basis are of order 2, and the three elements transpose Ẽ1 and D̃23. The
actions of g1, g2 and g3 on the set of irreducible components of the singular fibres
of π are then respectively
(E2 D12)(E3 D13)(E4 E5)(D14 D15)(E6 E7)(D16 D17),
(E2 D13)(E3 D12)(E4 D14)(E5 D15)(E6 E7)(D16 D17),
(E2 D13)(E3 D12)(E4 E5)(D14 D15)(E6 D16)(E7 D17).
This implies that the action of the element g1g2g3 is
(E2 D12)(E3 D13)(E4 D14)(E5 D15)(E6 D17)(E7 D16),
and thus it twists six singular fibres of the conic bundle and fixes a curve of genus 2
(Lemma 6.1), which contradicts the hypothesis. (In fact, one can also show that
the group generated by g1, g2 and g3 is not Abelian, see [Bla2], page 66.)
After studying the groups that do not contain a twisting involution, we now
study those which contain such elements. Since these twisting involutions cannot
fix a curve of positive genus, they twist exactly two fibres (Lemma 6.1).
Proposition 8.2. Let G ⊂ Aut(S, π) be a finite Abelian group of automorphisms
of a conic bundle (S, π) such that:
1. If g ∈ G, g 6= 1, then g does not fix a curve of positive genus.
2. The group G contains at least one involution that twists the conic bundle
(S, π).
3. The triple (G,S, π) is minimal.
Then, S is a del Pezzo surface of degree 5 or 6.
Proof. If the number of singular fibres is at most 3, then the surface is a del Pezzo
surface of degree 5 or 6 (Lemma 3.12).
We now assume that the number of singular fibres is at least 4 and show that
this situation is not compatible with the hypotheses. We recall once again the
exact sequence of Remark 3.13
1 → G′ → G
→ π(G) → 1, (1)
and prove the following important assertions:
(a) No element of G twists more than two singular fibres.
(b) Any twisting involution that belongs to G belongs to G′ and twists exactly
two singular fibres.
(c) Any singular fibre is twisted by an element of G.
(d) No non-trivial element preserves every component of every singular fibre.
(e) Any twisting element of G is a root of (or equal to) a twisting involution
that belongs to G′.
Corollary 6.4 shows that an element that twists more than two fibres fixes a
curve of positive genus; since this possibility is excluded by hypothesis, we obtain
assertion (a). Lemma 6.1 shows that any twisting involution contained in G be-
longs to G′ and twists an even number of fibres; using assertion (a), we thus obtain
assertion (b). Assertion (c) follows from the minimality of the triple (G,S, π) (see
Lemma 3.8). Let us prove assertion (d). Suppose that there exists a non-trivial
element g ∈ G that leaves every component of every singular fibre invariant, and
denote by h ∈ G′ a twisting involution (which exists by hypothesis). Since g and h
commute, Lemma 6.7 shows that each singular fibre invariant by h – there are
at least 4 – is twisted by h, which contradicts assertion (a). Therefore, such an
element g doesn’t exist and assertion (d) is proved. Finally, Proposition 6.5 shows
that any twisting element that does not act trivially on the basis of the fibration
is a root of an involution that belongs to G′, and assertion (d) shows that this
involution is twisting, and we obtain assertion (e).
Now that assertions (a) through (e) are proved, we deduce the proposition from
them. Let us denote by σ ∈ G′ a twisting involution, which twists two singular
fibres that we denote by F1 and F2. There are at least two other singular fibres
F3 and F4 that are twisted by other elements of G.
If G′ =< σ >, the fibres F3 and F4 are twisted by roots of σ belonging to G
(assertions (c) and (e)). The description of these elements (Proposition 6.5, and
in particular Corollary 6.6) shows that the roots must be square roots that twist
exactly one singular fibre and permute the two fibres F1 and F2 twisted by σ.
There thus exist two elements h3, h4 ∈ G that twist respectively the fibres F3 and
F4. Since h3 commutes with h4, it must leave invariant the unique fibre twisted
by h4, i.e. F4. Similarly, h4 must leave F3 invariant. Therefore, h3h4 leaves the
four fibres F1,...,F4 invariant and twists the two fibres F3 and F4; it is thus an
involution that belongs to G′, which contradicts the fact that G′ =< σ >.
If G′ 6=< σ >, since σ has no root in G′ (Corollary 6.2), the Abelian groupG′ ⊂
PGL(2,C(x)) is isomorphic to (Z/2Z)2 and contains (using (d)) three twisting
involutions σ, ρ and σρ. Note that two of these three involutions do not twist
singular fibres which are all distinct, otherwise the product of the two involutions
would give an involution that twists 4 singular fibres, contradicting (a). We may
thus suppose that ρ twists F1 and F3, which implies that σρ twists F2 and F3.
The fibre F4 is then twisted by an element which is a square root of one of the
three twisting involutions (assertion (e) and Corollary 6.6). Denote this square
root by h and suppose that h2 6= σ. Note that h exchanges the two singular fibres
twisted by h2. One of these is twisted by σ and the other is not, so h and σ do
not commute.
The only remaining possible finite Abelian groups of automorphisms of conic
bundles satisfying property (F ) are thus del Pezzo surfaces of degree 6 or 5 (studied
in Sections 4 and 5), the triple (Cs24, Ŝ4, π) studied in Section 7, and Hirzebruch
surfaces. We now describe this last case and prove that it is birationally reduced
to the case of P1 × P1.
Proposition 8.3. Let G ⊂ Aut(Fn) be a finite Abelian subgroup of automorphisms
of Fn, for some integer n ≥ 1. Then, a birational map of conic bundles conjugates
G to a finite group of automorphisms of F0 = P
1 × P1 that leaves one ruling
invariant.
Proof. Let G ⊂ Aut(Fn) be a finite Abelian group, with n ≥ 1. Note that G
preserves the unique ruling of Fn. We denote by E ⊂ Fn the unique section
of self-intersection −n, which is necessarily invariant by G. We have the exact
sequence (see Remark 3.13)
1 → G′ → G
→ π(G) → 1. (1)
Since the group π(G) ⊂ PGL(2,C) is Abelian, it is isomorphic to a cyclic group
or to (Z/2Z)2.
If π(G) is a cyclic group, at least two fibres are invariant by G. The group
G fixes two points in one such fibre. We can blow-up the point that does not lie
on E and blow-down the corresponding fibre to get a group of automorphisms of
Fn−1. We do this n times and finally obtain a birational map of conic bundles
that conjugates G to a group of automorphisms of F0 = P
1 × P1.
If π(G) is isomorphic to (Z/2Z)2, there exist two fibres F, F ′ of π whose union
is invariant by G. Let GF ⊂ G be the subgroup of G of elements that leave F
invariant. This group is of index 2 in G and hence is normal. Since GF fixes the
point F ∩ E in F , it acts cyclically on F . There exists another point P ∈ F ,
P /∈ E, which is fixed by GF . The orbit of P by G consists of two points, P and
P ′, such that P ′ ∈ F ′, P ′ /∈ E. We blow-up these two points and blow-down the
strict transforms of F and F ′ to get a group of automorphisms of Fn−2. We do
this ⌊n/2⌋ times to obtain G as a group of automorphisms of F0 or F1.
If n is even, we get in this manner a group of automorphisms of F0 = P
1 × P1.
Note that n cannot be odd, if the group π(G) is not cyclic. Otherwise, we
could conjugate G to a group of automorphisms of F1 and then to a group of
automorphisms of P2 by blowing-down the exceptional section on a point Q ∈
P2. We would get an Abelian subgroup of PGL(3,C) that fixes Q, and thus a
group with at least three fixed points. In this case, the action on the set of lines
passing through Q would be cyclic (see Proposition 2.2), which contradicts our
hypothesis.
We can now prove the main result of this section:
Proposition 8.4. Let G ⊂ Aut(S, π) be some finite Abelian group of automor-
phisms of the conic bundle (S, π) such that the triple (G,S, π) is minimal and no
non-trivial element of G fixes a curve of positive genus. Then, one of the following
situations occurs:
1. S is a Hirzebruch surface Fn;
2. S is a del Pezzo surface of degree 5 or 6;
3. The triple (G,S, π) is isomorphic to the triple (Cs24, Ŝ4, π) of Section 7.
If we suppose that the pair (G,S) is minimal, then we are in case 1 with n 6= 1 or
in case 3. Moreover, cases 1 and 2 are birationally conjugate to automorphisms of
P1 × P1 whereas the third is not.
Proof. The fact that one of the three cases occurs follows directly from Proposi-
tions 8.1 and 8.2.
Case 1 is clearly minimal if and only if n 6= 1 and Proposition 8.3 shows that
it is conjugate to automorphisms of P1 × P1. In the case of del Pezzo surfaces
of degree 5 and 6, the pair (G,S) is not minimal and the group is respectively
birationally conjugate to a subgroup of Sym4 ⊂ Aut(P
2) (Lemma 5.3) or Aut(P1×
P1) (Lemma 4.2). If the first situation occurs, since the group is Abelian and
not isomorphic to (Z/3Z)2 it is diagonalisable and conjugate to a subgroup of
Aut(P1 × P1) (Proposition 2.2). Thus, we are done with case 2.
It remains to show that the pair (Cs24, Ŝ4) is not birationally conjugate to a
group of automorphisms of P1 × P1. Let us suppose the contrary, i.e. that there
exists some Cs24-equivariant birational map ϕ : Ŝ4 99K P
1 × P1 (that conjugates
Cs24 to a group of automorphisms). Then, ϕ is the composition of Cs24-equivariant
elementary links (see for example [Isk3, Theorem 2.5], or [Do-Iz, Theorem 7.7]).
Since our group preserves the conic bundle, the first link is of type II, III or IV
(in the classical notation of Mori theory). We now study these possibilities and
show that it is not possible to go to P1 × P1.
Link of type II - In our case, this link is a birational map of conic bundles,
which is the composition of the blow-up of an orbit of Cs24, no two points on the
same fibre, with the blow-down of the strict transforms of the fibres of the points
blown-up. The points must be fixed by the elements of Cs24 that act trivially on
the basis of the fibration, and thus an orbit has 4 points, two on Ẽ1 and two on
D̃23. This link conjugates the triple (Cs24, Ŝ4, π) to a triple isomorphic to it, by
Proposition 8.1.
Link of type III - It is the contraction of some set of skew exceptional curves, in-
variant by Cs24. This is impossible since the pair (Cs24, Ŝ4) is minimal (Lemma 7.4).
Link of type IV - It is a change of the fibration. This is not possible since the
surface Ŝ4 admits only one conic bundle fibration (Corollary 7.2).
9 Actions on del Pezzo surfaces with fixed part
of the Picard group of rank one
In this section we prove the following result (note that finiteness is not required
and that minimality of the action is implied by the condition on Pic(S)G).
Proposition 9.1. Let S be a del Pezzo surface, and let G ⊂ Aut(S) be an Abelian
group such that rk Pic(S)G = 1 and no non-trivial element of G fixes a curve of
positive genus. Then, one of the following occurs:
1. S ∼= P2 or S ∼= P1 × P1;
2. S is a del Pezzo surface of degree 5 and G ∼= Z/5Z;
3. S is a del Pezzo surface of degree 6 and G ∼= Z/6Z.
Furthermore, in cases 2 and 3, the group G is birationally conjugate to a diagonal
cyclic subgroup of Aut(P2).
This will be proved separately for each degree, in Lemmas 9.7, 9.8, 9.13, 9.15,
9.16 and 9.17.
Remark 9.2. A del Pezzo surface S is either P1 × P1 or the blow-up of 0 ≤ r ≤
8 points in general position on P2 (i.e. such that no irreducible curve of self-
intersection ≤ −2 appears on S). The group Pic(S) has dimension r + 1, and its
intersection form gives a decomposition Pic(S) ⊗ Q = QKS ⊕K
S ; the signature
is (1,−1, ...,−1).
The group Aut(S) of automorphisms of a del Pezzo surface S acts on Pic(S)
and preserves the intersection form. This gives an homomorphism of Aut(S) →
Aut(Pic(S)) which is injective if and only if r > 3, since the kernel is the lift of
automorphisms of P2 that fix the r blown-up points. Furthermore, the image is
contained in the Weyl group and is finite (see [Dol]). In particular, the group
Aut(S) is finite if and only if r > 3.
When we have some group action on a del Pezzo surface, we would like to
determine the rank of the fixed part of the Picard group. Here are some tools to
this end.
Lemma 9.3 (Size of the orbits). Let S be a del Pezzo surface, which is the blow-up
of 1 ≤ r ≤ 8 points of P2 in general position, and let G ⊂ Aut(S) be a subgroup
of automorphisms with rk Pic(S)G = 1. Then:
• G 6= {1};
• the size of any orbit of the action of G on the set of exceptional divisors is
divisible by the degree of S, which is 9− r;
• in particular, if the order of G is finite, it is divisible by the degree of S.
Proof. It is clear that G 6= {1}, since rk Pic(S) > 1. Let D1, D2, ..., Dk be k
exceptional divisors of S, forming an orbit of G (the orbit is finite, see Remark 9.2).
The divisor
i=1 Di is fixed by G and thus is a multiple of KS . We can write∑k
i=1 Di = aKS, for some a ∈ Q. In fact, since aKS is effective, we have a < 0 and
a ∈ Z. Since the Di’s are irreducible and rational, we deduce from the adjunction
formula Di(KS +Di) = −2 that Di ·KS = −1. Hence
i=1 Di =
i=1 KS ·Di = −k = KS · aKS = a(9− r).
Consequently, the degree 9− r divides the size k of the orbit.
Remark 9.4. This lemma shows in particular that rk Pic(S)G > 1 if S is the blow-
up of r = 1, 2 points of P2, a result which is obvious when r = 1, and is clear
when r = 2, since the line joining the two blown-up points is invariant by any
automorphism.
Lemma 9.5. Let S be some (smooth projective rational) surface, and let g ∈
Aut(S) be some automorphism of finite order. Then, the trace of g acting on
Pic(S) is equal to χ(Fix(g)) − 2, where Fix(g) ⊂ S is the set of fixed points of g
and χ is the Euler characteristic.
Proof. This follows from the topological Lefschetz fixed-point formula, which as-
serts that the trace of g acting on H∗(S,Z) is equal to χ(Fix(g)) (this uses the fact
that g is an homeomorphism of finite order). Since S is a complex rational surface,
H0(S,Z) and H4(S,Z) have dimension 1, H2(S,Z) ∼= Pic(S), and Hi(S,Z) = 0
for i 6= 0, 2, 4. Since the trace on H2 and H4 is 1, we obtain the result.
Remark 9.6. This lemma is false if the order of g is infinite. Take for example the
automorphism (x : y : z) 7→ (λx : y : z + y) of P2, for any λ ∈ C∗, λ 6= 1. It fixes
exactly two points, namely (1 : 0 : 0) and (0 : 0 : 1), but its trace on Pic(P2) = Z
is 1.
We now start the proof of Proposition 9.1 by studying the cases of del Pezzo
surfaces of degree 6 or 5.
Lemma 9.7 (Actions on the del Pezzo surface of degree 6). Let S6 = {
(x : y :
z), (u : v : w)
| ux = vy = wz} ⊂ P2 × P2 be the del Pezzo surface of degree 6
and let G ⊂ Aut(S6) be an Abelian group such that rk Pic(S6)
G = 1. Then, G is
conjugate in Aut(S6) to the cyclic group of order 6 generated by
(x : y : z), (u :
v : w)
(v : w : u), (y : z : x)
. Furthermore, G is birationally conjugate to a
diagonal subgroup of Aut(P2).
Proof. Lemma 9.3 implies that the sizes of the orbits of the action of G on the ex-
ceptional divisors are divisible by 6. The action of G on the hexagon of exceptional
divisors is thus transitive, so G contains an element of the form
(x : y : z), (u : v : w)
(αv : βw : u), (βy : αz : αβx)
where α, β ∈ C∗. As the only element of (C∗)2 that commutes with g is the
identity (see the description of Aut(S6) = (C
∗)2 ⋊ (Sym3 × Z/2Z) in Section 4),
G must be cyclic, generated by g. Conjugating it by
(x : y : z), (u : v : w)
(βx : y : αz), (αu : αβv : βw)
we may assume that α = β = 1, as stated in the lemma (this shows in particular
thatG is of finite order). It remains to prove that this automorphism is birationally
conjugate to a linear automorphism of the plane.
Denote by p : S → P2 the restriction of the projection on the first factor.
This is a birational morphism which is the blow-up of the three diagonal points
A1, A2, A3 of P
2. Consider the birational map ĝ = pgp−1 of P2, which is explicitly
ĝ : (x : y : z) 99K (xz : xy : yz). Since g is an automorphism of the surface, it
fixes the canonical divisor KS , so the birational map ĝ leaves the linear system
of cubics of P2 passing through A1, A2 and A3 invariant (this can also be verified
directly).
Note that ĝ fixes exactly one point of P2, namely P = (1 : 1 : 1), and that its
action on the projective tangent space P(TP (P
2)) of P2 at P is of order 3, with two
fixed points, corresponding to the lines (x− y) + ωk(z − y) = 0, where ω = e2iπ/3,
k = 1, 2. Hence, the birational map ĝ preserves the linear system of cubics of P2
passing through A1, A2 and A3, which have a double point at P and are tangent
to the line (x− y) + ω(z − y) = 0 at this point. This linear system thus induces a
birational transformation of P2 that conjugates ĝ to a linear automorphism.
Lemma 9.8 (Actions on the del Pezzo surface of degree 5). Let S5 be the del
Pezzo surface of degree 5 and let G ∈ Aut(S5) = Sym5 be an Abelian group such
that rk Pic(S5)
G = 1. Then, G is cyclic of order 5. Furthermore, G is birationally
conjugate to a diagonal subgroup of Aut(P2).
Proof. We use the description of the surface S5 and its automorphisms group
Aut(S5) = Sym5 given in Section 5. Lemma 9.3 implies that the order of G is
divisible by 5, and thus that G is a cyclic subgroup of Sym5 of order 5. Since
all such subgroups are conjugate in Aut(S5) = Sym5, we may suppose that G is
generated by the lift of the birational transformation h : (x : y : z) 99K (xy :
y(x− z) : x(y− z)) of P2, that fixes two points of P2, namely (ζ +1 : ζ : 1), where
ζ2 − ζ − 1 = 0. Denoting one of them by P , the linear system of cubics passing
through the four blown-up points and having a double point at P is invariant
by h. The birational transformation associated to this system thus conjugates h
to a linear automorphism of P2.
Remark 9.9. The fact that (x : y : z) 99K (xy : y(x− z) : x(y − z)) is linearisable
was proved in [Be-Bl], using the same argument as above.
Corollary 9.10. Let S be a rational surface with (KS)
2 ≥ 5 and let G ⊂ Aut(S)
be a finite Abelian group. Then G is birationally conjugate to a subgroup of Aut(P2)
or Aut(P1 × P1).
Proof. We may assume that the pair (G,S) is minimal; consequently there are two
possibilities (see [Man], [Isk2] or [Do-Iz]):
1. S is a del Pezzo surface and rk Pic(S)G = 1. Then S is either P2, P1×P1 or
a del Pezzo surface of degree 6 or 5 (Remark 9.4); we apply Lemmas 9.7 and 9.8
to conclude.
2. G preserves a conic bundle structure on S. Here the number of fibres is at
most 3, hence no element of G fixes a curve of positive genus (Corollary 6.4); we
apply Proposition 8.4 to conclude.
To study del Pezzo surfaces of degree 4, let us describe their group of auto-
morphisms (note that we do not use the notation Sd for the del Pezzo surfaces of
degree d ≤ 4, because there are many different surfaces of the same degree):
Lemma 9.11 (Automorphism group of del Pezzo surfaces of degree 4). Let S
be a del Pezzo surface of degree 4 given by the blow-up η : S → P2 of five points
A1, ..., A5 ∈ P
2 such that no three are collinear. Setting Ei = η
−1(Ai) and denoting
by L the pull-back by η of a general line of P2, we have:
1. There are exactly 10 conic bundle structures on S, whose fibres are respec-
tively L− Ei, −KS − (L− Ei), for i = 1, ..., 5.
2. The action of Aut(S) on the five pairs of divisors {L−Ei,−KS− (L−Ei)},
i = 1, ..., 5 gives rise to a split exact sequence
0 → F → Aut(S)
→ Sym5,
where F = {(a1, ..., a5) ∈ (F2)
ai = 0} ∼= (F2)
4, and the automorphism
(a1, ..., a5) permutes the pair {L−Ei,−KS − (L−Ei)} if and only if ai = 1.
3. We have
Aut(S) = F⋊Aut(S, η),
where Aut(S, η) is the lift of the group of automorphisms of P2 that leave
the set {A1, ..., A5} invariant, and Aut(S, η) acts on F = {(a1, ..., a5) ∈
ai = 0} by permutation of the ai’s, as it acts on {A1, ..., A5}, and
as ρ(Aut(S)) = ρ(Aut(S, η)) ⊂ Sym5 acts on the exceptional pairs.
4. The elements of F with two ”ones” correspond to quadratic involutions of
P2 and fix exactly 4 points of S.
5. The elements of F with four ”ones” correspond to cubic involutions of P2
and the points of S fixed by these elements form a smooth elliptic curve.
Remark 9.12. The group F ⊂ Aut(S) has been studied intensively since 1895 (see
[Kan], Theorem XXXIII). A modern description of the group as the 2-torsion of
PGL(5,C) can be found in [Bea2, (4.1)], together with a study of the conjugacy
classes of such groups in the Cremona group. For further descriptions of the auto-
morphism groups of these surfaces, see [Do-Iz, section 6.4] and [Bla2, section 8.1].
Proof. Let A = mL −
i=1 aiEi be the divisor of the fibre of some conic bundle
structure on S, for some m, a1, ..., a5 ∈ Z. From the relations A
2 = 0 (the fibres
are disjoint) and AKS = −2 (adjunction formula) we get:
i=1 ai
2 = m2,
i=1 ai = 3m− 2.
As in Lemma 7.1, we have (
i=1 ai)
2 ≤ 5
i=1 ai
2, which implies here that
(3m− 2)2 ≤ 5m2, that is 4(m2 − 3m+ 1) ≤ 0. As m is an integer, we must have
1 ≤ m ≤ 2. If m = 1, we replace it in (3) and see that there exists i ∈ {1, ..., 5}
such that A = L − Ei. Otherwise, taking m = 2 and replacing it in (3), we see
that four of the aj ’s are equal to 1, and one is equal to 0. This gives the ten conic
bundles of assertion 1, which are the lift on S of the lines of P2 passing through
one of the Ai’s or of the conic passing through four of the Ai’s.
The group Aut(S) acts on the set ∪5i=1{L−Ei,−KS − (L−Ei)}; since KS is
fixed, this induces an action on the set of five pairs {L−Ei,−KS − (L−Ei)}. We
denote by ρ : Aut(S) → Sym5 the corresponding homomorphism. The action of
the kernel of ρ on the pairs of conic bundles gives a natural embedding of Ker(ρ)
into (F2)
We now prove that Ker(ρ) = {(a1, ..., a5) |
ai = 0} = F. Acting by a
linear automorphism of P2, we may assume that the points blown-up by η are
A1 = (1 : 0 : 0), A2 = (0 : 1 : 0), A3 = (0 : 0 : 1), A4 = (1 : 1 : 1), A5 = (a : b : c),
for some a, b, c ∈ C∗. Then, the birational involution τ : (x0 : x1 : x2) 99K (ax1x2 :
bx0x2 : cx0x1) of P
2 lifts as an automorphism η−1τη ∈ Aut(S) that acts on Pic(S)

0 −1 −1 0 0 −1
−1 0 −1 0 0 −1
−1 −1 0 0 0 −1
0 0 0 0 1 0
0 0 0 1 0 0
1 1 1 0 0 2

with respect to the basis (E1, E2, E3, E4, E5, L). It follows from this observation
that η−1τη belongs to the kernel of ρ, and acts on the pairs of conic bundles
as (0, 0, 0, 1, 1) ∈ (F2)
5. Permuting the roles of the points A1, ..., A5, we get 10
involutions whose representations in (F2)
5 have two ”ones” and three ”zeros”.
These involutions generate the group {(a1, ..., a5) |
ai = 0} = F. To prove that
this group is equal to Ker(ρ), it suffices to show that (1, 1, 1, 1, 1) does not belong
to Ker(ρ). This follows from the fact that (1, 1, 1, 1, 1) would send L = 1
(KS +∑5
i=1(L − Ei)) on the divisor
(KS +
i=1(−KS − L + Ei)) =
(−2L − 3KS),
which doesn’t belong to Pic(S). This concludes the proof of assertion 2 (except
the fact that the exact sequence is split, which will be proved by assertion 3).
We now prove assertion 3. Let σ ∈ Sym5 be a permutation of the set {1, ..., 5}
in the image of ρ and g be an automorphism of S such that ρ(g) = σ. Let α be
the element of Aut(Pic(S)) that sends Ei on Eσ(i) and fixes L. Viewing Aut(S)
as a subgroup of Aut(Pic(S)), the element gα−1 ∈ Aut(Pic(S)) fixes the five pairs
of conic bundles. There exists some element h ∈ F ⊂ Aut(S) such that hgα−1
either fixes the divisor of every conic bundle or permutes the divisors of conic
bundles in each pair. The same argument as in the above paragraph shows that
this latter possibility cannot occur. Hence hgα−1 fixes L−E1, ..., L−E5 and KS.
It follows that hgα−1 acts trivially on Pic(S), so α = hg ∈ Aut(S), and α is by
construction the lift of an automorphism of P2 that acts on the set {A1, ..., A5} as
σ does on {1, ..., 5}. Conversely, it is clear that every automorphism r of P2 which
leaves the set {A1, A2, A3, A4, A5} invariant lifts to the automorphism η
−1rη of
S whose action on the pairs of conic bundles is the same as that of r on the set
{A1, A2, A3, A4, A5}. This gives assertion 3.
Assertion 4 follows from the above description of some element of F ⊂ Aut(S)
with two ”ones” as the lift of a birational map of the form τ : (x0 : x1 : x2) 99K
(ax1x2 : bx0x2 : cx0x1). As the automorphism η
−1τη ∈ Aut(S) does not leave any
exceptional divisor invariant, its fixed points are the same as those of τ , which are
the four points (α : β : γ), where α2 = a, β2 = b, γ2 = c.
It remains to prove the last assertion. Note that the element h = (0, 1, 1, 1, 1) ∈
Aut(S) fixes the divisor L−E1, hence acts on the associated conic bundle structure.
Furthermore, the four singular fibres of this conic bundle, {L − E1 − Ei, Ei}, for
i = 2, ..., 5, are invariant by h and this element switches the two components of
each fibre. This shows that the action of h on the basis of the fibration is trivial,
so the restriction of h on each fibre is an involution of P1 which fixes two points.
On each singular fibre, exactly one point is fixed, which is the singular point of
the fibre. The situation is similar for the other elements with four ”ones” (in fact,
the involutions described here are twisting involutions, see Lemma 6.1).
Lemma 9.13 (Actions on the del Pezzo surfaces of degree 4). Let S be a del
Pezzo surface of degree 4, and let G ∈ Aut(S) be an Abelian group such that
rk Pic(S)G = 1. Then, G contains an involution that fixes an elliptic curve.
Proof. We keep the notation of Lemma 9.11 for η : S → P2,Aut(S, η), ρ,F, ... and
denote by H the group G ∩ F = G ∩ Kerρ. We will prove that H contains an
element of F with four ”ones”, which is an involution that fixes an elliptic curve
(Lemma 9.11).
The group ρ(G) ⊂ ρ(Aut(S)) ∼= Aut(S, η) is isomorphic to a subgroup of
Aut(S, η). The group Aut(S, η) is the lift of the group of automorphisms of P2 that
leave the set {A1, ..., A5} invariant (Lemma 9.11). The restriction of this group to
the conic of P2 passing through the five points is a subgroup of PGL(2,C) that
leaves five points invariant. Since ρ(G) is finite and Abelian, it is cyclic, of order
at most 5. We consider the different possibilities.
The order of ρ(G) is 1. This implies that G ⊂ F. If G contains an element
with four ”ones”, we are done. Otherwise, up to conjugation G is a subset of the
group generated by (1, 1, 0, 0, 0) and (1, 0, 1, 0, 0), and fixes L − E4 and L − E5
(thus rk Pic(S)G > 1).
The order of ρ(G) is 2. Up to a change of numbering, ρ(G) is generated by
(1 2)(3 4); since G is Abelian, we find that H ⊂ V = {(a, a, b, b, 0) | a, b ∈ F2}.
Let g = ((a, b, c, d, e), (1 2)(3 4)) ∈ G be such that ρ(g) = (1 2)(3 4). We may
suppose that e = 1 (otherwise, the group G would fix L − E5 and we would
have rk Pic(S)G ≥ 2.) Conjugating by ((0, b, 0, d, b + c), id) we may assume that
g = ((a+ b, 0, c+ d, 0, 1), (1 2)(3 4)). In fact, since a+ b+ c+ d+ e = 0, we have
g = ((α, 0, 1+α, 0, 1), (1 2)(3 4)), where α = a+b = c+d+1 ∈ F2. If α = 1, then g
has order 4 and fixes the divisor 2L− E3 − E4, thus G cannot be equal to < g >
and it follows that V ⊂ G; in particular the element (1, 1, 1, 1, 0) is contained in
G. If α = 0, then < g > fixes 2L− E1 − E2, so once again G contains V .
The order of ρ(G) is 3. In this case, ρ(G) is generated by a 3-cycle, namely
(1 2 3); then H must be a subgroup of V = {(a, a, a, b, a + b) | a, b ∈ F2}. The
order of G must be a multiple of 4, by Lemma 9.3, hence H = V , and thus G
contains the element (1, 1, 1, 1, 0).
The order of ρ(G) is 4. Then ρ(G) is generated by (1 2 3 4), so H must be a
subgroup of V =< (1, 1, 1, 1, 0) >. Let g = ((a, b, c, d, e), (1 2 3 4)) ∈ G be such
that ρ(g) = (1 3 2 4). Conjugating the group by ((a, a+b, a+b+c, 0, a+c), id), we
may suppose that g = ((0, 0, 0, e, e), (1 3 2 4)). If e = 1, then g4 = (1, 1, 1, 1, 0) ∈ G.
If e = 0, the element g belongs to HS , so it fixes the divisors L and E5. As the
group V fixes L− E5, the rank of Pic(S)
G cannot be 1.
The order of ρ(G) is 5. Then, ρ(G) is generated by a 5-cycle and H = {1}.
The rank of Pic(S)H cannot be 1, by Lemma 9.3.
Before studying the case of del Pezzo surfaces of degree ≤ 3, we remind the
reader of some classical embeddings of these surfaces.
Remark 9.14. Recall ([Kol], Theorem III.3.5) that a del Pezzo surface of degree 3
(respectively 2, 1) is isomorphic to a smooth hypersurface of degree 3 (respectively
4, 6) in the projective space P3 (respectively in P(1, 1, 1, 2), P(1, 1, 2, 3)). Further-
more, in each of the 3 cases, any automorphism of the surface is the restriction of
an automorphism of the ambient space. We will use these classical embeddings,
take w, x, y, z as the variables on the projective spaces, and denote by [α : β : γ : δ]
the automorphism (w : x : y : z) 7→ (αw : βx : γy : δz). Note that a del Pezzo
surface of degree 4 is isomorphic to the intersection of two quadrics in P4, but we
will not use this here.
Lemma 9.15 (Actions on the del Pezzo surfaces of degree 3). Let S be a del
Pezzo surface of degree 3, and let G ∈ Aut(S) be an Abelian group such that
rk Pic(S)G = 1. Then, G contains an element of order 2 or 3 that fixes an elliptic
curve of S.
Proof. Lemma 9.3 implies that the order of G is divisible by 3, so G contains an
element of order 3. We view S as a cubic surface in P3, and Aut(S) as a subgroup
of PGL(4,C) (see Remark 9.14). There are three kinds of elements of order 3 in
PGL(4,C), depending on the nature of their eigenvalues. Setting ω = e2iπ/3, there
are elements with one eigenvalue of multiplicity 3 (conjugate to [1 : 1 : 1 : ω], or its
inverse), elements with two eigenvalues of multiplicity 2 (conjugate to [1 : 1 : ω : ω])
and elements with three distinct eigenvalues (conjugate to [1 : 1 : ω : ω2]). We
consider the three possibilities.
Case a: G contains an element of order 3 with one eigenvalue of multiplicity 3.
The element [1 : 1 : 1 : ω] fixes the hyperplane z = 0, whose intersection with the
surface S is an elliptic curve (because Fix(g) ⊂ S is smooth). Thus, we are done.
Case b: G contains an element of order 3 with two eigenvalues of multiplicity 2.
With a suitable choice of coordinates, we may assume that this element is
g = [1 : 1 : ω : ω].
Since S is smooth, its equation F is of degree at least 2 in each variable, which
implies that F (w, x, ωy, ωz) = F (w, x, y, z) (the eigenvalue is 1); up to a change
of coordinates F = w3 + x3 + y3 + z3, which means that S is the Fermat cubic
surface. The group of automorphisms of S is (Z/3Z)3 ⋊ Sym4 and the centraliser
of g in it is (Z/3Z)3 ⋊ V , where V ∼= (Z/2Z)2 is the subgroup of Sym4 generated
by the two transpositions (w, x) and (y, z). The structure of the centraliser gives
rise to an exact sequence
1 → (Z/3Z)3 → (Z/3Z)3 ⋊ V
→ V → 1
∪ ∪ ∪
1 → G ∩ (Z/3Z)3 → G → γ(G) → 1.
We may suppose that G contains no element of order 3 with an eigenvalue of
multiplicity 3, since this case has been studied above (case a). There are then
three possibilities for G ∩ (Z/3Z)3, namely < g >, < g, [1 : ω : 1 : ω] > and
< g, [1 : ω : ω : 1] >. The last is conjugate to the second by the automorphism
(y, z). Note that g preserves exactly 9 of the 27 lines on the surface; these
are {w + ωix = y + ωjz = 0}, for 0 ≤ i, j ≤ 2. If G ∩ (Z/3Z)3 is equal to
< g >, then G/ < g >∼= γ(G) has order 1, 2 or 4 and thus G leaves at least
one of the 9 lines invariant, whence rk Pic(S)G > 1. If G ∩ (Z/3Z)3 is the group
H =< g, [1 : ω : 1 : ω] > we have G = H , since the centraliser of H in (Z/3Z)3⋊V
is the group (Z/3Z)3. As the set of three skew lines {w + ωix = y + ωiz = 0} for
0 ≤ i ≤ 2 is an orbit of H , the rank of Pic(S)G is strictly larger than 1.
Case c: G contains an element g of order 3 with three distinct eigenvalues.
We may suppose that g = [1 : 1 : ω : ω2]. Note that the action of g on P3
fixes the line Lyz of equation y = z = 0 and thus the whole group G leaves this
line invariant. If Lyz ⊂ S, the rank of rk Pic(S)
G is at least 2. Otherwise, the
equation of S is of the form L3(w, x)+L1(w, x)yz+ y
3+ z3 = 0, where L3 and L1
are homogeneous forms of degree respectively 3 and 1, and L3 has three distinct
roots, so Fix(g) = S ∩ Lyz. Since g fixes exactly three points, the trace of its
action on Pic(S) ∼= Z7 is 1 (Lemma 9.5) and thus rk Pic(S)g > 1, which implies
that G 6=< g >.
Note that every subgroup of PGL(4,C) isomorphic to (Z/3Z)2 contains an
element with only two distinct eigenvalues, so we may assume that G contains
only two elements of order 3, which are g and g2. This implies that the action of
G on the three points of Lyz ∩ S gives an exact sequence
1 →< g >→ G → Sym3,
where the image on the right is a transposition. The group G thus contains an
element of order 2, that we may assume to be diagonal of the form (w : x : y :
z) 7→ (−w : x : y : z) and that fixes the elliptic curve which is the trace on S of
the plane w = 0.
Lemma 9.16 (Actions on the del Pezzo surfaces of degree 2). Let S be a del
Pezzo surface of degree 2, and let G ∈ Aut(S) be an Abelian group such that
rk Pic(S)G = 1. Then, G contains either the Geiser involution (that fixes a curve
isomorphic to a smooth quartic curve) or an element of order 2 or 3 that fixes an
elliptic curve.
Proof. We view S as a surface of degree four in the weighted projective space
P(2, 1, 1, 1) (see Remark 9.14). Note that the projection on the last three coordi-
nates gives S as a double covering of P2 ramified over a smooth quartic curve Q.
Lemma 9.3 implies that the order of G is divisible by 2, so G contains an
element g of order 2.
If the element g is the involution induced by the double covering (classically
called the Geiser involution), we are done; otherwise we may assume that g acts
on P(2, 1, 1, 1) as g : (w : x : y : z) 7→ (ǫw : x : y : −z), where ǫ = ±1, and the
equation of S is w2 = z4 + L2(x, y)z
2 + L4(x, y), where Li is a form of degree i,
and L4 has four distinct roots. The trace on S of the equation z = 0 defines an
elliptic curve Lz ⊂ S. If ǫ = 1, then g fixes the curve Lz and we are done; we
therefore assume that ǫ = −1.
If G contains another involution, we diagonalise the group generated by these
two involutions and see that one element of the group fixes either an elliptic curve
or the smooth quartic curve, so we may assume that g is the only involution of G.
Note that g fixes exactly four points of S, which are the points of intersection
of Lz with the quartic Q (of equation w = 0). The trace of g on Pic(S) ∼= Z
thus equal to 2 (Lemma 9.5), whence rk Pic(S)g = 5 and G 6=< g >.
The group G acts on the line z = 0 of P2 and on the four points of Lz ∩ Q.
Since g is the only element of order 2 of G, the action of G on these four aligned
points has order 3 and thus, we may assume that L4(x, y) = x(x
3 +λy3) and that
there exists an element h of G that acts as (w : x : y : z) 7→ (αw : x : e2iπ/3y : βz),
with α2 = β4 = 1. We find that h4 is an element of order 3 that fixes the elliptic
curve which is the trace on S of the equation y = 0.
Lemma 9.17 (Actions on the del Pezzo surfaces of degree 1). Let S be a del
Pezzo surface of degree 1, and let G ∈ Aut(S) be an Abelian group such that
rk Pic(S)G = 1. Then, some non-trivial element of G fixes a curve of S of positive
genus.
Proof. We view S as a surface of degree six in the weighted projective space
P(3, 1, 1, 2) (see Remark 9.14). Up to a change of coordinates, we may assume
that the equation is
w2 = z3 + zL4(x, y) + L6(x, y),
where L4 and L6 are homogeneous forms of degree 4 and 6 respectively. The
embedding of S into P(3, 1, 1, 2) is given by | − 3KS| × | −KS | × | − 2KS|, which
implies that G is a subgroup of P (GL(1,C)×GL(2,C)×GL(1,C)). The projection
(w : x : y : z) 99K (x : y) is an elliptic fibration generated by | − KS |, and has
one base-point, namely (1 : 0 : 0 : 1), which is fixed by Aut(S). This projection
induces an homomorphism ρ : Aut(S) → Aut(P1) = PGL(2,C). Note that the
kernel of ρ is generated by the Bertini involution w 7→ −w (and the element z 7→ ωz
(ω = e2iπ/3) if L4 = 0) and is hence cyclic of order 2 (or 6). Furthermore, any
element of this kernel fixes a curve of positive genus.
We assume that no non-trivial element of G fixes a curve of positive genus.
This implies that G is isomorphic to ρ(G) ⊂ Aut(P1), and thus is either cyclic or
isomorphic to (Z/2Z)2. Since the lift of this latter group in Aut(S) is not Abelian,
G is cyclic. We use the Lefschetz fixed-point formula (Lemma 9.5) to deduce the
eigenvalues of the action of elements of G on Pic(S) ∼= Z9. For any element g ∈ G,
g 6= 1, Fix(g) contains the point (1 : 0 : 0 : 1) and is the disjoint union of points
and lines. Thus χ(Fix(g)) ≥ 1 and so the trace of g on Pic(S) is at least −1
(Lemma 9.5).
Elements of order 2: The eigenvalues are < 1a, (−1)b > with a ≥ 4, b ≤ 5.
Elements of order 3: The eigenvalues are < 1a, (ω)b, (ω2)b > with a ≥ 3, b ≤ 3.
Elements of order 4: The eigenvalues are < 1a, (−1)b, (i)c, (−i)c > with a ≥
b−1. Furthermore, the information on the square induces that a+b ≥ 4, so a ≥ 2.
Elements of order 5: The eigenvalues are < 15, l1, l2, l3, l4 >, where l1, ..., l4 are
the four primitive 5-th roots of unity.
Elements of order 6: The eigenvalues are< 1a, (−1)b, (ω)c, (ω2)c, (−ω)d, (−ω2)d >,
where a − b − c + d ≥ −1. Computing the square and the third power, we find
respectively a + b ≥ 3, c + d ≤ 3 and a + 2c ≥ 4, b + 2d ≤ 5. This implies that
a ≥ 2. Indeed, if a = 1, we get b, c ≥ 2 and thus d ≤ 1, which contradicts the fact
that the trace a− b− c+ d is at least −1.
Since rk Pic(S)G = 1, the order of the cyclic group G is at least 7. As the
action of G leaves L4 and L6 invariant, both L6 and L4 are monomials. If some
double root of L6 is a root of L4, the surface is singular, so up to an exchange of
coordinates we may suppose that L4 = x
4 and either L6 = xy
5 or L6 = y
In the first case, the equation of the surface is w2 = z3+x4z+xy5 whose group
of automorphisms Aut(S) is isomorphic to Z/20Z, generated by [i : 1 : ζ10 : −1],
and contains the Bertini involution. No subgroup of Aut(S) fullfills our hypotheses.
In the second case, the equation of the surface is w2 = z3 + x4z + y6, whose
group of automorphisms is isomorphic to Z/2Z×Z/12Z, generated by the Bertini
involution and g = [i : 1 : ζ12 : −1]. The only possibility for G is to be equal to
< g >. Since g4 = [1 : 1 : ω : 1] fixes an elliptic curve, we are done.
Proposition 9.1 now follows, using all the lemmas proved above.
10 The results
We now prove the five theorems stated in the introduction.
Proof of Theorem 4. Since the pair (G,S) is minimal, either rk Pic(S)G = 1 and
S is a del Pezzo surface, or G preserves a conic bundle structure (see [Man], [Isk2]
or [Do-Iz]).
In the first case, either S ∼= P2, or S ∼= P1 × P1 or S is a del Pezzo surface of
degree d = 5 or 6 and G ∼= Z/dZ (Proposition 9.1).
In the second case, either S is a Hirzebruch surface or the pair (G,S) is the
pair (Cs24, Ŝ4) of Section 7 (Proposition 8.4).
Proof of Theorem 2. No non-trivial element of Aut(P2),Aut(P1 × P1) or Cs24
fixes a non-rational curve (the first two cases are clear, the last one follows from
Lemma 7.4).
Conversely, suppose that G is a finite Abelian subgroup of the Cremona group
such that no non-trivial element fixes a curve of positive genus. Since G is finite,
it is birationally conjugate to a group of automorphisms of a rational surface S
(see for example [dF-Ei, Theorem 1.4] or [Do-Iz]). Then, we assume that the pair
(G,S) is minimal and use the classification of Theorem 4.
If S is an Hirzebruch surface, the group is birationally conjugate to a subgroup
of Aut(P1 ×P1) (Proposition 8.3). If S is a del Pezzo surface, the group G is bira-
tionally conjugate to a subgroup of Aut(P1 × P1) or Aut(P2), by Proposition 9.1.
Otherwise, the pair (G,S) is isomorphic to the pair (Cs24, Ŝ4).
It remains to show that the group Cs24 is not birationally conjugate to a
subgroup of Aut(P1 × P1) or Aut(P2). Since the group is isomorphic to Z/2Z ×
Z/4Z, only the case of Aut(P1 × P1) need be considered (see Section 2). This was
proved in Proposition 8.4.
Proof of Theorem 5. By Theorem 2, G is birationally conjugate either to a sub-
group of Aut(P2), or of Aut(P1 × P1), or to the group Cs24.
The group Cs24 is case [8]. The finite Abelian subgroups of Aut(P
2) are conju-
gate to the groups of case [1] or [9] (Proposition 2.2). The finite Abelian subgroups
of Aut(P1 × P1) are conjugate to the groups of cases [1] through [7] (Proposi-
tion 2.5).
It was proved in Proposition 2.5 that cases [1] through [7] are distinct. In
Proposition 8.4 we showed that [8] (Cs24) is not birationally conjugate to any
groups of cases [1] through [7]. Finally, the group [9] is isomorphic only to [1], but
is not birationally conjugate to it (Proposition 2.2). This completes the proof that
the distincts cases given above are not birationally conjugate.
The proof of Theorem 1 follows directly from Theorem 5, and Theorem 3 is a
corollary of Theorem 1.
11 Other kinds of groups
Our main interest up to now was in finite Abelian subgroups of the Cremona
group. In this section, we give some examples in the other cases, in order to show
why the hypothesis ”finite”, respectively ”Abelian”, is necessary to ensure that
condition (F ) (no curve of positive genus is fixed by a non-trivial element) implies
condition (M) (the group is birationally conjugate to a group of automorphisms
of a minimal surface). We refer to the introduction for more details.
Finiteness is important since it imposes that the group is conjugate to a group
of automorphisms of a projective rational surface. This is not the case if the group
is not finite (see for example [Bla2], Proposition 2.2.4).
Lemma 11.1. Let ϕ : P2 99K P2 be a quadratic birational transformation with
three proper base-points, and such that deg(ϕn) = 2n for each integer n ≥ 1.
Then, the following occur:
1. no pencil of curves is invariant by ϕ;
2. ϕ is not birationally conjugate to an automorphism of P2 or of P1 × P1.
Proof. Denote by A1, A2, A3 the three base-points of ϕ and by B1, B2, B3 those
of ϕ−1. Up to a change of coordinates, we may suppose that A1 = (1 : 0 : 0),
A2 = (0 : 1 : 0) and A3 = (0 : 0 : 1). The birational transformation ϕ is thus the
composition of the standard quadratic transformation σ : (x : y : z) 99K (yz : xz :
xy) with a linear automorphism τ ∈ Aut(P2) that sends Ai on Bi for i = 1, 2, 3.
Let Λ be some pencil of curves, and assume that ϕ(Λ) = Λ. We will prove
that some base-point of Λ is sent by ϕ on an orbit of infinite order. The con-
dition deg(ϕn) = 2n is equivalent to saying that for i = 1, 2, 3, the sequence
Bi, ϕ(Bi), ..., ϕ
n(Bi), ... is well-defined, i.e. that ϕ
m(Bi) is not equal to Aj for any
i, j ∈ {1, 2, 3},m ∈ N. Denote by α1, α2, α3, β1, β2, β3 the multiplicity of Λ at
respectively A1, A2, A3, B1, B2, B3 and by n the degree of the curves of Λ. The
curves of the pencil ϕ(Λ) thus have degree 2n−α1−α2−α3. Since Λ is invariant,
n = α1 + α2 + α3, so at least one of the αi’s is not equal to zero. The equality
n = α1+α2+α3 implies that the curves of σ(Λ) have multiplicity αi at Ai, so the
curves of ϕ(Λ) have multiplicity αi at Bi, whence αi = βi for i = 1, 2, 3. Since Λ
passes through Bi with multiplicity αi, the pencil ϕ(Λ) = Λ passes through ϕ(Bi)
with multiplicity αi for i = 1, 2, 3. Continuing in this way, we see that Λ passes
through ϕn(Bi) with multiplicity αi for each n ∈ N. Consequently, Λ has infinitely
many base-points, which is not possible. This establishes the first assertion.
The second assertion follows directly, as each automorphism of P2 or P1 × P1
leaves a pencil of rational curves invariant.
Corollary 11.2. The group generated by a very general quadratic transformation
is a infinite cyclic group satisfying (F ) but not (M).
Proof. The condition deg(ϕn) = 2n, n ∈ N is satisfied for all quadratic transfor-
mations, except for a countable set of proper subvarieties. Consequently condition
(F ) is not satisfied (Lemma 11.1) for a very general quadratic transformation.
Let n be some positive integer and write ϕn : (x : y : z) 99K (f1(x, y, z) :
f2(x, y, z) : f3(x, y, z)), for some homogeneous polynomials fi of degree 2
n. The
set of points fixed by ϕn belongs to the intersection of the curves with equations
xf2 − yf1, xf3 − zf1 and yf3 − zf2. In general, there is only a finite number of
points; this yields condition (F ).
In fact, the argument of Lemma 11.1 works for any very general birational
transformation of P2, since this is a composition of quadratic transformations.
We thus find infinitely many cyclic subgroups of the Cremona group that are not
birationally conjugate to a group of automorphisms of a minimal surface although
none of their non-trivial elements fixes a non-rational curve. The implication
(F ) ⇒ (M) is therefore false for general cyclic groups.
We now study the finite non-Abelian subgroups and provide, in this case, many
examples satisfying (F ) but not (M):
Lemma 11.3. Let S6 = {
(x : y : z), (u : v : w)
| ux = vy = wz} ⊂ P2 × P2
be the del Pezzo surface of degree 6. Let G ∼= Sym3 × Z/2Z be the subgroup of
automorphisms of S6 generated by
(x : y : z), (u : v : w)
(u : v : w), (x : y : z)
(x : y : z), (u : v : w)
(y : x : z), (v : u : w)
(x : y : z), (u : v : w)
(z : y : x), (w : v : u)
Then no non-trivial element of G fixes a curve of positive genus, and G is not
birationally conjugate to a group of automorphisms of a minimal surface.
Proof. Since every non-trivial element of finite order of Aut(S6) is birationally
conjugate to a linear automorphism of P2 (Corollary 9.10), no such element fixes
a curve of positive genus. The description of every G-equivariant elementary link
starting from S6 was given by Iskovskikh in [Isk4]. This shows that this group is
not birationally conjugate to a group of automorphisms of a minimal surface.
Lemma 11.4. Let S5 be the del Pezzo surface of degree 5. Let G ∼= Sym5 be
the whole group Aut(S5). Then no non-trivial element of G fixes a curve of posi-
tive genus, and G is not birationally conjugate to a group of automorphisms of a
minimal surface.
Proof. Since every non-trivial element of Aut(S5) is birationally conjugate to a
linear automorphism of P2 (Corollary 9.10), such an element does not fix a curve
of positive genus. Suppose that there exists some G-equivariant birational trans-
formation ϕ : S5 99K S̃ where S̃ is equal to P
2 or P1 × P1. We decompose ϕ into
G-equivariant elementary links (see for example [Isk3], Theorem 2.5). The classi-
fication of elementary links ([Isk3], Theorem 2.6) shows that a link S5 99K S
either a Bertini or a Geiser involution (and in this case S′ = S5, and thus this link
conjugates G to itself), or the composition of the blow-up of one or two points,
and the contraction of 5 curves to respectively P1 × P1 or P2. It remains to show
that no orbit of G has size 2 or 1, to conclude that these links are not possible.
This follows from the fact that the actions of Sym5,Alt5 ⊂ G on S5 are fixed-point
free (Proposition 5.1).
Finally, the way to find more counterexamples is to look at groups acting on
conic bundles. The generalisation of the example Cs24 gives many examples of
non-Abelian finite groups. Here is the simplest family:
Lemma 11.5. Let n be some positive integer, and let G be the group of birational
transformations of P2 generated by
g1 : (x : y : z) 99K (yz : xy : −xz),
g2 : (x : y : z) 99K (yz(y − z) : xz(y + z) : xy(y + z)),
h : (x : y : z) 99K (e2iπ/2nx : y : z).
Then, G preserves the pencil Λ of lines passing through (1 : 0 : 0) and the corre-
sponding action gives rise to a non-split exact sequence
1 →< h >∼= Z/2nZ → G → (Z/2Z)2 → 1.
In particular, the group G has order 8n. Furthermore, no non-trivial element of
G fixes a curve of positive genus, and G is not birationally conjugate to a group
of automorphisms of a minimal surface.
Proof. Firstly, since g1 and g2 generate the group Cs24, which is not birationally
conjugate to a group of automorphisms of a minimal surface, this is also the case
for G.
Secondly, we compute that (g1)
2 = (g2)
2 = (h)n is the birational transforma-
tion (x : y : z) 7→ (−x : y : z). The maps g1 and g2 thus have order 4 and h has
order 2n.
Thirdly, every generator of G preserves the pencil Λ of lines passing through
(1 : 0 : 0). The action of g1, g2 and h on this pencil is respectively (y : z) 7→ (−y :
z), (y : z) 7→ (z : y) and (y : z) 7→ (y : z). The action of G on the pencil thus gives
an exact sequence
1 → G′ → G → (Z/2Z)2 → 0,
where G′ is the subgroup of elements of G that act trivially on the pencil Λ. It is
clear that < h >∼= Z/2nZ is a subgroup of G′. Since g1h(g1)
−1 = g2h(g2)
−1 = h−1
and g1 and g2 commute, the group < h > is equal to G
Finally, any element of G that fixes a curve of positive genus must act trivially
on the pencil Λ and thus belongs to < h >. Hence, only the identity is possible.
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	Introduction
	The main questions and results
	How to decide
	Linearisation of birational actions
	The approach and other results
	Comparison with other work
	Aknowledgements
	Automorphisms of P2 or P1P1
	Some facts about automorphisms of conic bundles
	The del Pezzo surface of degree 6
	The del Pezzo surface of degree 5
	Description of twisting elements
	The example Cs24
	Finite Abelian groups of automorphisms of conic bundles - birational representative elements
	Actions on del Pezzo surfaces with fixed part of the Picard group of rank one
	The results
	Other kinds of groups

  This article gives the proof of results announced in [J. Blanc, Finite
Abelian subgroups of the Cremona group of the plane, C.R. Acad. Sci. Paris,
S\'er. I 344 (2007), 21-26.] and some description of automorphisms of rational
surfaces.
  Given a finite Abelian subgroup of the Cremona group of the plane, we provide
a way to decide whether it is birationally conjugate to a group of
automorphisms of a minimal surface.
  In particular, we prove that a finite cyclic group of birational
transformations of the plane is linearisable if and only if none of its
non-trivial elements fix a curve of positive genus. For finite Abelian groups,
there exists only one surprising exception, a group isomorphic to Z/2ZxZ/4Z,
whose non-trivial elements do not fix a curve of positive genus but which is
not conjugate to a group of automorphisms of a minimal rational surface.
  We also give some descriptions of automorphisms (not necessarily of finite
order) of del Pezzo surfaces and conic bundles.

Introduction
1.1 The main questions and results
In this paper, every surface will be complex, rational, algebraic and smooth, and
except for C2, will also be projective. By an automorphism of a surface we mean
a biregular algebraic morphism from the surface to itself. The group of automor-
phisms (respectively of birational transformations) of a surface S will be denoted
by Aut(S) (respectively by Bir(S)).
The group Bir(P2) is classically called the Cremona group. Taking some sur-
face S, any birational map S 99K P2 conjugates Bir(S) to Bir(P2); any subgroup
of Bir(S) may therefore be viewed as a subgroup of the Cremona group, up to
conjugacy.
The minimal surfaces are P2, P1×P1 and the Hirzebruch surfaces Fn for n ≥ 2;
their groups of automorphisms are a classical object of study, and their structures
are well known (see for example [Bea1]). These groups are in fact the maximal
connected algebraic subgroups of the Cremona group (see [Mu-Um], [Um]).
Given some group acting birationally on a surface, we would like to determine
some geometric properties that allow us to decide whether the group is conjugate to
a group of automorphisms of a minimal surface, or equivalently to decide whether
http://arxiv.org/abs/0704.0537v2
it belongs to a maximal connected algebraic subgroup of the Cremona group. This
conjugation looks like a linearisation, as we will see below, and explains our title.
We observe that the set of points of a minimal surface which are fixed by a
non-trivial automorphism is the union of a finite number of points and rational
curves. Given a group G of birational transformations of a surface, the following
properties are thus related (note that for us the genus is the geometric genus, so
that a curve has positive genus if and only if it is not rational); property (F ) is
our candidate for the geometric property for which we require:
(F ) No non-trivial element of G fixes (pointwise) a curve of positive genus.
(M) The group G is birationally conjugate to a group of automorphisms of
a minimal surface.
The fact that a curve of positive genus is not collapsed by a birational trans-
formation of surfaces implies that property (F ) is a conjugacy invariant; it is
clear that the same is true of property (M). The above discussion implies that
(M) ⇒ (F ); we would like to prove the converse.
The implication (F ) ⇒ (M) is true for finite cyclic groups of prime order
(see [Be-Bl]). The present article describes precisely the case of finite Abelian
groups. We prove that (F ) ⇒ (M) is true for finite cyclic groups of any order,
and that we may restrict the minimal surfaces to P2 or P1 × P1. In the case of
finite Abelian groups, there exists, up to conjugation, only one counterexample
to the implication, which is represented by a group isomorphic to Z/2Z × Z/4Z
acting biregularly on a special conic bundle. Precisely, we will prove the following
results, announced without proof as Theorems 4.4 and 4.5 in [Bla3]:
Theorem 1. Let G be a finite cyclic subgroup of order n of the Cremona group.
The following conditions are equivalent:
• If g ∈ G, g 6= 1, then g does not fix a curve of positive genus.
• G is birationally conjugate to a subgroup of Aut(P2).
• G is birationally conjugate to a subgroup of Aut(P1 × P1).
• G is birationally conjugate to the group of automorphisms of P2 generated
by (x : y : z) 7→ (x : y : e2iπ/nz).
Theorem 2. Let G be a finite Abelian subgroup of the Cremona group. The
following conditions are equivalent:
• If g ∈ G, g 6= 1, then g does not fix a curve of positive genus.
• G is birationally conjugate to a subgroup of Aut(P2), or to a subgroup of
Aut(P1 × P1) or to the group Cs24 isomorphic to Z/2Z × Z/4Z, generated
by the two elements
(x : y : z) 99K (yz : xy : −xz),
(x : y : z) 99K (yz(y − z) : xz(y + z) : xy(y + z)).
Moreover, this last group is conjugate neither to a subgroup of Aut(P2), nor to a
subgroup of Aut(P1 × P1).
Then, we discuss the case in which the group is infinite, respectively non-
Abelian (Section 11) and provide many examples of groups satisfying (F ) but not
Note that many finite groups which contain elements that fix a non-rational
curve are known, see for example [Wim] or more recently [Bla2] and [Do-Iz]. This
can also occur if the group is infinite, see [BPV] and [Bla5]. In fact, the set of
non-rational curves fixed by the elements of a group is a conjugacy invariant very
useful in describing conjugacy classes (see [Ba-Be], [dFe], [Bla4]).
1.2 How to decide
Given a finite Abelian group of birational transformations of a (rational) surface,
we thus have a good way to determine whether the group is birationally conjugate
to a group of automorphisms of a minimal surface (in fact to P2 or P1 × P1). If
some non-trivial element fixes a curve of positive genus (i.e. if condition (F ) is not
satisfied), this is false. Otherwise, if the group is not isomorphic to Z/2Z×Z/4Z,
it is birationally conjugate to a subgroup of Aut(P2) or of Aut(P1 × P1). There
are exactly four conjugacy classes of groups isomorphic to Z/2Z×Z/4Z satisfying
condition (F ) (see Theorem 5); three are conjugate to a subgroup of Aut(P2) or
Aut(P1 × P1), and the fourth (the group Cs24 of Theorem 2, described in detail
in Section 7) is not.
1.3 Linearisation of birational actions
Our question is related to that of linearisation of birational actions on C2. This
latter question has been studied intensively for holomorphic or polynomial actions,
see for example [De-Ku], [Kra] and [vdE]. Taking some group acting birationally
on C2, we would like to know if we may birationally conjugate this action to have
a linear action. Note that working on P2 or C2 is the same for this question.
Theorem 1 implies that for finite cyclic groups, being linearisable is equivalent
to fulfilling condition (F ). This is not true for finite Abelian groups in general,
since some groups acting biregularly on P1 × P1 are not birationally conjugate to
groups of automorphisms of P2. Note that Theorem 1 implies the following result
on linearisation, also announced in [Bla3] (as Theorem 4.2):
Theorem 3. Any birational map which is a root of a non-trivial linear automor-
phism of finite order of the plane is conjugate to a linear automorphism of the
plane.
1.4 The approach and other results
Our approach – followed in all the modern articles on the subject – is to view the
finite subgroups of the Cremona group as groups of (biregular) automorphisms of
smooth projective rational surfaces and then to assume that the action is minimal
(i.e. that it is not possible to blow-down some curves and obtain once again a
biregular action on a smooth surface). Manin and Iskovskikh ([Man] and [Isk2])
proved that the only possible cases are action on del Pezzo surfaces or conic bun-
dles. We will clarify this classification, for finite Abelian groups fillfulling (F), by
proving the following result:
Theorem 4. Let S be some smooth projective rational surface and let G ⊂ Aut(S)
be a finite Abelian group of automorphisms of S such that
• the pair (G,S) is minimal;
• if g ∈ G, g 6= 1, then g does not fix a curve of positive genus.
Then, one of the following occurs:
1. The surface S is minimal, i.e. S ∼= P2, or S ∼= Fn for some integer n 6= 1.
2. The surface S is a del Pezzo surface of degree 5 and G ∼= Z/5Z.
3. The surface S is a del Pezzo surface of degree 6 and G ∼= Z/6Z.
4. The pair (G,S) is isomorphic to the pair (Cs24, Ŝ4) defined in Section 7.
We will then prove that all the pairs in cases 1, 2 and 3 are birationally equiv-
alent to a group of automorphisms of P1 × P1 or P2, and that this is not true
for case 4. In fact, we are able to provide the precise description of all conjugacy
classes of finite Abelian subgroups of Bir(P2) satisfying (F ):
Theorem 5. Let G be a finite Abelian subgroup of the Cremona group such that
no non-trivial element of G fixes a curve of positive genus. Then, G is birationally
conjugate to one and only one of the following:
[1] G ∼= Z/nZ× Z/mZ g.b. (x, y) 7→ (ζnx, y) and (x, y) 7→ (x, ζmy)
[2] G ∼= Z/2Z× Z/2nZ g.b. (x, y) 7→ (x−1, y) and (x, y) 7→ (−x, ζ2ny)
[3] G ∼= (Z/2Z)2 × Z/2nZ g.b. (x, y) 7→ (±x±1, y) and (x, y) 7→ (x, ζ2ny)
[4] G ∼= (Z/2Z)3 g.b. (x, y) 7→ (±x,±y) and (x, y) 7→ (x−1, y−1)
[5] G ∼= (Z/2Z)4 g.b. (x, y) 7→ (±x±1,±y±1)
[6] G ∼= Z/2Z× Z/4Z g.b. (x, y) 7→ (x−1, y−1) and (x, y) 7→ (−y, x)
[7] G ∼= (Z/2Z)3 g.b. (x, y) 7→ (−x,−y), (x, y) 7→ (x−1, y−1),
and (x, y) 7→ (y, x)
[8] G ∼= (Z/2Z)× (Z/4Z) g.b. (x : y : z) 99K (yz(y − z) : xz(y + z) : xy(y + z))
and (x : y : z) 99K (yz : xy : −xz)
[9] G ∼= (Z/3Z)2 g.b. (x : y : z) 7→ (x : ζ3y : (ζ3)
and (x : y : z) 7→ (y : z : x)
(where n,m are positive integers, n divides m and ζn = e
2iπ/n).
Furthermore, the groups in cases [1] through [7] are birationally conjugate to sub-
groups of Aut(P1×P1), but the others are not. The groups in cases [1] and [9] are
birationally conjugate to subgroups of Aut(P2), but the others are not.
To prove these results, we will need a number of geometric results on automor-
phisms of rational surfaces, and in particular on automorphisms of conic bundles
and del Pezzo surfaces (Sections 3 to 9). We give for example the classification of
all the twisting elements (that exchange the two components of a singular fibre)
acting on conic bundles in Proposition 6.5 (for the elements of finite order) and
Proposition 6.8 (for those of infinite order); these are the most important elements
in this context (see Lemma 3.8). We also prove that actions of (possibly infinite)
Abelian groups on del Pezzo surfaces satifying (F ) are minimal only if the degree
is at least 5 (Section 9) and describe these cases precisely (Sections 4, 5 and 9).
We also show that a finite Abelian group acting on a projective smooth surface
S such that (KS)
2 ≥ 5 is birationally conjugate to a group of automorphisms of
P1 × P1 or P2 (Corollary 9.10) and in particular satisfies (F ).
1.5 Comparison with other work
Many authors have considered the finite subgroups of Bir(P2). Among them,
S. Kantor [Kan] gave a classification of the finite subgroups, which was incomplete
and included some mistakes; A. Wiman [Wim] and then I.V. Dolgachev and V.A.
Iskovskikh [Do-Iz] successively improved Kantor’s results. The long paper [Do-Iz]
expounds the general theory of finite subgroups of Bir(P2) according to the modern
techniques of algebraic geometry, and will be for years to come the reference on
the subject. Our viewpoint and aim differ from those of [Do-Iz]: we are only
interested in Abelian groups in relation with the above conditions (F) and (M);
this gives a restricted setting in which the theoretical approach is simplified and the
results obtained are more accurate. In the study of del Pezzo surfaces, using the
classification [Do-Iz] of subgroups of automorphisms would require the examination
of many cases; for the sake of readibility we prefered a direct proof. The two
main theorems of [Do-Iz] on automorphism of conic bundles (Proposition 5.3 and
Theorem 5.7(2)) do not exclude groups satisfying property (F ) and do not give
explicit forms for the generators of the groups or the surfaces.
1.6 Aknowledgements
This article is part of my PhD thesis [Bla2]; I am grateful to my advisor T. Vust
for his invaluable help during these years, to I. Dolgachev for helpful discussions,
and thank J.-P. Serre and the referees for their useful remarks on this paper.
2 Automorphisms of P2 or P1 × P1
Note that a linear automorphism of C2 may be extended to an automorphism of
either P2 or P1 × P1. Moreover, the automorphisms of finite order of these three
surfaces are birationally conjugate. For finite Abelian groups, the situation is quite
different. We give here the birational equivalence of these groups.
Notation 2.1. The element [a : b : c] denotes the diagonal automorphism (x : y :
z) 7→ (ax : by : cz) of P2, and ζm = e
2iπ/m.
Proposition 2.2 (Finite Abelian subgroups of Aut(P2)). Every finite Abelian
subgroup of Aut(P2) = PGL(3,C) is conjugate, in the Cremona group Bir(P2), to
one and only one of the following:
1. A diagonal group, isomorphic to Z/nZ×Z/mZ, where n divides m, generated
by [1 : ζn : 1] and [ζm : 1 : 1]. (The case n = 1 gives the cyclic groups).
2. The special group V9, isomorphic to Z/3Z×Z/3Z, generated by [1 : ζ3 : (ζ3)
and (x : y : z) 7→ (y : z : x).
Thus, except for the group V9, two isomorphic finite Abelian subgroups of PGL(3,C)
are conjugate in Bir(P2).
Proof. First of all, a simple calculation shows that every finite Abelian subgroup
of PGL(3,C) is either diagonalisable or conjugate to the group V9. Furthermore,
since this last group does not fix any point, it is not diagonalisable, even in Bir(P2)
[Ko-Sz, Proposition A.2].
Let T denote the torus of PGL(3,C) constituted by diagonal automorphisms
of P2. Let G be a finite subgroup of T ; as an abstract group it is isomorphic to
Z/nZ× Z/mZ, where n divides m. Now we can conjugate G by a birational map
of the form h : (x, y) 99K (xayb, xcyd) so that it contains [ζm : 1 : 1] (see [Be-Bl]
and [Bla1]). Since h normalizes the torus T , the group G remains diagonal and
contains the n-torsion of T , hence it contains [1 : ζn : 1].
Corollary 2.3. Every finite Abelian group of linear automorphisms of C2 is bi-
rationally conjugate to a diagonal group, isomorphic to Z/nZ × Z/mZ, where n
divides m, generated by (x, y) 7→ (ζnx, y) and (x, y) 7→ (x, ζmy).
Proof. This follows from the fact that the group GL(2,C) of linear automorphisms
of C2 extends to a group of automorphisms of P2 that leaves the line at infinity
invariant and fixes one point.
Example 2.4. Note that Aut(P1 × P1) contains the group (C∗)2 ⋊ Z/2Z, where
(C∗)2 is the group of automorphisms of the form (x, y) 7→ (αx, βy), α, β ∈ C∗, and
Z/2Z is generated by the automorphism (x, y) 7→ (y, x).
The birational map (x, y) 99K (x : y : 1) from P1×P1 to P2 conjugates (C∗)2⋊
Z/2Z to the group of automorphisms of P2 generated by (x : y : z) 7→ (αx : βy : z),
α, β ∈ C∗ and (x : y : z) 7→ (y : x : z).
Proposition 2.5 (Finite Abelian subgroups of Aut(P1 × P1)). Up to birational
conjugation, every finite Abelian subgroup of Aut(P1×P1) is conjugate to one and
only one of the following:
[1] G ∼= Z/nZ× Z/mZ g.b. (x, y) 7→ (ζnx, y) and (x, y) 7→ (x, ζmy)
[2] G ∼= Z/2Z× Z/2nZ g.b. (x, y) 7→ (x−1, y) and (x, y) 7→ (−x, ζ2ny)
[3] G ∼= (Z/2Z)2 × Z/2nZ g.b. (x, y) 7→ (±x±1, y) and (x, y) 7→ (x, ζ2ny)
[4] G ∼= (Z/2Z)3 g.b. (x, y) 7→ (±x,±y) and (x, y) 7→ (x−1, y−1)
[5] G ∼= (Z/2Z)4 g.b. (x, y) 7→ (±x±1,±y±1)
[6] G ∼= Z/2Z× Z/4Z g.b. (x, y) 7→ (x−1, y−1) and (x, y) 7→ (−y, x)
[7] G ∼= (Z/2Z)3 g.b. (x, y) 7→ (−x,−y), (x, y) 7→ (x−1, y−1),
and (x, y) 7→ (y, x)
(where n,m are positive integers, n divides m and ζn = e
2iπ/n).
Furthermore, the groups in [1] are conjugate to subgroups of Aut(P2), but the others
are not.
Proof. Recall that Aut(P1 × P1) = (PGL(2,C) × PGL(2,C)) ⋊ Z/2Z. Let G be
some finite Abelian subgroup of Aut(P1 × P1); we now prove that G is conjugate
to one of the groups in cases [1] through [7].
First of all, if G is a subgroup of the group (C∗)2⋊Z/2Z given in Example 2.4,
then it is conjugate to a subgroup of Aut(P2) and hence to a group in case [1].
Assume that G ⊂ PGL(2,C)×PGL(2,C) and denote by π1 and π2 the projec-
tions πi : PGL(2,C)×PGL(2,C) → PGL(2,C) on the i-th factor. Since π1(G) and
π2(G) are finite Abelian subgroups of PGL(2,C) each is conjugate to a diagonal
cyclic group or to the group x 99K ±x±1, isomorphic to (Z/2Z)2. We enumerate
the possible cases.
If both groups π1(G) and π2(G) are cyclic, the group G is conjugate to a
subgroup of the diagonal torus (C∗)2 of automorphisms of the form (x, y) 7→
(αx, βy), α, β ∈ C∗.
If exactly one of the two groups π1(G) and π2(G) is cyclic we may assume, up
to conjugation in Aut(P1 × P1), that π2(G) is cyclic, generated by y 7→ ζmy, for
some integer m ≥ 1, and that π1(G) is the group x 99K ±x
±1. We use the exact
sequence 1 → G ∩ kerπ2 → G → π2(G) → 1 and find, up to conjugation, two
possibilities for G:
(a) G is generated by (x, y) 7→ (x−1, y) and (x, y) 7→ (−x, ζmy).
(b) G is generated by (x, y) 7→ (±x±1, y) and (x, y) 7→ (x, ζmy).
If m is even, we obtain respectively [2] and [3] for n = m/2. If m is odd, the two
groups are equal; conjugating by ϕ : (x, y) 99K (x, y(x + x−1)) (which conjugates
(x, y) 7→ (−x, y) to (x, y) 7→ (−x,−y)) we obtain the group [2] for n = m.
If both groups π1(G) and π2(G) are isomorphic to (Z/2Z)
2, then up to conju-
gation, we obtain three groups, namely
(a) G is generated by (x, y) 7→ (−x,−y) and (x, y) 7→ (x−1, y−1).
(b) G is generated by (x, y) 7→ (±x,±y) and (x, y) 7→ (x−1, y−1).
(c) G is given by (x, y) 7→ (±x±1,±y±1).
The group [2] with n = 1 is conjugate to (a) by (x, y) 99K (x, x y+x
y+x−1
). The groups
(b) and (c) are respectively equal to [4] and [5].
We now suppose that the group G is not contained in PGL(2,C)×PGL(2,C).
Any element ϕ ∈ Aut(P1×P1) not contained in PGL(2,C)×PGL(2,C) is conjugate
to ϕ : (x, y) 7→ (α(y), x), where α ∈ Aut(P1), and if ϕ is of finite order, α may be
chosen to be y 7→ λy with λ ∈ C∗ a root of unity.
Thus, up to conjugation, G is generated by the group H = G ∩ (PGL(2,C)×
PGL(2,C)) and one element (x, y) 7→ (λy, x), for some λ ∈ C∗ of finite order. Since
the group G is Abelian, every element of H is of the form (x, y) 7→ (β(x), β(y)),
for some β ∈ PGL(2,C) satisfying β(λx) = λβ(x). Three possibilities occur,
depending on the value of λ which may be 1, −1 or something else.
If λ = 1, we conjugate the group by some element (x, y) 7→ (γ(x), γ(y)) so that
H is either diagonal or equal to the group generated by (x, y) 7→ (−x,−y) and
(x, y) 7→ (x−1, y−1). In the first situation, the group is contained in (C∗)2 ⋊Z/2Z
(which gives [1]); the second situation gives [7].
If λ = −1, the group H contains the square of (x, y) 7→ (−y, x), which is
(x, y) 7→ (−x,−y) and is either cyclic or generated by (x, y) 7→ (−x,−y) and
(x, y) 7→ (x−1, y−1). If H is cyclic, it is diagonal, since it contains (x, y) 7→
(−x,−y), so G is contained in (C∗)2 ⋊ Z/2Z. The second possibility gives [6].
If λ 6= ±1, the group H is diagonal and then G is contained in (C∗)2 ⋊ Z/2Z.
We now prove that distinct groups of the list are not birationally conjugate.
First of all, each group of case [1] fixes at least one point of P1 × P1. Since the
other groups of the list don’t fix any point, they are not conjugate to [1] [Ko-Sz,
Proposition A.2].
Consider the other groups. The set of isomorphic groups are those of cases [3]
(with n = 1), [4] and [7] (isomorphic to (Z/2Z)3), and of cases [2] (with n = 2)
and [6] (isomorphic to Z/2Z× Z/4Z).
The groups of cases [2] to [5] leave two pencils of rational curves invariant (the
fibres of the two projections P1 × P1 → P1) which intersect freely in exactly one
point. We prove that this is not the case for [6] and [7]; this shows that these
two groups are not birationally conjugate to any of the previous groups. Take
G ⊂ Aut(P1×P1) to be either [6] or [7]. We have then Pic(P1×P1)G = Zd, where
d = − 1
KP1×P1 is the diagonal of P
1×P1. Suppose that there exist two G-invariant
pencils Λ1 = n1d and Λ2 = n2d of rational curves, for some positive integers n1, n2
(we identify here a pencil with the class of its elements in Pic(P1 × P1)G). The
intersection Λ1 · Λ2 = 2n1n2 is an even integer. Note that the fixed part of the
intersection is also even, since G is of order 8 and acts without fixed points on
P1 × P1. The free part of the intersection is then also an even integer and hence
is not 1.
Let us now prove that [4] is not birationally conjugate to [3] (with n = 1).
This follows from the fact that [4] contains three subgroups that are fixed-point
free (the groups generated by (x, y) 7→ (x−1, y−1) and one of the three involutions
of the group (x, y) 7→ (±x,±y)), whereas [3] (with n = 1) contains only one such
subgroup, which is (x, y) 7→ (±x±1, y).
We now prove the last assertion. The finite Abelian groups of automorphisms
of P2 are conjugate either to [1] or to the group V9, isomorphic to (Z/3Z)
2 (see
Proposition 2.2). As no group of the list [2] through [7] is isomorphic to (Z/3Z)2,
we are done.
Summary of this section. We have found that the groups common to the
three surfaces C2,P2 and P1 × P1 are the ”diagonal” ones (generated by (x, y) 7→
(ζnx, y) and (x, y) 7→ (x, ζmy)). On P
2 there is only one more group, which is the
special group V9, and on P
1 × P1 there are 2 families ([2] and [3]) and 4 special
groups ([4], [5], [6] and [7]).
3 Some facts about automorphisms of conic bun-
We first consider conic bundles without mentioning any group action on them. We
recall some classical definitions:
Definition 3.1. Let S be a rational surface and π : S → P1 be a morphism. We
say that the pair (S, π) is a conic bundle if a general fibre of π is isomorphic to P1,
with a finite number of exceptions: these singular fibres are the union of smooth
rational curves F1 and F2 such that (F1)
2 = (F2)
2 = −1 and F1 · F2 = 1.
Let (S, π) and (S̃, π̃) be two conic bundles. We say that ϕ : S 99K S̃ is a
birational map of conic bundles if ϕ is a birational map which sends a general fibre
of π on a general fibre of π̃.
We say that a conic bundle (S, π) is minimal if any birational morphism of
conic bundles (S, π) → (S̃, π̃) is an isomorphism.
We remind the reader of the following well-known result:
Lemma 3.2. Let (S, π) be a conic bundle. The following conditions are equivalent:
• (S, π) is minimal.
• The fibration π is smooth, i.e. no fibre of π is singular.
• S is a Hirzebruch surface Fm, for some integer m ≥ 0. �
Blowing-down one irreducible component in any singular fibre of a conic bundle
(S, π), we obtain a birational morphism of conic bundles S → Fm for some integer
m ≥ 0. Note that m depends on the choice of the blown-down components. The
following lemma gives some information on the possibilities. Note first that since
the sections of Fm have self-intersection≥ −m, the self-intersections of the sections
of π are also bounded from below.
Lemma 3.3. Let (S, π) be a conic bundle on a surface S 6∼= P1 × P1. Let −n be
the minimal self-intersection of sections of π and let r be the number of singular
fibres of π. Then n ≥ 1 and:
1. There exists a birational morphism of conic bundles p− : S → Fn such that:
(a) p− is the blow-up of r points of Fn, none of which lies on the exceptional
section En.
(b) The strict pull-back Ẽn of En by p− is a section of π with self-intersection
2. If there exist two different sections of π with self-intersection −n, then r ≥
2n. In this case, there exist birational morphisms of conic bundles p0 : S →
F0 = P
1 × P1 and p1 : S → F1.
Proof. We denote by s a section of π of minimal self-intersection −n, for some
integer n (this integer is in fact positive, as will appear in the proof). Note that
this curve intersects exactly one irreducible component of each singular fibre.
If r = 0, the lemma is trivially true: take p− to be the identity map. We now
suppose that r ≥ 1, and denote by F1, ..., Fr the irreducible components of the
singular fibres which do not intersect s. Blowing these down, we get a birational
morphism of conic bundles p− : S → Fm, for some integerm ≥ 0. The image of the
section s by p− is a section of the conic bundle of Fm of minimal self-intersection,
so we get m = n, and n ≥ 0. If we had n = 0, then taking some section s̃ of
P1×P1 of self-intersection 0 passing through at least one blown-up point, its strict
pull-back by p− would be a section of negative self-intersection, which contradicts
the minimality of s2 = −n = 0. We find finally that m = n > 0, and that p−(s)
is the unique section Fn of self-intersection −n. This proves the first assertion.
We now prove the second assertion. Suppose that some section t 6= s has self-
intersection −n. The Picard group of S is generated by s = p∗−(En), the divisor f
of a fibre of π and F1, ..., Fr. Write t as t = s+ bf −
i=1 aiFi, for some integers
b, a1, ..., ar, with a1, ..., ar ≥ 0. We have t
2 = −n and t · (t+KS) = −2 (adjunction
formula), where KS = p
−(KFn) +
i=1 Fi = −(n + 2)f − 2s +
i=1 Fi. These
relations give:
s2 = t2 = s2 −
i=1 a
i + 2b,
n− 2 = t ·KS = −(n+ 2) + 2n− 2b+
i=1 ai,
whence
i=1 ai =
i=1 a
i = 2b, so each ai is equal to 0 or 1 and consequently
2b ≤ r. Since s · t = b− n ≥ 0, we find that r ≥ 2n, as announced.
Finally, by contracting f − F1, f − F2, ..., f − Fn, Fn+1, Fn+2, ..., Fr, we obtain
a birational morphism p0 of conic bundles which sends s on a section of self-
intersection 0 and whose image is thus F0. Similarly, the morphism p1 : S → F1
is given by the contraction of f − F1, f − F2, ..., f − Fn−1, Fn, Fn+1, ..., Fr.
We now add some group actions on the conic bundles, and give natural defi-
nitions (note that we will restrict ourselves to finite or Abelian groups only when
this is needed and will then say so):
Definition 3.4. Let (S, π) be some conic bundle.
• We denote by Aut(S, π) ⊂ Aut(S) the group of automorphisms of the conic
bundle, i.e. automorphisms of S that send a general fibre of π on another
general fibre.
Let G ⊂ Aut(S, π) be some group of automorphisms of the conic bundle (S, π).
• We say that a birational map of conic bundles ϕ : S 99K S̃ is G-equivariant
if the G-action on S̃ induced by ϕ is biregular (it is clear that it preserves
the conic bundle structure).
• We say that the triple (G,S, π) is minimal if any G-equivariant birational
morphism of conic bundles ϕ : S → S̃ is an isomorphism.
Remark 3.5. We insist on the fact that since a conic bundle is for us a pair (S, π),
an automorphism of S is not necessarily an automorphism of the conic bundle (i.e.
Aut(S) 6= Aut(S, π) in general). One should be aware that in the literature, conic
bundle sometimes means ”a variety admitting a conic bundle structure”.
Remark 3.6. If G ⊂ Aut(S, π) is such that the pair (G,S) is minimal, so is the
triple (G,S, π). The converse is not true in general (see Remark 4.7).
Note that any automorphism of the conic bundle acts on the set of singular
fibres and on its irreducible components. The permutation of the two components
of a singular fibre is very important (Lemma 3.8). For this reason, we introduce
some terminology:
Definition 3.7. Let g ∈ Aut(S, π) be an automorphism of the conic bundle (S, π).
Let F = {F1, F2} be a singular fibre. We say that g twists the singular fibre F if
g(F1) = F2 (and consequently g(F2) = F1).
If g twists at least one singular fibre of π, we will say that g twists the conic
bundle (S, π), or simply (if the conic bundle is implicit) that g is a twisting element.
Here is a simple but very important observation:
Lemma 3.8. Let G ⊂ Aut(S, π) be a group of automorphisms of a conic bundle.
The following conditions are equivalent:
1. The triple (G,S, π) is minimal.
2. Any singular fibre of π is twisted by some element of G. �
Remark 3.9. An automorphism of a conic bundle with a non-trivial action on
the basis of the fibration may twist at most two singular fibres. However, an
automorphism with a trivial action on the basis of the fibration may twist a large
number of fibres. We will give in Propositions 6.5 and 6.8 a precise description of
all twisting elements.
The following lemma is a direct consequence of Lemma 3.3; it provides infor-
mation on the structure of the underlying variety of a conic bundle admitting a
twisting automorphism.
Lemma 3.10. Suppose that some automorphism of the conic bundle (S, π) twists
at least one singular fibre. Then, the following occur.
1. There exist two birational morphisms of conic bundles p0 : S → F0 and
p1 : S → F1 (which are not g-equivariant).
2. Let −n be the minimal self-intersection of sections of π and let r be the
number of singular fibres of π. Then, r ≥ 2n ≥ 2.
Proof. Note that any section of π touches exactly one component of each singular
fibre. Since g twists some singular fibre, its action on the set of sections of S is
fixed-point-free. The number of sections of minimal self-intersection is then greater
than 1 and we apply Lemma 3.3 to get the result.
Remark 3.11. A result of the same kind can be found in [Isk1], Theorem 1.1.
Lemma 3.12. Let G ⊂ Aut(S, π) be a group of automorphisms of the conic bundle
(S, π), such that:
• π has at most 3 singular fibres (or equivalently (KS)
2 ≥ 5);
• the triple (G,S, π) is minimal.
Then, S is either a Hirzeburch surface or a del Pezzo surface of degree 5 or 6,
depending on whether the number of singular fibres is 0, 3 or 2 respectively.
Proof. Let −n be the minimal self-intersection of sections of π and let r ≤ 3 be
the number of singular fibres of π. If r = 0, we are done, so we may suppose that
r > 0. Since (G,S, π) is minimal, every singular fibre is twisted by some element
of G (Lemma 3.8). From Lemma 3.10, we get r ≥ 2n ≥ 2, whence r = 2 or 3 and
n = 1, and we obtain the existence of some birational morphism of conic bundles
(not G-equivariant) p1 : S → F1. So the surface S is obtained by the blow-up
of 2 or 3 points of F1, not on the exceptional section (Lemma 3.3), and thus by
blowing-up 3 or 4 points of P2, no 3 of which are collinear (otherwise we would
have a section of self-intersection ≤ −2). The surface is then a del Pezzo surface
of degree 6 or 5.
Remark 3.13. We conclude this section by mentioning an important exact se-
quence. Let G ⊂ Aut(S, π) be some group of automorphisms of a conic bundle
(S, π). We have a natural homomorphism π : G → Aut(P1) = PGL(2,C) that
satisfies π(g)π = πg, for every g ∈ G. We observe that the group G′ = kerπ of
automorphisms that leave every fibre invariant embeds in the group PGL(2,C(x))
of automorphisms of the generic fibre P1(C(x)). Then we get the exact sequence
1 → G′ → G
→ π(G) → 1. (1)
This restricts the structure of G; for example if G is Abelian and finite, so are
G′ and π(G), and we know that the finite Abelian subgroups of PGL(2,C) and
PGL(2,C(x)) are either cyclic or isomorphic to (Z/2Z)2.
We also see that the group G is birationally conjugate to a subgroup of the
group of birational transformations of P1 × P1 of the form (written in affine coor-
dinates):
(x, y) 99K
ax+ b
cx+ d
α(x)y + β(x)
γ(x)y + δ(x)
where a, b, c, d ∈ C, α, β, γ, δ ∈ C(x), and (ad− bc)(αδ − βγ) 6= 0.
This group, called the de Jonquières group, is the group of birational transfor-
mations of P1 ×P1 that preserve the fibration induced by the first projection, and
is isomorphic to PGL(2,C(x))⋊ PGL(2,C).
The subgroups of this group can be studied algebraically (as in [Bea2] and
[Bla4]) but we will not adopt this point of view here.
4 The del Pezzo surface of degree 6
There is a single isomorphism class of del Pezzo surfaces of degree 6, since all
sets of three non-collinear points of P2 are equivalent under the action of linear
automorphisms. Consider the surface S6 of degree 6 defined by the blow-up of
the points A1 = (1 : 0 : 0), A2 = (0 : 1 : 0) and A3 = (0 : 0 : 1). We may view
it in P2 × P2, defined as {
(x : y : z), (u : v : w)
| ux = vy = wz}, where the
blow-down p : S6 → P
2 is the restriction of the projection on one copy of P2,
explicitly p :
(x : y : z), (u : v : w)
7→ (x : y : z). There are exactly 6 exceptional
divisors, which are the pull-backs of the Ai’s by the two projection morphisms.
We write Ei = p
−1(Ai) and denote by Dij the strict pull-back by p of the line of
P2 passing through Ai and Aj .
The group of automorphisms of S6 is well known (see for example [Wim],
[Do-Iz]). It is isomorphic to (C∗)2 ⋊ (Sym3 × Z/2Z), where (C
∗)2 ⋊ Sym3 is the
lift on S6 of the group of automorphisms of P
2 that leave the set {A1, A2, A3}
invariant, and Z/2Z is generated by the permutation of the two factors (it is the
lift of the standard quadratic transformation (x : y : z) 99K (yz : xz : xy) of P2);
the action of Z/2Z on (C∗)2 sends an element on its inverse.
There are three conic bundle structures on the surface S6. Let π1 : S6 → P
be the morphism defined by
(x : y : z), (u : v : w)
(y : z) if (x : y : z) 6= (1 : 0 : 0),
(w : v) if (u : v : w) 6= (1 : 0 : 0).
Note that p sends the fibres of π1 on lines of P
2 passing through A1. There are
exactly two singular fibres of this fibration, namely
π−11 (1 : 0) = {E2, D12} and π
1 (0 : 1) = {E3, D13};
and E1, D23 are sections of π1.
Lemma 4.1. The group Aut(S6, π1) of automorphisms of the conic bundle (S6, π1)
acts on the hexagon {E1, E2, E3, D12, D13, D23} and leaves the set {E1, D23} in-
variant.
1. The action on the hexagon gives rise to the exact sequence
1 → (C∗)2 → Aut(S6, π1) → (Z/2Z)
2 → 1.
2. This exact sequence is split and Aut(S6, π1) = (C
∗)2 ⋊ (Z/2Z)2, where
(a) (C∗)2 is the group of automorphisms of the form(
(x : y : z), (u : v : w)
(x : αy : βz), (αβu : βv : αw)
, α, β ∈ C∗.
(b) The group (Z/2Z)2 is generated by the automorphisms
(x : y : z), (u : v : w)
(x : z : y), (u : w : v)
whose action on the set of exceptional divisors is (E2 E3)(D12 D13);
(x : y : z), (u : v : w)
(u : v : w), (x : y : z)
whose action is (E1 D23)(E2 D13)(E3 D12).
(c) The action of (Z/2Z)2 on (C∗)2 is generated by permutation of the
coordinates and inversion.
Proof. Since Aut(S6) acts on the hexagon, so does Aut(S6, π1) ⊂ Aut(S6). Since
the group Aut(S6, π1) sends a section on a section, the set {E1, D23} is invariant.
The group (C∗)2 leaves the conic bundle invariant, and is the kernel of the
action of Aut(S6, π1) on the hexagon. As the set {E1, D23} is invariant, the
image is contained in the group (Z/2Z)2 generated by (E2 E3)(D12 D13) and
(E1 D23)(E2 D13)(E3 D12). The rest of the lemma follows directly.
By permuting coordinates, we have two other conic bundle structures on the
surface S6, given by the following morphisms π2, π3 : S6 → P
(x : y : z), (u : v : w)
(x : z) if (x : y : z) 6= (0 : 1 : 0),
(w : u) if (u : v : w) 6= (0 : 1 : 0).
(x : y : z), (u : v : w)
(x : y) if (x : y : z) 6= (0 : 0 : 1),
(v : u) if (u : v : w) 6= (0 : 0 : 1).
The description of the exceptional divisors on S6 shows that π1, π2 and π3 are
the only conic bundle structures on S6.
Lemma 4.2. For i = 1, 2, 3, the pair (Aut(S6, πi), S6) is not minimal. More
precisely the morphism πj×πk : S6 → P
1×P1 conjugates Aut(S6, πi) to a subgroup
of Aut(P1 × P1), where {i, j, k} = {1, 2, 3}.
Proof. The union of the sections E1 and D23 is invariant by the action of the
whole group Aut(S6, π1). Since these two exceptional divisors don’t intersect, we
can contract both and get a birational Aut(S6, π1)-equivariant morphism from S6
to P1×P1: the pair (Aut(S6, π1), S6) is thus not minimal; explicitly, the birational
morphism is given by q 7→ (π2(q), π3(q)), as stated in the lemma. We obtain the
other cases by permuting coordinates.
Remark 4.3. The subgroup of Aut(P1×P1) obtained in this manner doesn’t leave
any of the two fibrations of P1 × P1 invariant.
Corollary 4.4. If (G,S6) is a minimal pair (where G ⊂ Aut(S6)), then G does
not preserve any conic bundle structure. �
We conclude this section with a fundamental example; we will use several times
the following automorphism κα,β of (S6, π1):
Example 4.5. For any α, β ∈ C∗, we define κα,β to be the following automorphism
of (S6, π1):
κα,β :
(x : y : z), (u : v : w)
(u : αw : βv), (x : α−1z : β−1y)
Note that κα,β twists the two singular fibres of π1 (see Lemma 4.6 below); its
action on the basis of the fibration is (x1 : x2) 7→ (αx1 : βx2) and
κ2α,β(
(x : y : z), (u : v : w)
(x : αβ−1y : α−1βz), (u : α−1βv : αβ−1w)
So κα,β is an involution if and only if its action on the basis of the fibration is
trivial.
Lemma 4.6. Let g ∈ Aut(S6, π1) be an automorphism of the conic bundle (S6, π1).
The following conditions are equivalent:
• the triple (< g >, S6, π1) is minimal;
• g twists the two singular fibres of π1;
• the action of g on the exceptional divisors of S6 is (E1 D23)(E2 D12)(E3 D13);
• g = κα,β for some α, β ∈ C
Proof. According to Lemma 4.1 the action of Aut(S6, π1) on the exceptional curves
is isomorphic to (Z/2Z)2 and hence the possible actions of g 6= 1 are these:
1. id, 2. (E2 E3)(D12 D13),
3. (E1 D23)(E2 D13)(E3 D12), 4. (E1 D23)(E2 D12)(E3 D13).
In the first three cases, the triple (< g >, S6, π1) is not minimal. Indeed, the
blow-down of {E2, E3} or {E2, D13} gives a g-equivariant birational morphism of
conic bundles.
Hence, if (< g >, S6, π1) is minimal, its action on the exceptional curves is the
fourth one above, as stated in the lemma, and it then twists the two singular fibres
of π1. Conversely if g twists the two singular fibres of π1, the triple (< g >, S6, π1)
is minimal (by Lemma 3.8).
It remains to see that the last assertion is equivalent to the others. This follows
from Lemma 4.1; indeed this lemma implies that (C∗)2κ1,1 is the set of elements
of Aut(S6, π1) inducing the permutation (E1 D23)(E2 D12)(E3 D13).
Remark 4.7. The pair (Aut(S6, π1), S6) is not minimal (Lemma 4.2). Consequently
< κα,β > is an example of a group whose action on the surface is not minimal,
but whose action on a conic bundle is minimal.
5 The del Pezzo surface of degree 5
As for the del Pezzo surface of degree 6, there is a single isomorphism class of del
Pezzo surfaces of degree 5. Consider the del Pezzo surface S5 of degree 5 defined
by the blow-up p : S5 → P
2 of the points A1 = (1 : 0 : 0), A2 = (0 : 1 : 0),
A3 = (0 : 0 : 1) and A4 = (1 : 1 : 1). There are 10 exceptional divisors on S5,
namely the divisor Ei = p
−1(Ai), for i = 1, ..., 4, and the strict pull-back Dij of
the line of P2 passing through Ai and Aj , for 1 ≤ i < j ≤ 4. There are 5 sets of 4
skew exceptional divisors on S5, namely
F1 = {E1, D23, D24, D34}, F2 = {E2, D13, D14, D34}, F3 = {E3, D12, D14, D24},
F4 = {E4, D12, D13, D23}, F5 = {E1, E2, E3, E4}.
Proposition 5.1. The action of Aut(S5) on the five sets F1, ..., F5 of four skew
exceptional divisors of S5 gives rise to an isomomorphism
ρ : Aut(S5) → Sym5.
Furthermore, the actions of Symn, Altm ⊂ Aut(S5) on S5 given by the canonical
embedding of these groups into Sym5 are fixed-point free if and only if n = 3, 4, 5,
respectively m = 4, 5.
Proof. Since any automorphism in the kernel of ρ leaves E1, E2, E3 and E4 invari-
ant and hence is the lift of an automorphism of P2 that fixes the 4 points, the
homomorphism ρ is injective.
We now prove that ρ is also surjective. Firstly, the lift of the group of au-
tomorphisms of P2 that leave the set {A1, A2, A3, A4} invariant is sent by ρ on
Sym4 = Sym{F1,F2,F3,F4}. Secondly, the lift of the standard quadratic transforma-
tion (x : y : z) 99K (yz : xz : xy) is an automorphism of S5, as its lift on S6 is an
automorphism, and as it fixes the point A4; its image by ρ is (F4 F5).
It remains to prove the last assertion. First of all, it is clear that the actions
of the cyclic groups Alt3 and Sym2 fix some points. The group Sym3 ⊂ Aut(P
of permutations of A1, A2 and A3 fixes exactly one point, namely (1 : 1 : 1). The
blow-up of this point gives a fixed-point free action on F1, and thus its lift on S5
is also fixed-point free. The group Alt4 ⊂ Aut(P
2) contains the element (x : y :
z) 7→ (z : x : y) (which corresponds to (1 2 3)) that fixes exactly three points, i.e.
(1 : a : a2) for a3 = 1. It also contains the element (x : y : z) 7→ (z − y : z − x : z)
(which corresponds to (1 2)(3 4)) that does not fix (1 : a : a2) for a3 = 1. Thus,
the action of Alt4 on P
2 is fixed-point free and the same is true on S5.
Remark 5.2. The structure of Aut(S5) is classical and can be found for example
in [Wim] and [Do-Iz].
Lemma 5.3. Let π : S5 → P
1 be some morphism inducing a conic bundle (S5, π).
There are exactly four exceptional curves of S5 which are sections of π; the blow-
down of these curves gives rise to a birational morphism p : S5 → P
2 which
conjugates the group Aut(S5, π) ∼= Sym4 to the subgroup of Aut(P
2) that leaves
invariant the four points blown-up by p. In particular, the pair (Aut(S5, π), S5) is
not minimal.
Proof. Blowing-down one component in any singular fibre, we obtain a birational
morphism of conic bundles (not Aut(S5, π)-equivariant) from S5 to some Hirze-
bruch surface Fn. Since S5 does not contain any curves of self-intersection ≤ −2, n
is equal to 0 or 1. Changing the component blown-down in a singular fibre performs
an elementary link Fn 99K Fn±1; we may then assume that n = 1, and that F1 is
the blow-up of A1 ∈ P
2. Consequently, the fibres of the conic bundles correspond
to the lines passing through A1. Denoting by A2, A3, A4 the other points blown-up
by the constructed birational morphism S5 → P
2 and using the same notation as
before, the three singular fibres are {Ei, D1i} for i = 2, ..., 4, and the other excep-
tional curves are four skew sections of the conic bundle, namely the elements of
F1 = {E1, D23, D24, D34}. The blow-down of F1 gives an Aut(S5, π)-equivariant
birational morphism (that is not a morphism of conic bundles) p : S5 → P
2 and
conjugates Aut(S5, π) to a subgroup of the group Sym4 ⊂ Aut(P
2) of automor-
phisms that leaves the four points blown-up by p invariant. The fibres of π are
sent on the conics passing through the four points, so the lift of the whole group
Sym4 belongs to Aut(S5, π).
Corollary 5.4. Let G be some group of automorphisms of a conic bundle (S, π)
such that the pair (G,S) is minimal and (KS)
2 ≥ 5 (or equivalently such that the
number of singular fibres of π is at most 3). Then, the fibration is smooth, i.e. S
is a Hirzebruch surface.
Proof. Since (G,S) is minimal, so is the triple (G,S, π). By Lemma 3.12, the
surface S is either a Hirzebruch surface, or a del Pezzo surface of degree 5 or 6.
Corollary 4.4 shows that the del Pezzo surface of degree 6 is not possible and
Lemma 5.3 eliminates the possibility of the del Pezzo surface of degree 5.
6 Description of twisting elements
In this section, we describe the twisting automorphisms of conic bundles, which
are the most important automorphisms (see Lemma 3.8).
Lemma 6.1 (Involutions twisting a conic bundle). Let g ∈ Aut(S, π) be a twist-
ing automorphism of the conic bundle (S, π). Then, the following properties are
equivalent:
1. g is an involution;
2. π(g) = 1, i.e. g has a trivial action on the basis of the fibration;
3. the set of points of S fixed by g is an irreducible hyperelliptic curve of genus
(k − 1) – a double covering of P1 by means of π, ramified over 2k points –
plus perhaps a finite number of isolated points, which are the singular points
of the singular fibres not twisted by g.
Furthermore, if the three conditions above are satisfied, the number of singular
fibres of π twisted by g is 2k ≥ 2.
Proof. 1 ⇒ 2: By contracting some exceptional curves, we may assume that the
triple (< g >, S, π) is minimal. Suppose that g is an involution and π(g) 6= 1.
Then g may twist only two singular fibres, which are the fibres of the two points
of P1 fixed by π(g). Hence, the number of singular fibres is ≤ 2. Lemma 3.12
tells us that S is a del Pezzo surface of degree 6 and then Lemma 4.6 shows that
g = κα,β (Example 4.5) for some α, β ∈ C
∗. But such an element is an involution
if and only if it acts trivially on the basis of the fibration.
(1 and 2) ⇒ 3: Suppose first that (< g >, S, π) is minimal. This implies that g
twists every singular fibre of π. Therefore, since π(g) = 1 and g2 = 1, on a singular
fibre there is one point fixed by g (the singular point of the fibre) and on a general
fibre there are two fixed points. The set of points of S fixed by g is thus a smooth
irreducible curve. The projection π gives it as a double covering of P1 ramified over
the points whose fibres are singular and twisted by g. By the Riemann-Hurwitz
formula, this number is even, equal to 2k and the genus of the curve is k − 1.
The situation when (< g >, S, π) is not minimal is obtained from this one, by
blowing-up some fixed points. This adds in each new singular fibre (not twisted
by the involution) an isolated point, which is the singular point of the singular
fibre. We then get the third assertion and the final remark.
3 ⇒ 2: This implication is clear.
2 ⇒ 1: If π(g) = 1, then, g2 leaves every component of every singular fibre of
π invariant. Let p1 : S → F1 be the birational morphism of conic bundles given by
Lemma 3.10; it is a g2-equivariant birational morphism which conjugates g2 to an
automorphism of F1 that necessarily fixes the exceptional section. The pull-back
by p1 of this section is a section C of π, fixed by g
2. Since C touches exactly
one component of each singular fibre (in particular those that are twisted by g),
g sends C on another section D also fixed by g2. The union of the sections D and
C intersects a general fibre in two points, which are exchanged by the action of g.
This implies that g has order 2.
We now give some further simple results on twisting involutions.
Corollary 6.2. Let (S, π) be some conic bundle. No involution twisting (S, π) has
a root in Aut(S, π) which acts trivially on the basis of the fibration.
Proof. Such a root must twist a singular fibre and so (Lemma 6.1) is an involution.
Remark 6.3. There may exist some roots in Aut(S, π) of twisting involutions which
act non trivially on the basis of the fibration.
Take for example four general points A1, ..., A4 of the plane and denote by g ∈
Aut(P2) the element of order 4 that permutes these points cyclically. The blow-up
of these points conjugates g to an automorphism of the del Pezzo surface S5 of
degree 5 (see Section 5). The pencil of conics of P2 passing through the four points
induces a conic bundle structure on S5, with three singular fibres which are the
lift of the pairs of two lines passing through the points. The lift on S5 of g is an
automorphism of the conic bundle whose square is a twisting involution.
Corollary 6.4. Let (S, π) be some conic bundle and let g ∈ Aut(S, π). The
following conditions are equivalent.
1. g twists more than 2 singular fibres of π.
2. g fixes a curve of positive genus.
And these conditions imply that g is an involution which acts trivially on the basis
of the fibration and twists at least 4 singular fibres.
Proof. The first condition implies that g acts trivially on the basis of the fibration,
and thus (by Lemma 6.1) that g is an involution which fixes a curve of positive
genus.
Suppose that g fixes a curve of positive genus. Then, g acts trivially on the
basis of the fibration, and fixes 2 points on a general fibre. Consequently, the curve
fixed by g is a smooth hyperelliptic curve; we get the remaining assertions from
Lemma 6.1.
As we mentioned above, the automorphisms that twist some singular fibre are
fundamental (Lemma 3.8). We now describe these elements and prove that the
only possibilities are twisting involutions, roots of twisting involutions (of even or
odd order) and elements of the form κα,β (see Example 4.5):
Proposition 6.5 (Classification of twisting elements of finite order). Let g ∈
Aut(S, π) be a twisting automorphism of finite order of a conic bundle (S, π).
Let n be the order of its action on the basis.
Then gn is an involution that acts trivially on the basis of the fibration and
twists an even number 2k of singular fibres; furthermore, exactly one of the fol-
lowing situations occurs:
1. n = 1.
2. n > 1 and k = 0; in this case n is even and there exists a g-equivariant bi-
rational morphism of conic bundles η : S → S6 (where S6 is the del Pezzo
surface of degree 6) such that ηgη−1 = κα,β for some α, β ∈ C
∗ (see Exam-
ple 4.5).
3. n > 1 is odd and k > 0; here g twists 1 or 2 fibres, which are the fibres twisted
by gn that are invariant by g.
4. n is even and k > 0; here g twists r = 1 or 2 singular fibres; none of them
are twisted by gn; moreover the action of g on the set of 2k fibres twisted by
gn is fixed-point free; furthermore, n divides 2k, and 2k/n ≡ r (mod 2).
Proof. Lemma 6.1 describes the situation when n = 1. We now assume that n > 1;
by blowing-down some components of singular fibres we may also suppose that the
triple (G,S, π) is minimal.
Denote by a1, a2 ∈ P
1 the two points fixed by π(g) ∈ Aut(P1). For i 6≡ 0
(mod n) the element π(gi) fixes only two points of P1, namely a1 and a2 (since
π(g) has order n); the only possible fibres twisted by gi are thus π−1(a1), π
−1(a2).
Suppose that gn does not twist any singular fibre. By minimality there are
at most 2 singular fibres (π−1(a1) and/or π
−1(a2)) of π and g twists each one.
Lemma 3.12 tells us that S is a del Pezzo surface of degree 6 and Lemma 4.6
shows that
g = κα,β :
(x : y : z), (u : v : w)
(u : αw : βv) , (x : α−1z : β−1y)
for some α, β ∈ C∗. We compute the square of g and find
(x : y : z), (u : v : w)
(x : αβ−1y : α−1βz) , (u : α−1βv : αβ−1w)
Consequently, the order of g is 2n. The fact that gi twists π−1(a1) and π
−1(a2)
when i is odd implies that n is even. Case 2 is complete.
If gn twists at least one singular fibre, it twists an even number of singular
fibres (Lemma 6.1) which we denote by 2k, and gn is an involution. If n is odd,
each fibre twisted by gn is twisted by g, and conversely; this yields case 3. It
remains to consider the more difficult case when n is even.
Firstly we observe that there are r + 2k singular fibres with r ∈ {1, 2}, cor-
responding to the points a1 and/or a2, c1, ..., c2k of P
1, the first r of them be-
ing twisted by g and the 2k others by gn. Under the permutation π(g), the
set {c1, ..., c2k} decomposes into disjoint cycles of length n (this action is fixed-
point-free); this shows that n divides 2k. We write t = 2k/n ∈ N and set
{c1, ..., c2k} = ∪
i=1Ci, where each Ci ⊂ P
1 is an orbit of π(g) of size n. To
deduce the congruence r ≡ t (mod 2), we study the action of g on Pic(S).
For i ∈ {1, ..., t}, choose Fi to be a component in the fibre of the singular fibre
of some point of Ci, and for i ∈ {1, r} choose Li to be a component in the fibre of
ai. Let us write
i=1(Fi + g(Fi) + ...+ g
n−1(Fi)) +
i=1 Li ∈ Pic(S).
Denoting by f ⊂ S a general fibre of π, we find the equalities g(Li) = f − Li
and gn(Fi) = f − Fi in Pic(S), which yield (once again in Pic(S)):
g(R) = R+ (r + t)f − 2(
i=1 Li +
i=1 Fi).
The contraction of the divisor R gives rise to a birational morphism of conic
bundles (not g-equivariant) ν : S → Fm for some integer m ≥ 0. Denote by s ⊂ S
the pull-back by ν of a general section of Fm of self-intersection m (which does
not pass through any of the base-points of ν−1). The canonical divisor KS of S is
then equal in Pic(S) to the divisor −2s+ (m − 2)f + R. We compute g(2s) and
2(g(s)− s) = g(2s)− 2s in Pic(S):
g(2s) = g(−KS + (m− 2)f +R) = −KS + (m− 2)f + g(R);
g(2s)− 2s = g(R)−R = (r + t)f − 2(
i=1 Li +
i=1 Fi).
This shows that (r + t)f ∈ 2Pic(S), which implies that r ≡ t (mod 2). Case 4 is
complete.
Corollary 6.6. If g ∈ Aut(S, π) is a root of a twisting involution h that fixes a
rational curve (i.e. that twists 2 singular fibres) and if g twists at least one fibre
not twisted by h, then g2 = h, g twists exactly one singular fibre, and it exchanges
the two fibres twisted by h.
Proof. We apply Proposition 6.5 and obtain case 4 with k = 1.
Corollary 6.6 and the following result will be useful in the sequel.
Lemma 6.7. Let g ∈ Aut(S, π) be a non-trivial automorphism of finite order
that leaves every component of every singular fibre of π invariant (i.e. that acts
trivially on Pic(S)) and let h ∈ Aut(S, π) be an element that commutes with g.
Then, either no singular fibre of π is twisted by h or each singular fibre of π which
is invariant by h is twisted by h.
Proof. If no twisting element belongs to Aut(S, π), we are done. Otherwise, the
birational morphism of conic bundles p0 : S → P
1 × P1 given by Lemma 3.10
conjugates g to an element of finite order of Aut(P1 × P1, π1) whose set of fixed
points is the union of two rational curves. The set of points of S fixed by g is thus
the union of two sections and a finite number of points (which are the singular
points of the singular fibres of π). Any element h ∈ Aut(S, π) that commutes
with g leaves the set of these two sections invariant. More precisely, the action on
one invariant singular fibre F implies the action on the two sections: h exchanges
the two sections if and only if it twists F . Since the situation is the same at any
other singular fibre, we obtain the result.
We conclude this section with some results on automorphisms of infinite order
of conic bundles, which will not help us directly here but seem interesting to
observe.
Proposition 6.8 (Classification of twisting elements of infinite order). Let (S, π)
be a conic bundle and g ∈ Aut(S, π) be a twisting automorphism of infinite order.
Then g twists exactly two fibres of π and there exists some g-equivariant bira-
tional morphism of conic bundles η : S → S6, where S6 is the del Pezzo surface of
degree 6 and ηgη−1 = κα,β for some α, β ∈ C
Proof. Assume that the triple (< g >, S, π) is minimal. Lemma 6.1 shows that
no twisting element of infinite order acts trivially on the basis of the fibration.
Consequently, gk acts trivially on the basis if and only if k = 0, whence gk twists a
fibre F if and only if k is odd and g twists F . There thus exist at most 2 singular
fibres of π, and Lemma 3.12 tells us that S is a del Pezzo surface of degree 6.
Lemma 4.6 shows that g = κα,β for some α, β ∈ C
Corollary 6.9. Let g ∈ Aut(S, π) be an element of infinite order; then a birational
morphism conjugates g to an automorphism of a Hirzebruch surface.
Proof. Assume that the triple (< g >, S, π) is minimal. If the fibration is smooth,
we are done. Otherwise, a birational morphism conjugates g to an automorphism
κα,β ∈ Aut(S6) of a conic bundle on the del Pezzo surface of degree 6 (Lemma 6.8).
We conclude by using Lemma 4.2.
7 The example Cs24
We now give the most important example of this paper. This is the only finite
Abelian subgroup of the Cremona group which is not conjugate to a group of
automorphisms of P2 or P1 × P1 but whose non-trivial elements do not fix any
curve of positive genus (Theorem 2).
Let S6 ⊂ P
2 × P2 be the del Pezzo surface of degree 6 (see Section 4) defined
S6 = {
(x : y : z), (u : v : w)
| ux = yv = zw};
we keep the notation of Section 4. We denote by η : Ŝ4 → S6 the blow-up of
A4, A5 ∈ S6 defined by
(0 : 1 : 1) , (1 : 0 : 0)
∈ D23,
(1 : 0 : 0) , (0 : 1 : −1)
∈ E1.
We again denote by E1, E2, E3, D12, D13, D23 the total pull-backs by η of these
divisors of S6. We denote by Ẽ1 and D̃23 the strict pull-backs of E1 and D23 by
η. (Note that for the other exceptional divisors, the strict and total pull-backs are
the same.) Let us illustrate the situations on the surfaces S6 and Ŝ4 respectively:
E2 D15 E4 E3
D12 E5 D14 D13
Let π1 denote the morphism S6 → P
1 defined in Section 4. The morphism
π = π1 ◦ η gives the surface Ŝ4 a conic bundle structure (Ŝ4, π). It has 4 singular
fibres, which are the fibres of (−1 : 1), (0 : 1), (1 : 1) and (1 : 0). We denote by f
the divisor of Ŝ4 corresponding to a fibre of π and set E4 = η
−1(A4), E5 = η
−1(A5).
Note that E4 is one of the components of the singular fibre of (1 : 1); we denote by
D14 = f−E4 the other component, which is the strict pull-back by η of π
1 (1 : 1).
Similarly, we denote by D15 the divisor f−E5, so that the singular fibre of (−1 : 1)
is {E5, D15}.
Lemma 7.1. On the surface Ŝ4 there are exactly 10 irreducible rational smooth
curves of negative self-intersection. Explicitly, the 8 curves
E2, E3, E4, E5, D12, D13, D14, D15
have self-intersection −1 and the two curves
Ẽ1 = E1 − E5 and D̃23 = D23 − E4
have self-intersection −2.
Proof. The difficult part is to show that every rational irreducible smooth curve
of negative self-intersection is one of the ten given above. Let C be such a curve.
Denote by L the pull-back of a general line of P2 by the blow-up pr1 ◦ η : Ŝ4 →
P2 of the five points. If C is collapsed by pr1 ◦ η, then C is one of the curves
Ẽ1, E2, E3, E4, E5. Otherwise, C = mL −
i=1 aiEi, where m, a1, ..., a5 are non-
negative integers, and m > 0. Since C is rational we have C · (C + K
) = −2,
and by hypothesis C2 = −r for some positive integer r. The relations C2 = −r
and C ·K
= r − 2 imply (since K
= −3L+
i=1 Ei) the equations
i=1 a
i = m
2 + r,
i=1 ai = 3m+ r − 2.
If m = r = 1, we find that C is the pull-back of a line passing through two of the
points, so C = D1i for some i ∈ {2, ..., 5}. If m = 2 and r = 1, C is the pull-back
of a conic passing through each blown-up point. The configuration of the points
eliminates this possibility. If m = 1 and r = 2, we obtain a line passing through
three blown-up points, so C = D̃23.
We now prove that if there is no integral solution to (2) for m, r ≥ 2. Since
i=1 ai)
2 ≤ 5(
i=1 a
i ) (by the Cauchy-Schwarz inequality with the vectors
(1, ..., 1) and (a1, ..., a5)), we obtain (3m+ (r − 2))
2 ≤ 5(m2 + r), and this gives
4m2 − 10 + (r − 2) · (6m+ r − 7) ≤ 0.
But this is not possible if m, r ≥ 2, since in this case 4m2 > 10 and 6m+r > 7.
Note that (K
)2 = 4, which is why we denote this surface by Ŝ4; the hat is
here because the surface is not a del Pezzo surface, since it contains irreducible
divisors of self-intersection −2.
Corollary 7.2. There is only one conic bundle structure on Ŝ4, which is the one
induced by π = π1 ◦ η.
Proof. Since (K
)2 = 4, the number of singular fibres of any conic bundle is 4,
and thus it consists of eight (−1)-curves C1, ..., C8. The divisor of a fibre of the
conic bundle is thus equal to 1
i=1 Ci. Since there are exactly eight (−1)-curves
on Ŝ4, there is no choice.
The group of automorphisms of Ŝ4 that leave every curve of negative self-
intersection invariant is isomorphic to C∗ and corresponds to automorphisms of P2
of the form (x : y : z) 7→ (αx : y : z), for α ∈ C∗. Indeed, such automorphisms are
the lifts of automorphisms of S6 leaving invariant every exceptional curve (which
are of the form
(x : y : z), (u : v : w)
(x : αy : βz), (u : α−1v : β−1w)
, for
α, β ∈ C∗) and which fix both points A4 and A5.
Definition 7.3. Let h1 and h2 be the following birational transformations of P
h1 : (x : y : z) 99K (yz : xy : −xz)
h2 : (x : y : z) 99K (yz(y − z) : xz(y + z) : xy(y + z))
and denote respectively by g1, g2 the lift of these elements on Ŝ4 and by Cs24 the
group generated by g1 and g2.
The following lemma shows that Cs24 ⊂ Aut(Ŝ4, π) and describes some of the
properties of this group.
Lemma 7.4. Let h1, h2, g1, g2,Cs24 be as in Definition 7.3. Then:
1. The group Cs24 is a group of automorphisms of Ŝ4 that preserve the conic
bundle (Ŝ4, π), i.e. Cs24 ⊂ Aut(Ŝ4, π).
2. The action of g1 and g2 on the set of irreducible rational curves of negative
self-intersection is respectively:
(Ẽ1 D̃23)(E2 D12)(E3 D13)(E4 E5)(D14 D15),
(Ẽ1 D̃23)(E2 D13)(E3 D12)(E4 D14)(E5 D15).
In particular, both g1 and g2 twist the conic bundle (Ŝ4, π).
3. Both g1 and g2 are elements of order 4 and
2 = (h2)
2 = (x : y : z) 7→ (−x : y : z).
Thus (g1)
2 = (g2)
2 ∈ kerπ is an automorphism of Ŝ4 which leaves every
divisor of negative self-intersection invariant.
4. The group Cs24 is isomorphic to Z/2Z × Z/4Z and the action on the basis
of the fibration π yields the exact sequence
1 →< (h1)
2 >∼= Z/2Z → Cs24
→< π(h1), π(h2) >∼= (Z/2Z)
2 → 1.
5. The group Cs24 contains no involution that twists the conic bundle (Ŝ4, π).
In particular, no element of Cs24 fixes a curve of positive genus.
6. The pair (Cs24, Ŝ4) and the triple (Cs24, Ŝ4, π) are both minimal.
Proof. Observe first that h1 and h2 preserve the pencil of lines of P
2 passing
through the point A1 = (1 : 0 : 0), so g1, g2 are birational transformations of Ŝ4
that send a general fibre of π on another fibre. Then, we compute (h1)
2 = (h2)
(x : y : z) 7→ (−x : y : z). This implies that both h1 and h2 are birational maps of
order 4.
Note that the lift of h1 on the surface S6 is the automorphism
κ1,−1 :
(x : y : z), (u : v : w)
(u : w : −v), (x : z : −y)
(see Example 4.5). Since this automorphism permutes A4 and A5, its lift on Ŝ4
is biregular. The action on the divisors with negative self-intersection is deduced
from that of κ1,−1 (see Lemma 4.6).
Compute the involution
h3 = h1h2 = (x : y : z) 99K (x(y + z) : z(y − z) : −y(y − z)).
Its linear system is
{ax(y + z) + (by + cz)(y − z) = 0 | (a : b : c) ∈ P2},
which is the linear sytem of conics passing through (0 : 1 : 1) and A1 = (1 : 0 : 0),
with tangent y + z = 0 at this point (i.e. passing through A5). Blowing-up these
three points (two on P2 and one in the blow-up of A1), we get an automorphism
g3 of some rational surface. As the points A2 = (0 : 1 : 0) and A3 = (0 : 0 : 1)
are permuted by h3, we can also blow them up and again get an automorphism.
The isomorphism class of the surface obtained is independent of the order of the
blown-up points. We may first blow-up A1, A2, A3 and get S6. Then, we blow-up
the two other base-points of h3, which are in fact A4 (the point (0 : 1 : −1)) and
A5 (the point infinitely near to A1 corresponding to the tangent y + z = 0). This
shows that g3, and therefore g2, belong to Aut(Ŝ4, π).
Since h3 permutes the points A2 and A3, g3 = g1g2 permutes the divisors E2
and E3. It also permutes D12 and D13, since h3 leaves the pencil of lines passing
through A1 invariant. It therefore leaves Ẽ1 and D̃23 invariant, since E2 and E3
touch D̃23 but not E1. The remaining exceptional divisors are E4, E5, D14, D15.
Either g1g2 leaves all four invariant, or it acts as (E4 D15)(E5 D14) (using the
intersection with Ẽ1 and D̃23). Since A4 and A5 are base-points of h1h2, E4 and
E5 are not invariant. Thus, g1g2 acts on the irreducible rational curves of negative
self-intersection as (E2 E3)(D12 D13)(E4 D15)(E5 D14). We obtain the action of
g2 by composing that of g1g2 with that of g1 and thus have proved assertions 1
through 3.
Assertion 4 follows from assertion 3 and the fact that g1 and g2 commute.
Let us prove that Cs24 contains no involution that twists the conic bundle
(Ŝ4, π). Recall that such elements are involutions acting trivially on the basis
of the fibration (see Lemma 6.1). Note that the 2-torsion of Cs24 is equal to
{1, g21, g1g2, g1g
2 }. The elements g1g2 and g1g
2 do not act trivially on the basis
of the fibration, and the element (g1)
2 does not twist any singular fibre since it
leaves every curve of negative self-intersection invariant. This proves assertion 5.
It remains to prove the last assertion. Observe that the orbits of the action of
Cs24 on the exceptional divisors of Ŝ4 are {E2, E3, D12, D13} and {E4, E5, D14, D15}.
Since these orbits cannot be contracted, the pair (Cs24, Ŝ4) is minimal, and so is
the triple (Cs24, Ŝ4, π).
Remark 7.5. The pair (Cs24, Ŝ4) was introduced in [Bla2] and was called Cs.24
because it is a group acting on a conic bundle, which is special, and isomorphic
to Z/2Z× Z/4Z.
8 Finite Abelian groups of automorphisms of conic
bundles - birational representative elements
In this section we use the tools prepared in the previous sections to describe the
finite Abelian groups of automorphisms of conic bundles such that no non-trivial
element fixes a curve of positive genus.
We first treat the case in which no involution twisting the conic bundle belongs
to the group:
Proposition 8.1. Let G ⊂ Aut(S, π) be a finite Abelian group of automorphisms
of the conic bundle (S, π) such that:
• no involution that twists the conic bundle (S, π) belongs to G;
• the triple (G,S, π) is minimal.
Then, one of the following occurs:
• The fibration is smooth, i.e. S is a Hirzebruch surface.
• S is the del Pezzo surface of degree 6.
• The triple (G,S, π) is isomorphic to the triple (Cs24, Ŝ4, π) of Section 7.
Proof. We assume that the fibration is not smooth. Recall that since the triple
(G,S, π) is minimal, any singular fibre of π is twisted by an element of G (by
Lemma 3.8). Since no twisting involution belongs to G, any element g ∈ G that
twists a fibre corresponds to case 2 of Proposition 6.5. In particular, g is the lift
on S of an automorphism of the form κα,β of the del Pezzo surface of degree 6 and
it twists 2 singular fibres, which correspond to the fibres of the two fixed points
of π(g) ∈ PGL(2,C). Furthermore, g is the root of an involution that leaves every
component of every singular fibre of π invariant.
If the number of singular fibres is exactly two, then S is the del Pezzo surface
of degree 6, and we are done.
Now suppose that the number of singular fibres is larger than two. This implies
that π(G) is not a cyclic group (otherwise the non-trivial elements of π(G) would
have the same two fixed points: there would then be at most two singular fibres);
therefore, π(G) is isomorphic to (Z/2Z)2. By a judicious choice of coordinates we
may suppose that
π(G) =
Since a singular fibre corresponds to a fixed point of one of the three elements
of order 2 of π(G), only the fibres of (0 : 1), (1 : 0), (1 : 1), (−1 : 1), (i : 1), (−i : 1)
can be singular. Since the group π(G) acts transitively on the sets {(1 : 0), (0 : 1)},
{(1 : ±1)} and {(1 : ±i)}, there are 4 or 6 singular fibres.
We denote by g1 an element of G which twists the two singular fibres of (1 : 0)
and (0 : 1). Let η : S → S6 denote the birational g1-equivariant morphism given
by Proposition 6.5, which conjugates g1 to the automorphism
−1 = κα,β :
(x : y : z), (u : v : w)
(u : αw : βv), (x : α−1z : β−1y)
of the del Pezzo surface S6 of degree 6, for some α, β ∈ C
∗. In fact, since π(g1)
has order 2, we have β = −α, so ηg1η
−1 = κα,−α. The points blown-up by η are
fixed by
η(g1)
2η−1 = (κα,−α)
(x : y : z), (u : v : w)
(x : −y : −z), (u : −v : −w)
and therefore belong to the curves
E1 = {
(1 : 0 : 0), (0 : a : b)
| (a : b) ∈ P1}
and D23 = {
(0 : a : b), (1 : 0 : 0)
| (a : b) ∈ P1}.
Since these points consist of orbits of ηg1η
−1, half of them lie in E1 and the
other half in D23. In fact, up to a change of coordinates,
(x, y, z), (u, v, w)
(u, v, w), (x, y, z)
, the points that may be blown-up by η are
(0 : 1 : 1) , (1 : 0 : 0)
∈ D23,
κα,−α(A4) = A5 =
(1 : 0 : 0) , (0 : 1 : −1)
∈ E1,
(0 : 1 : i) , (1 : 0 : 0)
∈ D23,
κα,−α(A6) = A7 =
(1 : 0 : 0) , (0 : 1 : i)
∈ E1.
The strict pull-backs Ẽ1 and D̃23 by η of E1 and D23 respectively thus have self-
intersection −2 or −3 in S, depending on the number of points blown-up. By
convention we again denote by E1, E2, E3, D12, D13, D23 the total pull-backs by
η of these divisors. (Note that for E2, E3, D12, D13, the strict and the total pull-
backs are the same.) We set E4 = η
−1(A4),..., E7 = η
−1(A7) and denote by f the
divisor class of the fibre of the conic bundle.
(a) Suppose that η is the blow-up of A4 and A5, which implies that S is the
surface Ŝ4 of Section 7. The Picard group of S is then generated by E1, E2, ..., E5
and f .
Since we assumed that (G,S, π) is minimal, the singular fibres of (1 : 1) and
(−1 : 1) must be twisted. One element g2 twists these two singular fibres and
acts with order 2 on the basis of the fibration, with action (x1 : x2) 7→ (x2 : x1).
Since g1 and g2 twist some singular fibre, both must invert the two curves of self-
intersection −2, namely Ẽ1 and D̃23. The action of g1 and g2 on the irreducible
rational curves of negative self-intersection is then respectively
(Ẽ1 D̃23)(E2 D12)(E3 D13)(E4 E5)(D14 D15),
(Ẽ1 D̃23)(E2 D13)(E3 D12)(E4 D14)(E5 D15).
The elements g1 and g2 thus have the same action on Pic(S) = Pic(Ŝ4) as the
two automorphisms with the same name in Definition 7.3 and Lemma 7.4, which
generate Cs24. Note that the group H of automorphisms of S that leave every
curve of negative self-intersection invariant is isomorphic to C∗ and corresponds
to automorphisms of P2 of the form (x : y : z) 7→ (αx : y : z), for any α ∈ C∗.
Then, g1 and g2 are equal to the lift of the the following birational maps of P
h1 : (x : y : z) 99K (µyz : xy : −xz),
h2 : (x : y : z) 99K (νyz(y − z) : xz(y + z) : xy(y + z)),
for some µ, ν ∈ C∗.
As h1h2(x : y : z) = (µx(y + z) : νz(y − z) : −νy(y − z)) and h2h1(x : y :
z) = (νx(y + z) : µz(y − z) : −µy(y − z)) must be the same by hypothesis, we get
µ2 = ν2.
We observe that π(g1) and π(g2) generate π(G) ∼= (Z/2Z)
2; on the other hand,
by hypothesis an element of G′ does not twist a singular fibre and hence belongs
to H . As the only elements of H which commute with g1 are id and (g1)
2 (which
is the lift of (h1)
2 : (x : y : z) 7→ (−x : y : z)), we see that g1 and g2 generate the
whole group G.
Conjugating h1 and h2 by (x : y : z) 7→ (αx : y : z), where α ∈ C
∗, α2 = µ, we
may suppose that µ = 1. So ν = ±1 and we get in both cases the same group,
because (h1)
2(x : y : z) = (−x : y : z). The triple (G,S, π) is hence isomorphic to
the triple (Cs24, Ŝ4, π) of Section 7.
(b) Suppose that η is the blow-up of A6 and A7. We get a case isomorphic
to the previous one, using the automorphism
(x : y : z), (u : v : w)
(x : y :
iz), (u : v : −iw)
of S6.
(c) Suppose that η is the blow-up of A4, A5, A6 and A7. The Picard group of
S is then generated by E1, E2, ..., E6, E7 and f . Since (G,S, π) is minimal, there
must be two elements g2, g3 ∈ G that twist respectively the fibres of (±1 : 1) and
those of (±i : 1). As in the previous example, the three actions of these elements
on the basis are of order 2, and the three elements transpose Ẽ1 and D̃23. The
actions of g1, g2 and g3 on the set of irreducible components of the singular fibres
of π are then respectively
(E2 D12)(E3 D13)(E4 E5)(D14 D15)(E6 E7)(D16 D17),
(E2 D13)(E3 D12)(E4 D14)(E5 D15)(E6 E7)(D16 D17),
(E2 D13)(E3 D12)(E4 E5)(D14 D15)(E6 D16)(E7 D17).
This implies that the action of the element g1g2g3 is
(E2 D12)(E3 D13)(E4 D14)(E5 D15)(E6 D17)(E7 D16),
and thus it twists six singular fibres of the conic bundle and fixes a curve of genus 2
(Lemma 6.1), which contradicts the hypothesis. (In fact, one can also show that
the group generated by g1, g2 and g3 is not Abelian, see [Bla2], page 66.)
After studying the groups that do not contain a twisting involution, we now
study those which contain such elements. Since these twisting involutions cannot
fix a curve of positive genus, they twist exactly two fibres (Lemma 6.1).
Proposition 8.2. Let G ⊂ Aut(S, π) be a finite Abelian group of automorphisms
of a conic bundle (S, π) such that:
1. If g ∈ G, g 6= 1, then g does not fix a curve of positive genus.
2. The group G contains at least one involution that twists the conic bundle
(S, π).
3. The triple (G,S, π) is minimal.
Then, S is a del Pezzo surface of degree 5 or 6.
Proof. If the number of singular fibres is at most 3, then the surface is a del Pezzo
surface of degree 5 or 6 (Lemma 3.12).
We now assume that the number of singular fibres is at least 4 and show that
this situation is not compatible with the hypotheses. We recall once again the
exact sequence of Remark 3.13
1 → G′ → G
→ π(G) → 1, (1)
and prove the following important assertions:
(a) No element of G twists more than two singular fibres.
(b) Any twisting involution that belongs to G belongs to G′ and twists exactly
two singular fibres.
(c) Any singular fibre is twisted by an element of G.
(d) No non-trivial element preserves every component of every singular fibre.
(e) Any twisting element of G is a root of (or equal to) a twisting involution
that belongs to G′.
Corollary 6.4 shows that an element that twists more than two fibres fixes a
curve of positive genus; since this possibility is excluded by hypothesis, we obtain
assertion (a). Lemma 6.1 shows that any twisting involution contained in G be-
longs to G′ and twists an even number of fibres; using assertion (a), we thus obtain
assertion (b). Assertion (c) follows from the minimality of the triple (G,S, π) (see
Lemma 3.8). Let us prove assertion (d). Suppose that there exists a non-trivial
element g ∈ G that leaves every component of every singular fibre invariant, and
denote by h ∈ G′ a twisting involution (which exists by hypothesis). Since g and h
commute, Lemma 6.7 shows that each singular fibre invariant by h – there are
at least 4 – is twisted by h, which contradicts assertion (a). Therefore, such an
element g doesn’t exist and assertion (d) is proved. Finally, Proposition 6.5 shows
that any twisting element that does not act trivially on the basis of the fibration
is a root of an involution that belongs to G′, and assertion (d) shows that this
involution is twisting, and we obtain assertion (e).
Now that assertions (a) through (e) are proved, we deduce the proposition from
them. Let us denote by σ ∈ G′ a twisting involution, which twists two singular
fibres that we denote by F1 and F2. There are at least two other singular fibres
F3 and F4 that are twisted by other elements of G.
If G′ =< σ >, the fibres F3 and F4 are twisted by roots of σ belonging to G
(assertions (c) and (e)). The description of these elements (Proposition 6.5, and
in particular Corollary 6.6) shows that the roots must be square roots that twist
exactly one singular fibre and permute the two fibres F1 and F2 twisted by σ.
There thus exist two elements h3, h4 ∈ G that twist respectively the fibres F3 and
F4. Since h3 commutes with h4, it must leave invariant the unique fibre twisted
by h4, i.e. F4. Similarly, h4 must leave F3 invariant. Therefore, h3h4 leaves the
four fibres F1,...,F4 invariant and twists the two fibres F3 and F4; it is thus an
involution that belongs to G′, which contradicts the fact that G′ =< σ >.
If G′ 6=< σ >, since σ has no root in G′ (Corollary 6.2), the Abelian groupG′ ⊂
PGL(2,C(x)) is isomorphic to (Z/2Z)2 and contains (using (d)) three twisting
involutions σ, ρ and σρ. Note that two of these three involutions do not twist
singular fibres which are all distinct, otherwise the product of the two involutions
would give an involution that twists 4 singular fibres, contradicting (a). We may
thus suppose that ρ twists F1 and F3, which implies that σρ twists F2 and F3.
The fibre F4 is then twisted by an element which is a square root of one of the
three twisting involutions (assertion (e) and Corollary 6.6). Denote this square
root by h and suppose that h2 6= σ. Note that h exchanges the two singular fibres
twisted by h2. One of these is twisted by σ and the other is not, so h and σ do
not commute.
The only remaining possible finite Abelian groups of automorphisms of conic
bundles satisfying property (F ) are thus del Pezzo surfaces of degree 6 or 5 (studied
in Sections 4 and 5), the triple (Cs24, Ŝ4, π) studied in Section 7, and Hirzebruch
surfaces. We now describe this last case and prove that it is birationally reduced
to the case of P1 × P1.
Proposition 8.3. Let G ⊂ Aut(Fn) be a finite Abelian subgroup of automorphisms
of Fn, for some integer n ≥ 1. Then, a birational map of conic bundles conjugates
G to a finite group of automorphisms of F0 = P
1 × P1 that leaves one ruling
invariant.
Proof. Let G ⊂ Aut(Fn) be a finite Abelian group, with n ≥ 1. Note that G
preserves the unique ruling of Fn. We denote by E ⊂ Fn the unique section
of self-intersection −n, which is necessarily invariant by G. We have the exact
sequence (see Remark 3.13)
1 → G′ → G
→ π(G) → 1. (1)
Since the group π(G) ⊂ PGL(2,C) is Abelian, it is isomorphic to a cyclic group
or to (Z/2Z)2.
If π(G) is a cyclic group, at least two fibres are invariant by G. The group
G fixes two points in one such fibre. We can blow-up the point that does not lie
on E and blow-down the corresponding fibre to get a group of automorphisms of
Fn−1. We do this n times and finally obtain a birational map of conic bundles
that conjugates G to a group of automorphisms of F0 = P
1 × P1.
If π(G) is isomorphic to (Z/2Z)2, there exist two fibres F, F ′ of π whose union
is invariant by G. Let GF ⊂ G be the subgroup of G of elements that leave F
invariant. This group is of index 2 in G and hence is normal. Since GF fixes the
point F ∩ E in F , it acts cyclically on F . There exists another point P ∈ F ,
P /∈ E, which is fixed by GF . The orbit of P by G consists of two points, P and
P ′, such that P ′ ∈ F ′, P ′ /∈ E. We blow-up these two points and blow-down the
strict transforms of F and F ′ to get a group of automorphisms of Fn−2. We do
this ⌊n/2⌋ times to obtain G as a group of automorphisms of F0 or F1.
If n is even, we get in this manner a group of automorphisms of F0 = P
1 × P1.
Note that n cannot be odd, if the group π(G) is not cyclic. Otherwise, we
could conjugate G to a group of automorphisms of F1 and then to a group of
automorphisms of P2 by blowing-down the exceptional section on a point Q ∈
P2. We would get an Abelian subgroup of PGL(3,C) that fixes Q, and thus a
group with at least three fixed points. In this case, the action on the set of lines
passing through Q would be cyclic (see Proposition 2.2), which contradicts our
hypothesis.
We can now prove the main result of this section:
Proposition 8.4. Let G ⊂ Aut(S, π) be some finite Abelian group of automor-
phisms of the conic bundle (S, π) such that the triple (G,S, π) is minimal and no
non-trivial element of G fixes a curve of positive genus. Then, one of the following
situations occurs:
1. S is a Hirzebruch surface Fn;
2. S is a del Pezzo surface of degree 5 or 6;
3. The triple (G,S, π) is isomorphic to the triple (Cs24, Ŝ4, π) of Section 7.
If we suppose that the pair (G,S) is minimal, then we are in case 1 with n 6= 1 or
in case 3. Moreover, cases 1 and 2 are birationally conjugate to automorphisms of
P1 × P1 whereas the third is not.
Proof. The fact that one of the three cases occurs follows directly from Proposi-
tions 8.1 and 8.2.
Case 1 is clearly minimal if and only if n 6= 1 and Proposition 8.3 shows that
it is conjugate to automorphisms of P1 × P1. In the case of del Pezzo surfaces
of degree 5 and 6, the pair (G,S) is not minimal and the group is respectively
birationally conjugate to a subgroup of Sym4 ⊂ Aut(P
2) (Lemma 5.3) or Aut(P1×
P1) (Lemma 4.2). If the first situation occurs, since the group is Abelian and
not isomorphic to (Z/3Z)2 it is diagonalisable and conjugate to a subgroup of
Aut(P1 × P1) (Proposition 2.2). Thus, we are done with case 2.
It remains to show that the pair (Cs24, Ŝ4) is not birationally conjugate to a
group of automorphisms of P1 × P1. Let us suppose the contrary, i.e. that there
exists some Cs24-equivariant birational map ϕ : Ŝ4 99K P
1 × P1 (that conjugates
Cs24 to a group of automorphisms). Then, ϕ is the composition of Cs24-equivariant
elementary links (see for example [Isk3, Theorem 2.5], or [Do-Iz, Theorem 7.7]).
Since our group preserves the conic bundle, the first link is of type II, III or IV
(in the classical notation of Mori theory). We now study these possibilities and
show that it is not possible to go to P1 × P1.
Link of type II - In our case, this link is a birational map of conic bundles,
which is the composition of the blow-up of an orbit of Cs24, no two points on the
same fibre, with the blow-down of the strict transforms of the fibres of the points
blown-up. The points must be fixed by the elements of Cs24 that act trivially on
the basis of the fibration, and thus an orbit has 4 points, two on Ẽ1 and two on
D̃23. This link conjugates the triple (Cs24, Ŝ4, π) to a triple isomorphic to it, by
Proposition 8.1.
Link of type III - It is the contraction of some set of skew exceptional curves, in-
variant by Cs24. This is impossible since the pair (Cs24, Ŝ4) is minimal (Lemma 7.4).
Link of type IV - It is a change of the fibration. This is not possible since the
surface Ŝ4 admits only one conic bundle fibration (Corollary 7.2).
9 Actions on del Pezzo surfaces with fixed part
of the Picard group of rank one
In this section we prove the following result (note that finiteness is not required
and that minimality of the action is implied by the condition on Pic(S)G).
Proposition 9.1. Let S be a del Pezzo surface, and let G ⊂ Aut(S) be an Abelian
group such that rk Pic(S)G = 1 and no non-trivial element of G fixes a curve of
positive genus. Then, one of the following occurs:
1. S ∼= P2 or S ∼= P1 × P1;
2. S is a del Pezzo surface of degree 5 and G ∼= Z/5Z;
3. S is a del Pezzo surface of degree 6 and G ∼= Z/6Z.
Furthermore, in cases 2 and 3, the group G is birationally conjugate to a diagonal
cyclic subgroup of Aut(P2).
This will be proved separately for each degree, in Lemmas 9.7, 9.8, 9.13, 9.15,
9.16 and 9.17.
Remark 9.2. A del Pezzo surface S is either P1 × P1 or the blow-up of 0 ≤ r ≤
8 points in general position on P2 (i.e. such that no irreducible curve of self-
intersection ≤ −2 appears on S). The group Pic(S) has dimension r + 1, and its
intersection form gives a decomposition Pic(S) ⊗ Q = QKS ⊕K
S ; the signature
is (1,−1, ...,−1).
The group Aut(S) of automorphisms of a del Pezzo surface S acts on Pic(S)
and preserves the intersection form. This gives an homomorphism of Aut(S) →
Aut(Pic(S)) which is injective if and only if r > 3, since the kernel is the lift of
automorphisms of P2 that fix the r blown-up points. Furthermore, the image is
contained in the Weyl group and is finite (see [Dol]). In particular, the group
Aut(S) is finite if and only if r > 3.
When we have some group action on a del Pezzo surface, we would like to
determine the rank of the fixed part of the Picard group. Here are some tools to
this end.
Lemma 9.3 (Size of the orbits). Let S be a del Pezzo surface, which is the blow-up
of 1 ≤ r ≤ 8 points of P2 in general position, and let G ⊂ Aut(S) be a subgroup
of automorphisms with rk Pic(S)G = 1. Then:
• G 6= {1};
• the size of any orbit of the action of G on the set of exceptional divisors is
divisible by the degree of S, which is 9− r;
• in particular, if the order of G is finite, it is divisible by the degree of S.
Proof. It is clear that G 6= {1}, since rk Pic(S) > 1. Let D1, D2, ..., Dk be k
exceptional divisors of S, forming an orbit of G (the orbit is finite, see Remark 9.2).
The divisor
i=1 Di is fixed by G and thus is a multiple of KS . We can write∑k
i=1 Di = aKS, for some a ∈ Q. In fact, since aKS is effective, we have a < 0 and
a ∈ Z. Since the Di’s are irreducible and rational, we deduce from the adjunction
formula Di(KS +Di) = −2 that Di ·KS = −1. Hence
i=1 Di =
i=1 KS ·Di = −k = KS · aKS = a(9− r).
Consequently, the degree 9− r divides the size k of the orbit.
Remark 9.4. This lemma shows in particular that rk Pic(S)G > 1 if S is the blow-
up of r = 1, 2 points of P2, a result which is obvious when r = 1, and is clear
when r = 2, since the line joining the two blown-up points is invariant by any
automorphism.
Lemma 9.5. Let S be some (smooth projective rational) surface, and let g ∈
Aut(S) be some automorphism of finite order. Then, the trace of g acting on
Pic(S) is equal to χ(Fix(g)) − 2, where Fix(g) ⊂ S is the set of fixed points of g
and χ is the Euler characteristic.
Proof. This follows from the topological Lefschetz fixed-point formula, which as-
serts that the trace of g acting on H∗(S,Z) is equal to χ(Fix(g)) (this uses the fact
that g is an homeomorphism of finite order). Since S is a complex rational surface,
H0(S,Z) and H4(S,Z) have dimension 1, H2(S,Z) ∼= Pic(S), and Hi(S,Z) = 0
for i 6= 0, 2, 4. Since the trace on H2 and H4 is 1, we obtain the result.
Remark 9.6. This lemma is false if the order of g is infinite. Take for example the
automorphism (x : y : z) 7→ (λx : y : z + y) of P2, for any λ ∈ C∗, λ 6= 1. It fixes
exactly two points, namely (1 : 0 : 0) and (0 : 0 : 1), but its trace on Pic(P2) = Z
is 1.
We now start the proof of Proposition 9.1 by studying the cases of del Pezzo
surfaces of degree 6 or 5.
Lemma 9.7 (Actions on the del Pezzo surface of degree 6). Let S6 = {
(x : y :
z), (u : v : w)
| ux = vy = wz} ⊂ P2 × P2 be the del Pezzo surface of degree 6
and let G ⊂ Aut(S6) be an Abelian group such that rk Pic(S6)
G = 1. Then, G is
conjugate in Aut(S6) to the cyclic group of order 6 generated by
(x : y : z), (u :
v : w)
(v : w : u), (y : z : x)
. Furthermore, G is birationally conjugate to a
diagonal subgroup of Aut(P2).
Proof. Lemma 9.3 implies that the sizes of the orbits of the action of G on the ex-
ceptional divisors are divisible by 6. The action of G on the hexagon of exceptional
divisors is thus transitive, so G contains an element of the form
(x : y : z), (u : v : w)
(αv : βw : u), (βy : αz : αβx)
where α, β ∈ C∗. As the only element of (C∗)2 that commutes with g is the
identity (see the description of Aut(S6) = (C
∗)2 ⋊ (Sym3 × Z/2Z) in Section 4),
G must be cyclic, generated by g. Conjugating it by
(x : y : z), (u : v : w)
(βx : y : αz), (αu : αβv : βw)
we may assume that α = β = 1, as stated in the lemma (this shows in particular
thatG is of finite order). It remains to prove that this automorphism is birationally
conjugate to a linear automorphism of the plane.
Denote by p : S → P2 the restriction of the projection on the first factor.
This is a birational morphism which is the blow-up of the three diagonal points
A1, A2, A3 of P
2. Consider the birational map ĝ = pgp−1 of P2, which is explicitly
ĝ : (x : y : z) 99K (xz : xy : yz). Since g is an automorphism of the surface, it
fixes the canonical divisor KS , so the birational map ĝ leaves the linear system
of cubics of P2 passing through A1, A2 and A3 invariant (this can also be verified
directly).
Note that ĝ fixes exactly one point of P2, namely P = (1 : 1 : 1), and that its
action on the projective tangent space P(TP (P
2)) of P2 at P is of order 3, with two
fixed points, corresponding to the lines (x− y) + ωk(z − y) = 0, where ω = e2iπ/3,
k = 1, 2. Hence, the birational map ĝ preserves the linear system of cubics of P2
passing through A1, A2 and A3, which have a double point at P and are tangent
to the line (x− y) + ω(z − y) = 0 at this point. This linear system thus induces a
birational transformation of P2 that conjugates ĝ to a linear automorphism.
Lemma 9.8 (Actions on the del Pezzo surface of degree 5). Let S5 be the del
Pezzo surface of degree 5 and let G ∈ Aut(S5) = Sym5 be an Abelian group such
that rk Pic(S5)
G = 1. Then, G is cyclic of order 5. Furthermore, G is birationally
conjugate to a diagonal subgroup of Aut(P2).
Proof. We use the description of the surface S5 and its automorphisms group
Aut(S5) = Sym5 given in Section 5. Lemma 9.3 implies that the order of G is
divisible by 5, and thus that G is a cyclic subgroup of Sym5 of order 5. Since
all such subgroups are conjugate in Aut(S5) = Sym5, we may suppose that G is
generated by the lift of the birational transformation h : (x : y : z) 99K (xy :
y(x− z) : x(y− z)) of P2, that fixes two points of P2, namely (ζ +1 : ζ : 1), where
ζ2 − ζ − 1 = 0. Denoting one of them by P , the linear system of cubics passing
through the four blown-up points and having a double point at P is invariant
by h. The birational transformation associated to this system thus conjugates h
to a linear automorphism of P2.
Remark 9.9. The fact that (x : y : z) 99K (xy : y(x− z) : x(y − z)) is linearisable
was proved in [Be-Bl], using the same argument as above.
Corollary 9.10. Let S be a rational surface with (KS)
2 ≥ 5 and let G ⊂ Aut(S)
be a finite Abelian group. Then G is birationally conjugate to a subgroup of Aut(P2)
or Aut(P1 × P1).
Proof. We may assume that the pair (G,S) is minimal; consequently there are two
possibilities (see [Man], [Isk2] or [Do-Iz]):
1. S is a del Pezzo surface and rk Pic(S)G = 1. Then S is either P2, P1×P1 or
a del Pezzo surface of degree 6 or 5 (Remark 9.4); we apply Lemmas 9.7 and 9.8
to conclude.
2. G preserves a conic bundle structure on S. Here the number of fibres is at
most 3, hence no element of G fixes a curve of positive genus (Corollary 6.4); we
apply Proposition 8.4 to conclude.
To study del Pezzo surfaces of degree 4, let us describe their group of auto-
morphisms (note that we do not use the notation Sd for the del Pezzo surfaces of
degree d ≤ 4, because there are many different surfaces of the same degree):
Lemma 9.11 (Automorphism group of del Pezzo surfaces of degree 4). Let S
be a del Pezzo surface of degree 4 given by the blow-up η : S → P2 of five points
A1, ..., A5 ∈ P
2 such that no three are collinear. Setting Ei = η
−1(Ai) and denoting
by L the pull-back by η of a general line of P2, we have:
1. There are exactly 10 conic bundle structures on S, whose fibres are respec-
tively L− Ei, −KS − (L− Ei), for i = 1, ..., 5.
2. The action of Aut(S) on the five pairs of divisors {L−Ei,−KS− (L−Ei)},
i = 1, ..., 5 gives rise to a split exact sequence
0 → F → Aut(S)
→ Sym5,
where F = {(a1, ..., a5) ∈ (F2)
ai = 0} ∼= (F2)
4, and the automorphism
(a1, ..., a5) permutes the pair {L−Ei,−KS − (L−Ei)} if and only if ai = 1.
3. We have
Aut(S) = F⋊Aut(S, η),
where Aut(S, η) is the lift of the group of automorphisms of P2 that leave
the set {A1, ..., A5} invariant, and Aut(S, η) acts on F = {(a1, ..., a5) ∈
ai = 0} by permutation of the ai’s, as it acts on {A1, ..., A5}, and
as ρ(Aut(S)) = ρ(Aut(S, η)) ⊂ Sym5 acts on the exceptional pairs.
4. The elements of F with two ”ones” correspond to quadratic involutions of
P2 and fix exactly 4 points of S.
5. The elements of F with four ”ones” correspond to cubic involutions of P2
and the points of S fixed by these elements form a smooth elliptic curve.
Remark 9.12. The group F ⊂ Aut(S) has been studied intensively since 1895 (see
[Kan], Theorem XXXIII). A modern description of the group as the 2-torsion of
PGL(5,C) can be found in [Bea2, (4.1)], together with a study of the conjugacy
classes of such groups in the Cremona group. For further descriptions of the auto-
morphism groups of these surfaces, see [Do-Iz, section 6.4] and [Bla2, section 8.1].
Proof. Let A = mL −
i=1 aiEi be the divisor of the fibre of some conic bundle
structure on S, for some m, a1, ..., a5 ∈ Z. From the relations A
2 = 0 (the fibres
are disjoint) and AKS = −2 (adjunction formula) we get:
i=1 ai
2 = m2,
i=1 ai = 3m− 2.
As in Lemma 7.1, we have (
i=1 ai)
2 ≤ 5
i=1 ai
2, which implies here that
(3m− 2)2 ≤ 5m2, that is 4(m2 − 3m+ 1) ≤ 0. As m is an integer, we must have
1 ≤ m ≤ 2. If m = 1, we replace it in (3) and see that there exists i ∈ {1, ..., 5}
such that A = L − Ei. Otherwise, taking m = 2 and replacing it in (3), we see
that four of the aj ’s are equal to 1, and one is equal to 0. This gives the ten conic
bundles of assertion 1, which are the lift on S of the lines of P2 passing through
one of the Ai’s or of the conic passing through four of the Ai’s.
The group Aut(S) acts on the set ∪5i=1{L−Ei,−KS − (L−Ei)}; since KS is
fixed, this induces an action on the set of five pairs {L−Ei,−KS − (L−Ei)}. We
denote by ρ : Aut(S) → Sym5 the corresponding homomorphism. The action of
the kernel of ρ on the pairs of conic bundles gives a natural embedding of Ker(ρ)
into (F2)
We now prove that Ker(ρ) = {(a1, ..., a5) |
ai = 0} = F. Acting by a
linear automorphism of P2, we may assume that the points blown-up by η are
A1 = (1 : 0 : 0), A2 = (0 : 1 : 0), A3 = (0 : 0 : 1), A4 = (1 : 1 : 1), A5 = (a : b : c),
for some a, b, c ∈ C∗. Then, the birational involution τ : (x0 : x1 : x2) 99K (ax1x2 :
bx0x2 : cx0x1) of P
2 lifts as an automorphism η−1τη ∈ Aut(S) that acts on Pic(S)

0 −1 −1 0 0 −1
−1 0 −1 0 0 −1
−1 −1 0 0 0 −1
0 0 0 0 1 0
0 0 0 1 0 0
1 1 1 0 0 2

with respect to the basis (E1, E2, E3, E4, E5, L). It follows from this observation
that η−1τη belongs to the kernel of ρ, and acts on the pairs of conic bundles
as (0, 0, 0, 1, 1) ∈ (F2)
5. Permuting the roles of the points A1, ..., A5, we get 10
involutions whose representations in (F2)
5 have two ”ones” and three ”zeros”.
These involutions generate the group {(a1, ..., a5) |
ai = 0} = F. To prove that
this group is equal to Ker(ρ), it suffices to show that (1, 1, 1, 1, 1) does not belong
to Ker(ρ). This follows from the fact that (1, 1, 1, 1, 1) would send L = 1
(KS +∑5
i=1(L − Ei)) on the divisor
(KS +
i=1(−KS − L + Ei)) =
(−2L − 3KS),
which doesn’t belong to Pic(S). This concludes the proof of assertion 2 (except
the fact that the exact sequence is split, which will be proved by assertion 3).
We now prove assertion 3. Let σ ∈ Sym5 be a permutation of the set {1, ..., 5}
in the image of ρ and g be an automorphism of S such that ρ(g) = σ. Let α be
the element of Aut(Pic(S)) that sends Ei on Eσ(i) and fixes L. Viewing Aut(S)
as a subgroup of Aut(Pic(S)), the element gα−1 ∈ Aut(Pic(S)) fixes the five pairs
of conic bundles. There exists some element h ∈ F ⊂ Aut(S) such that hgα−1
either fixes the divisor of every conic bundle or permutes the divisors of conic
bundles in each pair. The same argument as in the above paragraph shows that
this latter possibility cannot occur. Hence hgα−1 fixes L−E1, ..., L−E5 and KS.
It follows that hgα−1 acts trivially on Pic(S), so α = hg ∈ Aut(S), and α is by
construction the lift of an automorphism of P2 that acts on the set {A1, ..., A5} as
σ does on {1, ..., 5}. Conversely, it is clear that every automorphism r of P2 which
leaves the set {A1, A2, A3, A4, A5} invariant lifts to the automorphism η
−1rη of
S whose action on the pairs of conic bundles is the same as that of r on the set
{A1, A2, A3, A4, A5}. This gives assertion 3.
Assertion 4 follows from the above description of some element of F ⊂ Aut(S)
with two ”ones” as the lift of a birational map of the form τ : (x0 : x1 : x2) 99K
(ax1x2 : bx0x2 : cx0x1). As the automorphism η
−1τη ∈ Aut(S) does not leave any
exceptional divisor invariant, its fixed points are the same as those of τ , which are
the four points (α : β : γ), where α2 = a, β2 = b, γ2 = c.
It remains to prove the last assertion. Note that the element h = (0, 1, 1, 1, 1) ∈
Aut(S) fixes the divisor L−E1, hence acts on the associated conic bundle structure.
Furthermore, the four singular fibres of this conic bundle, {L − E1 − Ei, Ei}, for
i = 2, ..., 5, are invariant by h and this element switches the two components of
each fibre. This shows that the action of h on the basis of the fibration is trivial,
so the restriction of h on each fibre is an involution of P1 which fixes two points.
On each singular fibre, exactly one point is fixed, which is the singular point of
the fibre. The situation is similar for the other elements with four ”ones” (in fact,
the involutions described here are twisting involutions, see Lemma 6.1).
Lemma 9.13 (Actions on the del Pezzo surfaces of degree 4). Let S be a del
Pezzo surface of degree 4, and let G ∈ Aut(S) be an Abelian group such that
rk Pic(S)G = 1. Then, G contains an involution that fixes an elliptic curve.
Proof. We keep the notation of Lemma 9.11 for η : S → P2,Aut(S, η), ρ,F, ... and
denote by H the group G ∩ F = G ∩ Kerρ. We will prove that H contains an
element of F with four ”ones”, which is an involution that fixes an elliptic curve
(Lemma 9.11).
The group ρ(G) ⊂ ρ(Aut(S)) ∼= Aut(S, η) is isomorphic to a subgroup of
Aut(S, η). The group Aut(S, η) is the lift of the group of automorphisms of P2 that
leave the set {A1, ..., A5} invariant (Lemma 9.11). The restriction of this group to
the conic of P2 passing through the five points is a subgroup of PGL(2,C) that
leaves five points invariant. Since ρ(G) is finite and Abelian, it is cyclic, of order
at most 5. We consider the different possibilities.
The order of ρ(G) is 1. This implies that G ⊂ F. If G contains an element
with four ”ones”, we are done. Otherwise, up to conjugation G is a subset of the
group generated by (1, 1, 0, 0, 0) and (1, 0, 1, 0, 0), and fixes L − E4 and L − E5
(thus rk Pic(S)G > 1).
The order of ρ(G) is 2. Up to a change of numbering, ρ(G) is generated by
(1 2)(3 4); since G is Abelian, we find that H ⊂ V = {(a, a, b, b, 0) | a, b ∈ F2}.
Let g = ((a, b, c, d, e), (1 2)(3 4)) ∈ G be such that ρ(g) = (1 2)(3 4). We may
suppose that e = 1 (otherwise, the group G would fix L − E5 and we would
have rk Pic(S)G ≥ 2.) Conjugating by ((0, b, 0, d, b + c), id) we may assume that
g = ((a+ b, 0, c+ d, 0, 1), (1 2)(3 4)). In fact, since a+ b+ c+ d+ e = 0, we have
g = ((α, 0, 1+α, 0, 1), (1 2)(3 4)), where α = a+b = c+d+1 ∈ F2. If α = 1, then g
has order 4 and fixes the divisor 2L− E3 − E4, thus G cannot be equal to < g >
and it follows that V ⊂ G; in particular the element (1, 1, 1, 1, 0) is contained in
G. If α = 0, then < g > fixes 2L− E1 − E2, so once again G contains V .
The order of ρ(G) is 3. In this case, ρ(G) is generated by a 3-cycle, namely
(1 2 3); then H must be a subgroup of V = {(a, a, a, b, a + b) | a, b ∈ F2}. The
order of G must be a multiple of 4, by Lemma 9.3, hence H = V , and thus G
contains the element (1, 1, 1, 1, 0).
The order of ρ(G) is 4. Then ρ(G) is generated by (1 2 3 4), so H must be a
subgroup of V =< (1, 1, 1, 1, 0) >. Let g = ((a, b, c, d, e), (1 2 3 4)) ∈ G be such
that ρ(g) = (1 3 2 4). Conjugating the group by ((a, a+b, a+b+c, 0, a+c), id), we
may suppose that g = ((0, 0, 0, e, e), (1 3 2 4)). If e = 1, then g4 = (1, 1, 1, 1, 0) ∈ G.
If e = 0, the element g belongs to HS , so it fixes the divisors L and E5. As the
group V fixes L− E5, the rank of Pic(S)
G cannot be 1.
The order of ρ(G) is 5. Then, ρ(G) is generated by a 5-cycle and H = {1}.
The rank of Pic(S)H cannot be 1, by Lemma 9.3.
Before studying the case of del Pezzo surfaces of degree ≤ 3, we remind the
reader of some classical embeddings of these surfaces.
Remark 9.14. Recall ([Kol], Theorem III.3.5) that a del Pezzo surface of degree 3
(respectively 2, 1) is isomorphic to a smooth hypersurface of degree 3 (respectively
4, 6) in the projective space P3 (respectively in P(1, 1, 1, 2), P(1, 1, 2, 3)). Further-
more, in each of the 3 cases, any automorphism of the surface is the restriction of
an automorphism of the ambient space. We will use these classical embeddings,
take w, x, y, z as the variables on the projective spaces, and denote by [α : β : γ : δ]
the automorphism (w : x : y : z) 7→ (αw : βx : γy : δz). Note that a del Pezzo
surface of degree 4 is isomorphic to the intersection of two quadrics in P4, but we
will not use this here.
Lemma 9.15 (Actions on the del Pezzo surfaces of degree 3). Let S be a del
Pezzo surface of degree 3, and let G ∈ Aut(S) be an Abelian group such that
rk Pic(S)G = 1. Then, G contains an element of order 2 or 3 that fixes an elliptic
curve of S.
Proof. Lemma 9.3 implies that the order of G is divisible by 3, so G contains an
element of order 3. We view S as a cubic surface in P3, and Aut(S) as a subgroup
of PGL(4,C) (see Remark 9.14). There are three kinds of elements of order 3 in
PGL(4,C), depending on the nature of their eigenvalues. Setting ω = e2iπ/3, there
are elements with one eigenvalue of multiplicity 3 (conjugate to [1 : 1 : 1 : ω], or its
inverse), elements with two eigenvalues of multiplicity 2 (conjugate to [1 : 1 : ω : ω])
and elements with three distinct eigenvalues (conjugate to [1 : 1 : ω : ω2]). We
consider the three possibilities.
Case a: G contains an element of order 3 with one eigenvalue of multiplicity 3.
The element [1 : 1 : 1 : ω] fixes the hyperplane z = 0, whose intersection with the
surface S is an elliptic curve (because Fix(g) ⊂ S is smooth). Thus, we are done.
Case b: G contains an element of order 3 with two eigenvalues of multiplicity 2.
With a suitable choice of coordinates, we may assume that this element is
g = [1 : 1 : ω : ω].
Since S is smooth, its equation F is of degree at least 2 in each variable, which
implies that F (w, x, ωy, ωz) = F (w, x, y, z) (the eigenvalue is 1); up to a change
of coordinates F = w3 + x3 + y3 + z3, which means that S is the Fermat cubic
surface. The group of automorphisms of S is (Z/3Z)3 ⋊ Sym4 and the centraliser
of g in it is (Z/3Z)3 ⋊ V , where V ∼= (Z/2Z)2 is the subgroup of Sym4 generated
by the two transpositions (w, x) and (y, z). The structure of the centraliser gives
rise to an exact sequence
1 → (Z/3Z)3 → (Z/3Z)3 ⋊ V
→ V → 1
∪ ∪ ∪
1 → G ∩ (Z/3Z)3 → G → γ(G) → 1.
We may suppose that G contains no element of order 3 with an eigenvalue of
multiplicity 3, since this case has been studied above (case a). There are then
three possibilities for G ∩ (Z/3Z)3, namely < g >, < g, [1 : ω : 1 : ω] > and
< g, [1 : ω : ω : 1] >. The last is conjugate to the second by the automorphism
(y, z). Note that g preserves exactly 9 of the 27 lines on the surface; these
are {w + ωix = y + ωjz = 0}, for 0 ≤ i, j ≤ 2. If G ∩ (Z/3Z)3 is equal to
< g >, then G/ < g >∼= γ(G) has order 1, 2 or 4 and thus G leaves at least
one of the 9 lines invariant, whence rk Pic(S)G > 1. If G ∩ (Z/3Z)3 is the group
H =< g, [1 : ω : 1 : ω] > we have G = H , since the centraliser of H in (Z/3Z)3⋊V
is the group (Z/3Z)3. As the set of three skew lines {w + ωix = y + ωiz = 0} for
0 ≤ i ≤ 2 is an orbit of H , the rank of Pic(S)G is strictly larger than 1.
Case c: G contains an element g of order 3 with three distinct eigenvalues.
We may suppose that g = [1 : 1 : ω : ω2]. Note that the action of g on P3
fixes the line Lyz of equation y = z = 0 and thus the whole group G leaves this
line invariant. If Lyz ⊂ S, the rank of rk Pic(S)
G is at least 2. Otherwise, the
equation of S is of the form L3(w, x)+L1(w, x)yz+ y
3+ z3 = 0, where L3 and L1
are homogeneous forms of degree respectively 3 and 1, and L3 has three distinct
roots, so Fix(g) = S ∩ Lyz. Since g fixes exactly three points, the trace of its
action on Pic(S) ∼= Z7 is 1 (Lemma 9.5) and thus rk Pic(S)g > 1, which implies
that G 6=< g >.
Note that every subgroup of PGL(4,C) isomorphic to (Z/3Z)2 contains an
element with only two distinct eigenvalues, so we may assume that G contains
only two elements of order 3, which are g and g2. This implies that the action of
G on the three points of Lyz ∩ S gives an exact sequence
1 →< g >→ G → Sym3,
where the image on the right is a transposition. The group G thus contains an
element of order 2, that we may assume to be diagonal of the form (w : x : y :
z) 7→ (−w : x : y : z) and that fixes the elliptic curve which is the trace on S of
the plane w = 0.
Lemma 9.16 (Actions on the del Pezzo surfaces of degree 2). Let S be a del
Pezzo surface of degree 2, and let G ∈ Aut(S) be an Abelian group such that
rk Pic(S)G = 1. Then, G contains either the Geiser involution (that fixes a curve
isomorphic to a smooth quartic curve) or an element of order 2 or 3 that fixes an
elliptic curve.
Proof. We view S as a surface of degree four in the weighted projective space
P(2, 1, 1, 1) (see Remark 9.14). Note that the projection on the last three coordi-
nates gives S as a double covering of P2 ramified over a smooth quartic curve Q.
Lemma 9.3 implies that the order of G is divisible by 2, so G contains an
element g of order 2.
If the element g is the involution induced by the double covering (classically
called the Geiser involution), we are done; otherwise we may assume that g acts
on P(2, 1, 1, 1) as g : (w : x : y : z) 7→ (ǫw : x : y : −z), where ǫ = ±1, and the
equation of S is w2 = z4 + L2(x, y)z
2 + L4(x, y), where Li is a form of degree i,
and L4 has four distinct roots. The trace on S of the equation z = 0 defines an
elliptic curve Lz ⊂ S. If ǫ = 1, then g fixes the curve Lz and we are done; we
therefore assume that ǫ = −1.
If G contains another involution, we diagonalise the group generated by these
two involutions and see that one element of the group fixes either an elliptic curve
or the smooth quartic curve, so we may assume that g is the only involution of G.
Note that g fixes exactly four points of S, which are the points of intersection
of Lz with the quartic Q (of equation w = 0). The trace of g on Pic(S) ∼= Z
thus equal to 2 (Lemma 9.5), whence rk Pic(S)g = 5 and G 6=< g >.
The group G acts on the line z = 0 of P2 and on the four points of Lz ∩ Q.
Since g is the only element of order 2 of G, the action of G on these four aligned
points has order 3 and thus, we may assume that L4(x, y) = x(x
3 +λy3) and that
there exists an element h of G that acts as (w : x : y : z) 7→ (αw : x : e2iπ/3y : βz),
with α2 = β4 = 1. We find that h4 is an element of order 3 that fixes the elliptic
curve which is the trace on S of the equation y = 0.
Lemma 9.17 (Actions on the del Pezzo surfaces of degree 1). Let S be a del
Pezzo surface of degree 1, and let G ∈ Aut(S) be an Abelian group such that
rk Pic(S)G = 1. Then, some non-trivial element of G fixes a curve of S of positive
genus.
Proof. We view S as a surface of degree six in the weighted projective space
P(3, 1, 1, 2) (see Remark 9.14). Up to a change of coordinates, we may assume
that the equation is
w2 = z3 + zL4(x, y) + L6(x, y),
where L4 and L6 are homogeneous forms of degree 4 and 6 respectively. The
embedding of S into P(3, 1, 1, 2) is given by | − 3KS| × | −KS | × | − 2KS|, which
implies that G is a subgroup of P (GL(1,C)×GL(2,C)×GL(1,C)). The projection
(w : x : y : z) 99K (x : y) is an elliptic fibration generated by | − KS |, and has
one base-point, namely (1 : 0 : 0 : 1), which is fixed by Aut(S). This projection
induces an homomorphism ρ : Aut(S) → Aut(P1) = PGL(2,C). Note that the
kernel of ρ is generated by the Bertini involution w 7→ −w (and the element z 7→ ωz
(ω = e2iπ/3) if L4 = 0) and is hence cyclic of order 2 (or 6). Furthermore, any
element of this kernel fixes a curve of positive genus.
We assume that no non-trivial element of G fixes a curve of positive genus.
This implies that G is isomorphic to ρ(G) ⊂ Aut(P1), and thus is either cyclic or
isomorphic to (Z/2Z)2. Since the lift of this latter group in Aut(S) is not Abelian,
G is cyclic. We use the Lefschetz fixed-point formula (Lemma 9.5) to deduce the
eigenvalues of the action of elements of G on Pic(S) ∼= Z9. For any element g ∈ G,
g 6= 1, Fix(g) contains the point (1 : 0 : 0 : 1) and is the disjoint union of points
and lines. Thus χ(Fix(g)) ≥ 1 and so the trace of g on Pic(S) is at least −1
(Lemma 9.5).
Elements of order 2: The eigenvalues are < 1a, (−1)b > with a ≥ 4, b ≤ 5.
Elements of order 3: The eigenvalues are < 1a, (ω)b, (ω2)b > with a ≥ 3, b ≤ 3.
Elements of order 4: The eigenvalues are < 1a, (−1)b, (i)c, (−i)c > with a ≥
b−1. Furthermore, the information on the square induces that a+b ≥ 4, so a ≥ 2.
Elements of order 5: The eigenvalues are < 15, l1, l2, l3, l4 >, where l1, ..., l4 are
the four primitive 5-th roots of unity.
Elements of order 6: The eigenvalues are< 1a, (−1)b, (ω)c, (ω2)c, (−ω)d, (−ω2)d >,
where a − b − c + d ≥ −1. Computing the square and the third power, we find
respectively a + b ≥ 3, c + d ≤ 3 and a + 2c ≥ 4, b + 2d ≤ 5. This implies that
a ≥ 2. Indeed, if a = 1, we get b, c ≥ 2 and thus d ≤ 1, which contradicts the fact
that the trace a− b− c+ d is at least −1.
Since rk Pic(S)G = 1, the order of the cyclic group G is at least 7. As the
action of G leaves L4 and L6 invariant, both L6 and L4 are monomials. If some
double root of L6 is a root of L4, the surface is singular, so up to an exchange of
coordinates we may suppose that L4 = x
4 and either L6 = xy
5 or L6 = y
In the first case, the equation of the surface is w2 = z3+x4z+xy5 whose group
of automorphisms Aut(S) is isomorphic to Z/20Z, generated by [i : 1 : ζ10 : −1],
and contains the Bertini involution. No subgroup of Aut(S) fullfills our hypotheses.
In the second case, the equation of the surface is w2 = z3 + x4z + y6, whose
group of automorphisms is isomorphic to Z/2Z×Z/12Z, generated by the Bertini
involution and g = [i : 1 : ζ12 : −1]. The only possibility for G is to be equal to
< g >. Since g4 = [1 : 1 : ω : 1] fixes an elliptic curve, we are done.
Proposition 9.1 now follows, using all the lemmas proved above.
10 The results
We now prove the five theorems stated in the introduction.
Proof of Theorem 4. Since the pair (G,S) is minimal, either rk Pic(S)G = 1 and
S is a del Pezzo surface, or G preserves a conic bundle structure (see [Man], [Isk2]
or [Do-Iz]).
In the first case, either S ∼= P2, or S ∼= P1 × P1 or S is a del Pezzo surface of
degree d = 5 or 6 and G ∼= Z/dZ (Proposition 9.1).
In the second case, either S is a Hirzebruch surface or the pair (G,S) is the
pair (Cs24, Ŝ4) of Section 7 (Proposition 8.4).
Proof of Theorem 2. No non-trivial element of Aut(P2),Aut(P1 × P1) or Cs24
fixes a non-rational curve (the first two cases are clear, the last one follows from
Lemma 7.4).
Conversely, suppose that G is a finite Abelian subgroup of the Cremona group
such that no non-trivial element fixes a curve of positive genus. Since G is finite,
it is birationally conjugate to a group of automorphisms of a rational surface S
(see for example [dF-Ei, Theorem 1.4] or [Do-Iz]). Then, we assume that the pair
(G,S) is minimal and use the classification of Theorem 4.
If S is an Hirzebruch surface, the group is birationally conjugate to a subgroup
of Aut(P1 ×P1) (Proposition 8.3). If S is a del Pezzo surface, the group G is bira-
tionally conjugate to a subgroup of Aut(P1 × P1) or Aut(P2), by Proposition 9.1.
Otherwise, the pair (G,S) is isomorphic to the pair (Cs24, Ŝ4).
It remains to show that the group Cs24 is not birationally conjugate to a
subgroup of Aut(P1 × P1) or Aut(P2). Since the group is isomorphic to Z/2Z ×
Z/4Z, only the case of Aut(P1 × P1) need be considered (see Section 2). This was
proved in Proposition 8.4.
Proof of Theorem 5. By Theorem 2, G is birationally conjugate either to a sub-
group of Aut(P2), or of Aut(P1 × P1), or to the group Cs24.
The group Cs24 is case [8]. The finite Abelian subgroups of Aut(P
2) are conju-
gate to the groups of case [1] or [9] (Proposition 2.2). The finite Abelian subgroups
of Aut(P1 × P1) are conjugate to the groups of cases [1] through [7] (Proposi-
tion 2.5).
It was proved in Proposition 2.5 that cases [1] through [7] are distinct. In
Proposition 8.4 we showed that [8] (Cs24) is not birationally conjugate to any
groups of cases [1] through [7]. Finally, the group [9] is isomorphic only to [1], but
is not birationally conjugate to it (Proposition 2.2). This completes the proof that
the distincts cases given above are not birationally conjugate.
The proof of Theorem 1 follows directly from Theorem 5, and Theorem 3 is a
corollary of Theorem 1.
11 Other kinds of groups
Our main interest up to now was in finite Abelian subgroups of the Cremona
group. In this section, we give some examples in the other cases, in order to show
why the hypothesis ”finite”, respectively ”Abelian”, is necessary to ensure that
condition (F ) (no curve of positive genus is fixed by a non-trivial element) implies
condition (M) (the group is birationally conjugate to a group of automorphisms
of a minimal surface). We refer to the introduction for more details.
Finiteness is important since it imposes that the group is conjugate to a group
of automorphisms of a projective rational surface. This is not the case if the group
is not finite (see for example [Bla2], Proposition 2.2.4).
Lemma 11.1. Let ϕ : P2 99K P2 be a quadratic birational transformation with
three proper base-points, and such that deg(ϕn) = 2n for each integer n ≥ 1.
Then, the following occur:
1. no pencil of curves is invariant by ϕ;
2. ϕ is not birationally conjugate to an automorphism of P2 or of P1 × P1.
Proof. Denote by A1, A2, A3 the three base-points of ϕ and by B1, B2, B3 those
of ϕ−1. Up to a change of coordinates, we may suppose that A1 = (1 : 0 : 0),
A2 = (0 : 1 : 0) and A3 = (0 : 0 : 1). The birational transformation ϕ is thus the
composition of the standard quadratic transformation σ : (x : y : z) 99K (yz : xz :
xy) with a linear automorphism τ ∈ Aut(P2) that sends Ai on Bi for i = 1, 2, 3.
Let Λ be some pencil of curves, and assume that ϕ(Λ) = Λ. We will prove
that some base-point of Λ is sent by ϕ on an orbit of infinite order. The con-
dition deg(ϕn) = 2n is equivalent to saying that for i = 1, 2, 3, the sequence
Bi, ϕ(Bi), ..., ϕ
n(Bi), ... is well-defined, i.e. that ϕ
m(Bi) is not equal to Aj for any
i, j ∈ {1, 2, 3},m ∈ N. Denote by α1, α2, α3, β1, β2, β3 the multiplicity of Λ at
respectively A1, A2, A3, B1, B2, B3 and by n the degree of the curves of Λ. The
curves of the pencil ϕ(Λ) thus have degree 2n−α1−α2−α3. Since Λ is invariant,
n = α1 + α2 + α3, so at least one of the αi’s is not equal to zero. The equality
n = α1+α2+α3 implies that the curves of σ(Λ) have multiplicity αi at Ai, so the
curves of ϕ(Λ) have multiplicity αi at Bi, whence αi = βi for i = 1, 2, 3. Since Λ
passes through Bi with multiplicity αi, the pencil ϕ(Λ) = Λ passes through ϕ(Bi)
with multiplicity αi for i = 1, 2, 3. Continuing in this way, we see that Λ passes
through ϕn(Bi) with multiplicity αi for each n ∈ N. Consequently, Λ has infinitely
many base-points, which is not possible. This establishes the first assertion.
The second assertion follows directly, as each automorphism of P2 or P1 × P1
leaves a pencil of rational curves invariant.
Corollary 11.2. The group generated by a very general quadratic transformation
is a infinite cyclic group satisfying (F ) but not (M).
Proof. The condition deg(ϕn) = 2n, n ∈ N is satisfied for all quadratic transfor-
mations, except for a countable set of proper subvarieties. Consequently condition
(F ) is not satisfied (Lemma 11.1) for a very general quadratic transformation.
Let n be some positive integer and write ϕn : (x : y : z) 99K (f1(x, y, z) :
f2(x, y, z) : f3(x, y, z)), for some homogeneous polynomials fi of degree 2
n. The
set of points fixed by ϕn belongs to the intersection of the curves with equations
xf2 − yf1, xf3 − zf1 and yf3 − zf2. In general, there is only a finite number of
points; this yields condition (F ).
In fact, the argument of Lemma 11.1 works for any very general birational
transformation of P2, since this is a composition of quadratic transformations.
We thus find infinitely many cyclic subgroups of the Cremona group that are not
birationally conjugate to a group of automorphisms of a minimal surface although
none of their non-trivial elements fixes a non-rational curve. The implication
(F ) ⇒ (M) is therefore false for general cyclic groups.
We now study the finite non-Abelian subgroups and provide, in this case, many
examples satisfying (F ) but not (M):
Lemma 11.3. Let S6 = {
(x : y : z), (u : v : w)
| ux = vy = wz} ⊂ P2 × P2
be the del Pezzo surface of degree 6. Let G ∼= Sym3 × Z/2Z be the subgroup of
automorphisms of S6 generated by
(x : y : z), (u : v : w)
(u : v : w), (x : y : z)
(x : y : z), (u : v : w)
(y : x : z), (v : u : w)
(x : y : z), (u : v : w)
(z : y : x), (w : v : u)
Then no non-trivial element of G fixes a curve of positive genus, and G is not
birationally conjugate to a group of automorphisms of a minimal surface.
Proof. Since every non-trivial element of finite order of Aut(S6) is birationally
conjugate to a linear automorphism of P2 (Corollary 9.10), no such element fixes
a curve of positive genus. The description of every G-equivariant elementary link
starting from S6 was given by Iskovskikh in [Isk4]. This shows that this group is
not birationally conjugate to a group of automorphisms of a minimal surface.
Lemma 11.4. Let S5 be the del Pezzo surface of degree 5. Let G ∼= Sym5 be
the whole group Aut(S5). Then no non-trivial element of G fixes a curve of posi-
tive genus, and G is not birationally conjugate to a group of automorphisms of a
minimal surface.
Proof. Since every non-trivial element of Aut(S5) is birationally conjugate to a
linear automorphism of P2 (Corollary 9.10), such an element does not fix a curve
of positive genus. Suppose that there exists some G-equivariant birational trans-
formation ϕ : S5 99K S̃ where S̃ is equal to P
2 or P1 × P1. We decompose ϕ into
G-equivariant elementary links (see for example [Isk3], Theorem 2.5). The classi-
fication of elementary links ([Isk3], Theorem 2.6) shows that a link S5 99K S
either a Bertini or a Geiser involution (and in this case S′ = S5, and thus this link
conjugates G to itself), or the composition of the blow-up of one or two points,
and the contraction of 5 curves to respectively P1 × P1 or P2. It remains to show
that no orbit of G has size 2 or 1, to conclude that these links are not possible.
This follows from the fact that the actions of Sym5,Alt5 ⊂ G on S5 are fixed-point
free (Proposition 5.1).
Finally, the way to find more counterexamples is to look at groups acting on
conic bundles. The generalisation of the example Cs24 gives many examples of
non-Abelian finite groups. Here is the simplest family:
Lemma 11.5. Let n be some positive integer, and let G be the group of birational
transformations of P2 generated by
g1 : (x : y : z) 99K (yz : xy : −xz),
g2 : (x : y : z) 99K (yz(y − z) : xz(y + z) : xy(y + z)),
h : (x : y : z) 99K (e2iπ/2nx : y : z).
Then, G preserves the pencil Λ of lines passing through (1 : 0 : 0) and the corre-
sponding action gives rise to a non-split exact sequence
1 →< h >∼= Z/2nZ → G → (Z/2Z)2 → 1.
In particular, the group G has order 8n. Furthermore, no non-trivial element of
G fixes a curve of positive genus, and G is not birationally conjugate to a group
of automorphisms of a minimal surface.
Proof. Firstly, since g1 and g2 generate the group Cs24, which is not birationally
conjugate to a group of automorphisms of a minimal surface, this is also the case
for G.
Secondly, we compute that (g1)
2 = (g2)
2 = (h)n is the birational transforma-
tion (x : y : z) 7→ (−x : y : z). The maps g1 and g2 thus have order 4 and h has
order 2n.
Thirdly, every generator of G preserves the pencil Λ of lines passing through
(1 : 0 : 0). The action of g1, g2 and h on this pencil is respectively (y : z) 7→ (−y :
z), (y : z) 7→ (z : y) and (y : z) 7→ (y : z). The action of G on the pencil thus gives
an exact sequence
1 → G′ → G → (Z/2Z)2 → 0,
where G′ is the subgroup of elements of G that act trivially on the pencil Λ. It is
clear that < h >∼= Z/2nZ is a subgroup of G′. Since g1h(g1)
−1 = g2h(g2)
−1 = h−1
and g1 and g2 commute, the group < h > is equal to G
Finally, any element of G that fixes a curve of positive genus must act trivially
on the pencil Λ and thus belongs to < h >. Hence, only the identity is possible.
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	Introduction
	The main questions and results
	How to decide
	Linearisation of birational actions
	The approach and other results
	Comparison with other work
	Aknowledgements
	Automorphisms of P2 or P1P1
	Some facts about automorphisms of conic bundles
	The del Pezzo surface of degree 6
	The del Pezzo surface of degree 5
	Description of twisting elements
	The example Cs24
	Finite Abelian groups of automorphisms of conic bundles - birational representative elements
	Actions on del Pezzo surfaces with fixed part of the Picard group of rank one
	The results
	Other kinds of groups
