Rationality of the quotient of $\mathbb{P}^2$ by finite group of automorphisms over arbitrary field of characteristic zero

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by  Andrey S. Trepalin

RATIONALITY OF THE QUOTIENT OF P2 BY FINITE
GROUP OF AUTOMORPHISMS OVER ARBITRARY
FIELD OF CHARACTERISTIC ZERO
1
1
0
ANDREY S. TREPALIN
2

t
c
Abstract. Let k be a field, chark = 0 and G be a finite group
O
of authomorphisms of P2. Castelnuovo’s Theorem implies that
k
8 the quotient variety P2/G is rational if the field k is algebraically
k
2 closed. In this paper we prove that the quotient P2/G is rational
k

for an arbitrary field k of characteristic zero.
]
G
A
.
h
t 1. Introduction
a
m
Let G be a finite group and k be a field. Consider a pure transcen-
[
dental extension K/k of transcendental degree n = ordG. We may

1 assume that K = k{(x )}, where g runs through all the elements of
g
v
group G. The group G naturally acts on K as h(x ) = x . Noether’s
8 g hg
3 problem asks whether the field of invariants KG is rational (i.e. pure
3
transcendental) over k or not. On the language of algebraic geometry,
6
. this is a question about the rationality of the quotient variety An/G.
0
The most complete answer to this question is known for abelian
1
1 groups, but even in this case quotient variety can be non-rational (see
1
[Swa69], [Vos-foi] [EM73], [Len74]).
:
v Noether’s problem can be generalized as follows. Let G be a finite
i
X group, let V be finite-dimensional vector space over an arbitrary field
r k and let ρ : G → GL(V) be a representation. The question is if the
a
quotient variety V/G is k-rational?
Note that V/G has a natural birational structure of a P1-fibration
over P(V)/G, which is locallytrivial in Zarisky topology. So rationality
of V/G follows from rationality of P(V)/G.
In this generalization it is natural to start with a low-dimensional
case.
The most general result is known for dimension 1 and 2.
The author was partially supported by AG Laboratory HSE, RF government
grant,ag. 11.G34.31.0023and the grantsRFFI 11-01-00336-a,MK-503.2010.1and
N.SH.-4713.2010.1.
1
Theorem 1.1 (Lu¨roth). Let k be an arbitrary field and let G ⊂
PGL (k) be a finite subgroup. Then P1/G is k-rational.
2 k
The next theorem is a consequence of Castelnuovo’s Theorem [Cast].
Theorem 1.2. Let k be an algebraically closed field of characteristic
zero and let G ⊂ PGL (k) be a finite subgroup. Then P2/G is k-
3 k
rational.
The main result of this paper is the following.
Theorem 1.3. Let k be an arbitrary field of characteristic zero and let
G ⊂ PGL (k) be a finite subgroup. Then P2/G is k-rational.
3 k
Corollary 1.4. Let k be an arbitrary field of characteristic zero and let
G ⊂ GL (k) be a finite subgroup. The field of invariants k(x ,x ,x )G
3 1 2 3
is k-rational.
To prove this statement we consider algebraical closure of the field
k. We have two groups acting on P2: the geometrical group G and
k
the Galois group Γ = Gal(k/k). Then we consider the quotient variety
P2/N where N isanormal subgroup ofG (ifsuch a subgroupN exists).
k
Next, weresolve thesingularities ofP2/N, runtheG/N×Γ-equivariant
k
minimalmodelprogramandgetasurfaceX. Thenwerepeattheabove
procedure applying this method to the surface X and the group G/N.
In the section 2 we describe notions and results of minimal model
program which are used in this work. In the section 3 we sketch the
classification of finite subgroups in PGL (k) where k is an algebraically
3
closed field of characteristic zero. In the section 4 we prove Theorem
1.3.
The author is grateful to scientific adviser Yu.G. Prokhorov, to
C.A. Shramov for useful discussions and to I.V. Netay for drawing
illustrations.
We use the following notation.
k denotes an arbitrary field of characteristic zero.
k denotes the algebraic closure of a field k.
C denotes the cyclic group of order n.
n
D denotes the dihedral group of order 2n.
2n
S denotes the symmetic group of degree n.
n
A denotes the alternating group of degree n.
n
2πi
ω = e 3 .
I denotes the identity matrix of dimension n.
n
α 0 0
diag(α,β,γ) =  0 β 0 .
0 0 γ
 
2
K denotes the canonical divisor of a variety X.
X
Pic(X) (resp. Pic(X)G) denotes the (invariant) Picard group of a
variety X.
ρ(X) (resp. ρ(X)G) denotes the (invariant) Picard number of a va-
riety X.
Fn denotes the minimal rational ruled (Hirzebruch) surface PP1(O⊕
O(n)).
X ≈ Y denotes birationally equivalence between varieties X and Y.
2. G-equivariant minimal model program
In this section we follow papers [DI-fs], [DI-po], [Isk79].
Definition 2.1. A rational surface X is a smooth projective surface
over k such that X = X ⊗k is birationally isomorphic to P2.
k
Definition 2.2. A G-surface is a pair (X,ρ) where X is a smooth
projective surface and ρ is a monomorphism G ֒→ Aut(X). A mor-
phism of surfaces f : X → X is called a morphism of G-surfaces
′
(X,ρ) → (X ,ρ) if ρ(G) = f ◦ρ(G)◦f 1.
′ ′ ′ −
Definition 2.3. A G-surface (X,ρ) is called minimal if any birational
morphism of G-surfaces (X,ρ) → (X ,ρ) is an isomorphism.
′ ′
Note that in our case there are two groups acting on X: the geomet-
rical group G and the Galois group Γ = Gal(k/k) and the action of G
is Γ-equivariant. It means that for each g ∈ G, γ ∈ Γ and x ∈ X one
has γρ(g)x = ρ(g)γx. Throughout this paper by minimal surface we
mean (G×Γ)-minimal surface.
The classification of minimal rational surfaces is well-known due to
S. Mori. We introduce some important notions before surveying it.
Definition 2.4. A rational G-surface (X,ρ) admits a structure of a
conic bundle if there exists a G-equivariant morphism φ : X → P1 such
that any fibre is isomorphic to a reduced conic in P2.
Note that a general fibre of φ is isomorphic to P1 and its self-
k
intersection equals 0. At the same time there may be singular fibres
each of which being a pair of intersecting (−1)-curves. It is clear that
if a conic bundle is G-minimal then two components of each singular
fibre are permuted by the group G.
We will use the next theorem to work with conic bundles.
Theorem 2.5. [Isk79, Theorem 4] Let X → P1 be a conic bundle.
Then X is not minimal if K2 ∈ {3,5,6,7}.
X
3
Definition 2.6. A Del Pezzo surface is a smooth projective surface X
such that the anticanonical divisor −K is ample.
X
Theorem 2.7. [Isk79, Theorem 1] Let X be a minimal rational G-
surface. Then either X admits a structure of conic bundle with
Pic(X)G ∼= Z2, or X is isomorphic to a Del Pezzo surface with
Pic(X)G ∼= Z.
The next theorem is an important criterium for proving k-rationality
over an arbitrary perfect field k (see [Isk96]).
Theorem 2.8. A minimal rational surface X over a perfect field k is
k-rational if and only if the following two conditions are satisfied:
(i) X(k) 6= ∅;
(ii) d = K2 ≥ 5.
X
Note that all surfaces in this work are rational by Theorem 1.2.
Taking a quotient, resolving singularities and running a minimal model
program don’t affect the existence of k-points, so there exists a k-point
on each considered surface. Therefore in this work if X is a smooth
surface and K2 ≥ 5 then X is k-rational.
X
The important type of rational surfaces is toric surfaces.
Definition 2.9. Toric variety is a normal variety containing an alge-
braic torus as a dense subset.
Remark 2.10. A minimal rational surface X is toric if and only if K2 ≥
X
6.
A minimal rational surface X with K2 ≥ 6 is P2, P1×P1, del Pezzo
X k k k
surface of degree 6 or a minimal rational ruled surface F (n ≥ 2).
n
3. Finite subgroups in PGL (C)
3
Definition 3.1. Any finite subgroup of GL (k) is called a linear group
n
in n variables.
We will use detailed classification of finite linear subgroups in 3 vari-
ables over an algebraically closed field of characteristic zero.
Let a linear group G act on the space V = k3 where k is algebraically
closed.
Definition 3.2. IftheactionofthegroupGonV isreduciblethegroup
G is called intransitive. Otherwise the group G is called transitive.
Definition 3.3. Let G be a transitive group. If there exists a decom-
position V = V ⊕···⊕V to subspaces such that for any element g ∈ G
1 l
one has gV = V then the group G is called imprimitive. Otherwise
i j
the group G is called primitive.
4
Lemma 3.4. Let k be an algebraicallyclosed field of characteristic zero.
Then any representation of a finite group G in GL (k) is conjugate to
n
a representation of the group G in GL (Q)
n
According Lemma 3.4 for our purpose it is sufficient to know the
classification of finite subgroups of GL (C). In the classification we
3
do not distinguish between groups which are equivalent modulo scalar
multiplications because they define the same subgroup in PGL (C).
3
Thus we need to know the classification of finite subgroups of SL (C)
3
modulo scalar multiplications.
We use the following notation: S = diag(1,ω,ω2), T =
0 1 0 1 1 1
 0 0 1 , V = 1  1 ω ω2 , U = diag(ε,ε,εω), ε3 = ω2.
√ 3
1 0 0 − 1 ω2 ω
   
The finite subgroups of SL (C) were completely classified in [Bl17,
3
Chapter V] and [MBD16, Chapter XII].
Theorem 3.5. Any finite subgroup in SL (C) (modulo scalar multi-
3
plications) is conjugate to one of the following.
Intransitive group:
(A) A diagonal abelian group.
(B) A group having a unique invariant subspace of dimension 2.
Imprimitive group:
∼
(C) A group having a normal abelian subgroup N such that G/N =
C .
3
∼
(D) A group having a normal abelian subgroup N such that G/N =
S .
3
Primitive groups having normal subgroups:
(E) Group of order 108 generated by S, T and V.
(F) Group of order 216 generated by (E) and P = UVU 1.
−
(G) Group of order 648 generated by (E) and U.
Simple groups:
(H) Group of order 60 isomorphic to the group A .
5
(I) Klein group of order 168 isomorphic to the group PSL (F ).
2 7
(K) Valentiner group G of order 1080 i.e., its quotient G/F is iso-
morphic to the group A , where F is the group generated by ωI .
6 3
5
4. Rationality of the quotient variety
In this section for each finite group G ⊂ PGL (C) (see Theorem
3
3.5) we prove that the quotient variety P2/G is k-rational. The main
k
method is the following. We consider the algebraic closure k of the field
k. TwogroupsactonP2: thegeometricalgroupGandtheGaloisgroup
k
Γ = Gal(k/k). In addition the action of the group G is Γ-equivariant.
If the group G is cyclic or simple, k-rationality of P2/G is proved in
k
the first and the last cases of this section.
Otherwise thereisanormalsubgroupN inthegroupG. Weconsider
the quotient variety P2/N, resolve singularities, run the G/N × Γ-
k
equivariantminimalmodelprogramandgetaG/N×Γ-minimalsurface
X. One has
P2/G = (P2/N)/(G/N) ≈ X/(G/N),
k k
therefore it is sufficient to prove that X/(G/N) is k-rational. If there
is a normal group M in the group G/N we can repeat this method.
We will use the following definition for convenience.
Definition 4.1. Let S be a G × Γ-surface, S → S be its minimal
resolution of singularities and Y be a G×Γ-equivariant minimal model
e
of S. We denote the surface Y by G×Γ-MMP-reduction of S.
Feor short we will write an MMP-reduction instead of a G×Γ-MMP
reduction.
4.1. Diagonal abelian groups. Each abelian subgroup G ⊂ SL (k)
3
is conjugate to a diagonal subgroup, so its action on P2 can be consid-
k
ered as the action of finite subgroup in an open torus in P2.
k
Lemma 4.2. Let X be a toric variety over an algebraic closed field k,
chark = 0 and let G be a finite subgroup of a torus. Then the quotient
X/G is a toric variety.
Proof. Let Tn be an open torus in X. The regular function’s algebra of
Tn is k[x ,...,x , 1 ,..., 1 ] and its monoms form a lattice Zn. The
1 n x1 xn
action of the group G on this algebra is monomial so monoms of the
algebra of G-invariants form a sublattice in this lattice. It means that
Tn/G is a torus in X/G, so X/G is a toric variety. (cid:3)
Remark 4.3. Note that the resolution of singularities and minimal
model programm don’t affect toric structure on a surface. So for a
toric surface X and finite subgroup G of a torus one has an MMP-
reduction of X/G is a minimal toric surface. Therefore if there exists
a k-point on X/G then it is k-rational by Theorem 2.8.
6
Let G be a finite abelian subgroup in PGL (k). Then the MMP-
3
reduction of P2/G is a minimal toric surface by Lemma 4.2, it is k-
k
rational by Theorem 2.8.
4.2. Groups having a unique fixed point. In this case the groupG
3 3 2
acts on k and there exists decomposition k = k ⊕k into G-invariant
linear spaces. Moreover, the one-dimensional subspace k is Γ-invariant
because the action of the group G is Γ-equivariant and there are no
other one-dimensional G-invariant subspaces. Therefore there exists
decomposition k3 = k2 ⊕k into G-invariant linear subspaces.
It means that there is a unique G-fixed point p ∈ P2 and a unique G-
k
invariant line l. Let F be the blowup of P2 at the point p. The surface
1 k
F admits a G-equivariant P1-bundle structure F → P1 which fibres
1 k 1 k
are proper transforms of lines passing through the point p. Obviously
this P1-bundle has G-invariant sections: the exceptional divisor of the
k
blowup at p and the proper transform of l. So one has F /G ≈ P1 ×
1 k
P1/G. Therefore our problem is reduced to a one-dimensional case.
k
4.3. Imprimitive groups. Each imprimitive group G contains a nor-
mal abelian subgroup N conjugate to a diagonal abelian subgroup in
GL (k). The quotient group G/N is isomorphic to C or S (it cor-
3 3 3
responds to cases (C) and (D) of Theorem 3.5). Moreover a MMP-
reduction of P2/N is a k-rational minimal toric surface by Lemma 4.2.
k
In this subsection we prove the following proposition:
Proposition 4.4. Let X be a k-rational minimal toric surface and let
G be a group C or S Γ-equivariantly acting on X. Then X/G is
3 3
k-rational.
Intheproofofthispropositionquotient singularities ofdefinite types
play important role. The following remark is useful to work with them.
Remark 4.5. Let a group G act on a surface X and fix a point p ∈ X.
Let f : X → X/G is the quotient map. Then one has:
For the action of the group G = C on a tangent space at the point
2
p as −I the point f(p) is a du Val singularity of type A ,
2 1
For the action of the group G = C on a tangent space at the point
3
p as diag(ω,ω2) the point f(p) is a du Val singularity of type A .
2
Moreover these singularities have the following properties:
(a) The minimal resolution π : Y → X/C of a singularity A gives
2 1
the exceptional divisor which is a (−2)-curve; one has K2 = K2 .
Y X/G
For nonsingular curve C passing through the singularity and
X/C2
C = π C we have C2 = C2 − 1.
Y ∗ X/C2 Y X/C2 2
7
(b) The minimal resolution π : Y → X/C of a singularity A gives
3 2
theexceptional divisorwhichconsistsoftwocomponents. Eachofthem
is a (−2)-curve and they intersect transversally at one point; one has
K2 = K2 .
Y X/C3
Let C be a C -invariant nonsingular irreducible curve on X passing
X 3
through the point p, C = f(C ) and C = π C . Then C2 =
X/C3 X Y ∗ X/C3 Y
C2 − 2.
X/C3 3
Let C and D be two C -invariant nonsingular irreducible curves
X X 3
on X which intersect transversally at the point p. Then the curves
C = π f(C )andD = π f(D )intersect withdifferentcomponents
Y ∗ X Y ∗ X
of the exceptional divisor.
Now we come to the proof of Proposition 4.4.
Lemma 4.6. Let X be a k-rational minimal toric surface, p be prime
and the group C Γ-equivariantly act on X. Then an MMP-reduction
p
of X/C is a k-rational minimal toric surface.
p
The Proposition 4.4 follows from this Lemma. If the group G = C
3
it directly follows from Lemma 4.6. If the group G = S then a surface
3
Y, which is C × Γ-MMP reduction of X/C , is a k-rational minimal
2 3
toric surface by Lemma 4.6 and MMP-reduction of Y/C is a k-rational
2
minimal toric surface by Lemma 4.6.
To prove Lemma 4.6 we case-by-case consider P2, P1×P1, del Pezzo
k k k
surface of degree 6 and minimal rational ruled surfaces F (n ≥ 2) with
n
an action of the group C .
p
4.3.1. Case 1: P2. Each cyclic group C is a finite subgroup of an open
k n
torus in P2. Therefore an MMP-reduction of P2/C is a k-rational
k k n
minimal toric surface by Lemma 4.2.
4.3.2. Case 2: P1 × P1. The automorphism group of P1 × P1 is iso-
k k k k
morphic to the group (PGL (k)×PGL (k))⋊C . We have the exact
2 2 2
sequence:
1 → PGL (k)×PGL (k) ֒→ (PGL (k)×PGL (k))⋊C ։ C → 1.
2 2 2 2 2 2
Let p be a prime and C be a cyclic subgroup of the group Aut(P1×
p k
P1) = (PGL (k)×PGL (k))⋊C . The composition of maps
2 2 2
k
C ֒→ (PGL (k)×PGL (k))⋊C ։ C
p 2 2 2 2
takes C to identity if p 6= 2. Thus one has C ֒→ PGL (k)×PGL (k).
p p 2 2
So the group C is a finite subgroup of an open torus in P1 ×P1. An
p k k
MMP-reduction of (P1 × P1)/C is a k-rational minimal toric surface
k k p
by Lemma 4.2.
8
If p = 2 and C is not a subgroup of PGL (k)×PGL (k) then the
2 2 2
action of the group C is conjugate to
2
(x : x ;y : y ) 7→ (y : y ;x : x )
1 0 1 0 1 0 1 0
where (x : x ;y : y ) are homogeneous coordinates on P1×P1. There
1 0 1 0
k k
is a fixed curve x1 = y1 on P1 ×P1 which class in Pic(P1 ×P1) equals
x0 y0 k k k k
−21KP1 P1. The quotient variety Y = (Pk1 ×Pk1)/C2 is nonsingular. By
k× k
Hurvitz formula one has K2 = 1(3K )2 = 9. Thus the surface Y is
Y 2 2 X
isomorphic to P2 and toric.
k
4.3.3. Case 3: Minimal ruled surfaces F (n ≥ 2). Let the group G
n
act on F . Then the conic bundle structure F → P1 is G equivariant.
n n k
It means that there exists the exact sequence:
1 → G ֒→ G ։ G → 1
F B
where G is a group of authomorphisms of general fibre and G is a
F B
group of authomorphisms of the base B = P1. Therefore for a prime p
k
the action of the group C on the base is either faithful or trivial.
p
In the first case there are two fixed points on the base P1. The
k
corresponding fibres of F → P1 are C -invariant. In the second case
n k p
all fibres of F → P1 are C -invariant. So in the both cases we can
n k p
choose C -invariant fibre F .
p 1
The action of the group C on F is either faithful or trivial. There-
p 1
fore there are at least two C -fixed points. So we can choose a fixed
p
point p which don’t lay on the (−n)-section. Let us blow up the point
p and contract the proper transform of F . We get a surface F with
1 n 1
−
the action of the group C . By repeating of this procedure n times we
p
can obtain a C -equivariant birational map f : F 99K P1 ×P1.
p n k k
Note that ρ(P1 ×P1)Cp = ρ(F )Cp = 2. Therefore if p = 2 the group
k k n
C can’t act as a permutation of the rulings of P1 ×P1. So the group
2
k k
C is a finite subgroup of an open torus in P1×P1. The birational map
p k k
f 1 : P1×P1 99K F preserves this torus. Therefore anMMP-reduction
− k k n
of F /C is a k-rational minimal toric surface by Lemma 4.2.
n p
4.3.4. Case 4: Del Pezzo surface of degree 6. A Del Pezzo surface of
degree 6 X is isomorphic over algebraically closed field k to blowup
6
P2 at three points in general position. It can be assumed that these
k
points have homogenous coordinates (1 : 0 : 0), (0 : 1 : 0), (0 : 0 : 1).
The automorphism group of X is isomorphic to T2⋊D where T2
6 12
is two-dimensional torus over k and D acts on the set of (−1)-curves
12
(the exceptional divisors of the blowup and the proper transforms of
lines passing through a pair of points of blowup).
9
Remark 4.7. To work with a Del Pezzo of degree 6 we will use coordi-
nates on P2. An equation in these coordinates defines a curve on the
k
open set, which is the Del Pezzo surface of degree 6 without 6 (−1)-
curves. At the same time it is clear how the curve intersects each of
(−1)-curves because the Del Pezzo surface of degree 6 is isomorphic to
blowup P2 at three points (1 : 0 : 0), (0 : 1 : 0), (0 : 0 : 1).
k
We have the exact sequence:
1 → T2 ֒→ T2 ⋊D ։ D → 1.
12 12
Let pbeaprimeandC bea cyclic subgroupofthegroupAut(X ) =
p 6
T2 ⋊D . The composition of maps
12
C ֒→ T2 ⋊D ։ D
p 12 12
takes C to identity if p > 3. Thus one has C ֒→ T2. So the group
p p
C is a finite subgroup of an open torus in X . An MMP-reduction of
p 6
X /C is a k-rational minimal toric surface by Lemma 4.2.
6 p
If p = 2 or p = 3 and C is not a subgroup of T2 then the group
p
C acts on the set of (−1)-curves forming a hexagon. There are four
p
nonconjugate cyclic subgroups of prime order in the group D . The
12
actions on the hexagon, which sides correspond to (−1)-curves and
vertices correspond to their intersection points, are the following:
(a)Areflection alongalinepassing throughmiddlesofoppositesides
of the hexagon;
(b) A reflection along a line passing through two opposite vertices of
the hexagon;
(c) The central symmetry;
(d) The rotation by an angle of π.
3
a. b. c. d.
In the case (a) we have two invariant disjointed (−1)-curves and the
others are not invariant. Therefore the action of C is not minimal and
2
we can contract this pair and get P1 ×P1 considered in the case 2.
k k
In the case (b) we have the invariant pair of disjointed (−1)-curves
(twootherpairsof(−1)-curvesarenotdisjointed). Thereforetheaction
of C is not minimal and we can contract this pair and get P1 × P1
2
k k
considered in the case 2.
10