Manin's conjecture for a quartic del Pezzo surface with A_4 singularity

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by  T. D. Browning

MANIN’S CONJECTURE FOR A QUARTIC DEL PEZZO
SURFACE WITH A SINGULARITY
4
T.D. BROWNINGANDU.DERENTHAL
Abstract. TheManin conjectureisestablished forasplit singulardel
9 Pezzo surface of degree four, with singularity typeA4.
0
0
2

n
Contents
a
J
1. Introduction 1
7
2. Calculation of Peyre’s constant 4
2
3. Arithmetic functions 6

] 4. The universal torsor 8
T
5. The main argument 11
N
5.1. Real-valued functions 12
.
h 5.2. Estimating N (B) and N (B) — first step 14
a b
t
a 5.3. Estimating N (B) — second step 16
a
m
5.4. Estimating N (B) — second step 21
b
[ 5.5. The final step 24

2 References 29
v
0
6
5
1. Introduction
1
.
0 The distribution of rational points on del Pezzo surfaces is a challeng-
1
ing topic that has enjoyed a surge of activity in recent years. Guided by
7
the largely unverified conjectures of Manin [11] and his collaborators, the
0
: primary aim of this paper is to investigate further the situation for split
v
i singular del Pezzo surfaces of degree 4 in P4, that are defined over Q. Our
X
main achievement will be a proof of the Manin conjecture for the surface
r
a x x x x = x x +x x +x2 = 0, (1.1)
0 1− 2 3 0 4 1 2 3
which we denote by S P4. This surface contains a unique singularity of
⊂
type A and exactly three lines, all of which are defined over Q.
4
LetU be theZariskiopen subsetformedbydeleting thelines fromS, and
let
N (B) := # x U(Q) H(x) 6 B ,
U,H
{ ∈ | }
for any B > 1. Here H is the usual height on P4, in which the height H(x)
is defined as max x ,..., x for a point x = (x : ... : x ) U(Q),
0 4 0 4
{| | | |} ∈
provided that x = (x ,...,x ) has integral coordinates that are relatively
0 4
coprime. Bearing this in mind, the following is our principal result.
2000 Mathematics Subject Classification. 11G35 (14G05, 14G10).
1
2 T.D.BROWNINGANDU.DERENTHAL
Theorem. We have
N (B)= c B(logB)5+O B(logB)5−2/7 ,
U,H S,H
where (cid:0) (cid:1)
6
1 1 6 1
c = ω 1 1+ +
S,H 21600 · ∞· − p p p2
p (cid:18) (cid:19) (cid:18) (cid:19)
Y
and
ω = dt dt dt . (1.2)
∞ 2 6 7
Z|t2|,|t2t6t7|,|t7(t36t7+t22)|,|t26t7|61,0<t661
It is easily checked that the surface S is not toric, and we shall see in
Lemma 3 that it is not an equivariant compactification of G2. Thus our
a
resultdoesnotfollowfromtheworkofTschinkelandhiscollaborators [1,4].
As the minimal desingularisation X of a split del Pezzo surface X
d d
⊂
Pd of degree d is the blow-up of P2 in 9 d points, it has Picard group
−
Pic(X ) = Z10−d. In the setting d =f4, Manin’s conjecture [11] therefore
d ∼
predicts that
f N (B) α(X )ω (X )B(logB)5, (1.3)
U,H 4 H 4
∼
as B , where the exponent of logB is rankPic(X ) 1. Moreover, the
4
→ ∞ f f −
constants α(X ) and ω (X ) are those predicted by Peyre [14]. Note that
4 H 4
the exponent of logB agrees with the statement of fthe theorem. We shall
verify in 2 tfhat c =αf(S)ω (S) in this result.
S,H H
§
An overview of progress relating to the Manin conjecture for arbitrary
del Pezzo surfaces can be foeund inethe first author’s survey [3]. The present
paper should be seen as a modest step on the path to its resolution for the
singular del Pezzo surfaces of degree 4 that are split over Q. According
to the classification of such surfaces found in Coray and Tsfasman [6], it
transpires that there are 15 possible singularity types for split singular del
Pezzosurfacesofdegree4. ItfollowsfromtheworkofBatyrevandTschinkel
[1], la Bret`eche and the first author [2], and the second author’s joint work
with Tschinkel [9], that the Manin conjecture is already known to hold for
5 explicit surfaces from this catalogue. In view of our theorem, which deals
withasurfaceofsingularitytypeA ,itremainstodealwiththesplitquartic
4
del Pezzo surfaces that have singularity types
A for n 1,2,3 , 3A , A +A for n 1,2,3 . (1.4)
n 1 1 n
∈ { } ∈ { }
Here one should note that there are two types of surfaces that have sin-
gularity type A , one containing four lines and one containing five lines.
3
Similarly, there are two types that have 2A singularities.
1
The surface that we have chosen to focus on in the present investigation
satisfies the property that the cone of effective divisors associated to the
minimal desingularisation S is not merely generated by the divisors that
form a basis for the Picard group Pic(S), but requires one further divisor to
generate it. This leads to soeme additional considerations in the proof, as we
will see shortly. e
The proof of the theorem uses a universal torsor. For each split del Pezzo
surface of degree d, there is one (essentially unique) universal torsor, which
is always an open subset of a (12 d)-dimensional affine variety. For toric
−
MANIN’S CONJECTURE FOR A QUARTIC DEL PEZZO SURFACE 3
varieties, universal torsors are open subsets of affine space. Salberger [15]
has shown how to establish Manin’s conjecture using universal torsors for
split toric varieties defined over Q. As a step towards handlingnon-toric del
Pezzo surfaces that still have a relatively simple universaltorsor, the second
author [8] has determined which del Pezzo surfaces of degree at least 3 have
a universal torsor that can be described as a hypersurface in A13−d. Out of
the singularity types in (1.4), these include those surfaces of type A +A ,
1 2
A +A , 3A and the A surface with five lines. The surfaces of type D
1 3 1 3 5
and D considered in [2] and [9] also belong to this class, as does the A
4 4
surface S considered here. In fact we will see in 4 that the universal torsor
§
for the present problem is an open subset of the hypersurface
η α +η α2+η η2η3η = 0, (1.5)
5 1 1 2 3 4 6 7
which is embedded in A9 = SpecQ[η ,...,η ,α ,α ]. Note that one of the
∼ 1 7 1 2
variables does not explicitly appear in the equation.
OurbasicstrategyissimilartotheoneusedfortheD andD quarticdel
5 4
Pezzo surfaces. The first step is to establish an explicit bijection between
the rational points outside the lines on S and certain integral points on
the universal torsor. We adopt the approach of Tschinkel and the second
author [9] in order to obtain this bijection in an elementary way, motivated
by the structure of the minimal desingularisation S as a blow-up of P2 in
five points. The integral points on the universal torsor are counted in 5,
§
using the method developed by la Bret`eche and thee first author [2]. The
torsor variables η ,...,η ,α ,α must satisfy (1.5), together with certain
1 7 1 2
coprimality and height conditions. The first step is to fix the variables
η ,...,η and to estimate the relevant number of α ,α by viewing the
1 7 1 2
equationas acongruencemoduloη . Theresultingestimate isthensummed
5
over the remaining variables.
Theorderinwhichwehandletheremainingvariablesiscrucialandsubtle.
Whenitcomestosummingoverη andη wewillrunintotroublecontrolling
6 7
the overall contribution from the error term each time, because both η and
6
η can be rather big. Summing the number of α ,α over η , for example,
7 1 2 7
leads to an error term that we cannot estimate in a way that is sufficiently
small when summed over η ,...,η and large values of η . In line with
1 5 6
this we shall let the order of summation depend on which of η or η has
6 7
largest absolute value. When it comes to summing the integral points on
the universal torsor that satisfy η > η , we sum first over η and then
6 7 6
| | | |
over η . For the alternative contribution we sum first over η and then over
7 7
η . This process leads to two main terms that we put back together to get
6
something of the general shape
B
M(η ,...,η ) := ω (S) , (1.6)
1 5 H
· η η η η η
1 2 3 4 5
e
where ω (S) is as in (1.3). The final task is to sum this quantity over the
H
remaining variables η ,...,η .
1 5
While esesentially routine, it is in this final analysis that a further inter-
esting feature of the proof of the theorem is revealed. For k Z5 , define
∈ >0
4 T.D.BROWNINGANDU.DERENTHAL
the simplex
Pk := (x1,...,x5) R5 xi > 0, k1x1+ +k5x5 6 1 , (1.7)
∈ | ···
whose volume is easily determined as
(cid:8) (cid:9)
1
vol(Pk)= .
5! k k k k k
1 2 3 4 5
· · · · ·
In 2 we will see that α(S) = vol(P ) vol(P ), whence
(2,4,3,2,3) (3,6,4,2,5)
§ −
1 1 1
α(S)= e = . (1.8)
5! 2 4 3 2 3 − 5! 3 6 4 2 5 21600
· · · · · · · · · ·
Returning to the summation of (1.6) over η ,...,η Z , which is subject
e 1 5 ∈ >0
to η2η4η3η2η3 6 B, it will transpire that there is a negligible contribu-
1 2 3 4 5
tion from those η ,...,η for which η3η6η4η2η5 > B. Summing over the
1 5 1 2 3 4 5
η ,...,η Z that are remaining therefore leads to the final main term
1 5 >0
∈
vol(P ) vol(P ) ω (S)B(logB)5,
(2,4,3,2,3) (3,6,4,2,5) H
− ·
as expected. Thus the main term in the asymptotic formula is really a
(cid:0) (cid:1)
e
difference of two main terms that conspire to give the predicted value for
α(S). It would be interesting to see whether the same sort of phenomenon
occurs for other split del Pezzo surfaces of degree 4, with singularity type
ameong the list (1.4).
Acknowledgements. The authors are extremely grateful to the anony-
mous referee for his careful reading of the manuscript and numerous helpful
comments. While working on this paper the first author was supported
by EPSRC grant number EP/E053262/1. The second author was partially
supported by a Feodor Lynen Research Fellowship of the Alexander von
Humboldt Foundation.
2. Calculation of Peyre’s constant
In this section we wish to show that the value of the constant c ob-
S,H
tained in our theorem is in agreement with the prediction (1.3) of Peyre
[14]. Beginning with the value of ω (S), whose precise definition we will
H
not include here but which corresponds to a product of local densities, we
have e
1 6
ω (S) = ω 1 ω , (2.1)
H ∞ p
− p
Yp (cid:16) (cid:17)
where ω and ω are the reeal and p-adic densities, respectively. The calcu-
∞ p
lation of ω is routine and leads to the conclusion that
p
6 1
ω = 1+ + .
p p p2
The reader is referred to [2, 2] for an analogous calculation. We now turn
§
to the calculation of ω , which needs to agree with (1.2).
∞
Recalltheequations(1.1)forthesurfaceS,andwritef (x) = x x x x
1 0 1 2 3
−
and f (x) = x x +x x +x2. To compute ω , we parametrise the points
2 0 4 1 2 3 ∞
by writing x ,x as functions of x ,x ,x . Thus we have
1 4 0 2 3
x x x x +x2 x2x +x x2
x = 2 3, x = 1 2 3 = 2 3 0 3,
1 x 4 − x − x2
0 0 0
MANIN’S CONJECTURE FOR A QUARTIC DEL PEZZO SURFACE 5
and furthermore,
∂f1 ∂f2 x x
det ∂x1 ∂x1 = det 0 2 = x2.
∂∂xf14 ∂∂xf24! (cid:18)0 x0(cid:19) 0
Since x and x have the same image in P4, we have
−
1
ω = x−2dx dx dx
∞ 2Z|x0|,˛˛˛x2xx03˛˛˛,|x2|,|x3|,˛˛˛˛x22x3x+20x0x23˛˛˛˛61 0 0 2 3
1
= dt dt dt ,
2 6 7
2
Z|t6|,|t2t6t7|,|t2|,|t26t7|,|t22t7+t36t27|61
on carrying out the change of variables x = t ,x = t and x = t2t . But
0 6 2 2 3 6 7
the range of integration is symmetric with respect to the transformation
(t ,t ,t ) (t , t , t ), and so we may restrict to the range t > 0. This
2 6 7 2 6 7 6
7→ − −
therefore confirms the equality in (1.2).
It remains to deal with the constant α(S) that appears in (1.3). As we’ve
already commented, the Picard group Pic(S) of S has rank 6. Distinguished
elements of Pic(S) are the classes of irreeducible curves with negative self
intersection number. As described in 4,etheseeare the classes of four ex-
§
ceptional divisorseE ,...,E coming from the A -singularity of S and the
1 4 4
transforms E ,E ,E of the three lines on S. By the work of the second au-
5 6 7
thor[8, 7],E ,...,E formabasisofPic(S). Intermsofthisbasiswehave
1 6
§
E = E +2E +E +2E E and K = 2E +4E +3E +2E +3E +E .
7 1 2 3 5− 6 − Se 1 2 3 4 5 6
The convex cone in Pic(S) := Pic(S) e R generated by classes of ef-
R Z
⊗
fective divisors is generated by E ,...,E (see [10, Theorem 3.10]). The
1 7
intersection of its dual withethe hyperpleane
x Pic(S) (x, K )= 1
{ ∈ R | − Se }
is a polytope P whose volume is the constant α(S) defined by Peyre [14].
e
By definition
e
x > 0, x +2x +x +2x x > 0,
P = (x1,...,x6)∈ Pic(S)R | 2xi1+4x2+1 3x3+2 2x43+3x55+−x66= 1 .
(cid:26) (cid:27)
Eliminating the last coordeinate shows that P is isomorphic to
x > 0, 2x +4x +3x +2x +3x 6 1,
P′ = (x ,...,x ) R5 i 1 2 3 4 5 .
1 5 ∈ | 3x1+6x2+4x3+2x4+5x5 > 1
(cid:26) (cid:27)
Analyzing the volume form with respect to which we must compute the
volume of P in order to obtain α(S) (see [10, Section 2], for example), we
see that
α(S)= vol(P′) = vol(eP ) vol(P ),
(2,4,3,2,3 (3,6,4,2,5)
−
in the notation of (1.7). This therefore establishes (1.8).
e
An alternative approach to calculating α(S) is available to us through
recent work of Joyce, Teitler and the second author [10]. Recall from [7,
Table 1] that α(S ) = 1/180 for any non-sinegular split del Pezzo surface
0
S of degree 4. Since the order of the Weyl group associated to the root
0
6 T.D.BROWNINGANDU.DERENTHAL
system A is (n+1)!, as recorded in [10, Table 2], so it follows from [10,
n
Theorem 1.3] that
1 1 1
α(S) = = .
180 · 5! 21600
This completes the verification that our theorem confirms the Manin con-
e
jecture for the split A surface (1.1).
4
3. Arithmetic functions
In this section we present some elementary facts about certain arithmetic
functions and their average order, as required for our argument. Define the
multiplicative arithmetic functions
1 1
φ∗(n) := 1 , φ†(n) := 1+ .
− p p
Yp|n(cid:16) (cid:17) Yp|n(cid:16) (cid:17)
Both of these functions have average order O(1), and one has
φ†(n)j
logx, (3.1)
j
n ≪
n6x
X
for any x > 1 and any j Z . To see this we note that φ†(n) 6 1/d,
∈ >0 d|n
whence
P
φ†(n)j 1 1 ∞ 1 1
6 6 ,
n n d ...d d ...d [d ,...,d ] e
1 j 1 j 1 j
nX6x nX6x d1,X...,dj|n d1,.X..,dj=1 Xe6x
where [d ,...,d ] denotes the least common multiple of d ,...,d . The
1 j 1 j
required bound (3.1) then follows from the estimate
∞ ∞
1 1
6 1.
d ...d [d ,...,d ] (d ...d )1+1/j ≪j
1 j 1 j 1 j
d1,.X..,dj=1 d1,.X..,dj=1
For given positive integers a,b, our work will lead us to work with the
function
φ∗(n)/φ∗(gcd(n,a)), if gcd(n,b) = 1,
f (n) := (3.2)
a,b
(0, if gcd(n,b) > 1.
We begin by establishing the following result.
Lemma 1. Let I = [t ,t ], for t < t . Let α Z such that gcd(α,q) = 1.
1 2 1 2
∈
Then we have
t t
f (n)= 2− 1c +O 2ω(b)log I ,
a,b 0
q | |
n∈I∩Z
n≡αX(mod q) (cid:0) (cid:1)
where I := 2+max t , t and
1 2
| | {| | | |}
φ∗(b) 1 −1
c = 1 . (3.3)
0 φ∗(gcd(b,q))ζ(2) − p2
pY|abq(cid:16) (cid:17)
MANIN’S CONJECTURE FOR A QUARTIC DEL PEZZO SURFACE 7
Proof. We will follow the convention that µ( n) = µ(n) and µ(0) = 0. We
−
begin by calculating the Dirichlet convolution
(f µ)(n) = f (d)µ(n/d) = f (pν) f (pν−1) .
a,b a,b a,b a,b
∗ −
d|n pνkn
X Y (cid:0) (cid:1)
ν>1
It is clear that f (1) = 1 and
a,b
1 1/p, if p ∤ab,
−
f (pj)= f (p) = 1, if p ∤b and p a,
a,b a,b
 |
0, if p b,

|
for any j > 1. Hence it follows that

µ(n)gcd(b,n)/n , if gcd(a,n) b,
(fa,b∗µ)(n) = 0, | | otherwise. |
(cid:26)
In particular
gcd(b,n)µ(n)
(f µ)(n) 6 | | 2ω(b)logN,
a,b
| ∗ | n ≪
n6N n6N | |
X X
for any N > 1. Since f = (f µ) 1, we therefore deduce that
a,b a,b
∗ ∗
∞
f (n)= (f µ)(d) 1
a,b a,b
∗
nX∈I∩Z Xd=1 m∈dX−1I∩Z
n≡α (mod q) gcd(d,q)=1 md≡α (mod q)
∞
t t (f µ)(d)
= 2− 1 a,b∗ +O 2ω(b)log I .
q d | |
d=1
gcdX(d,q)=1 (cid:0) (cid:1)
Here we have observed that the outer sum in the first line is really a sum
over d 6 I , making the previous bound applicable for dealing with the
| |
error term. We have then extended the summation over d to infinity, with
acceptable error. Finally, it remains to observe that
∞
(f µ)(d) 1 1 −1 1
a,b
∗ = 1 1 1 = c ,
d − p2 − p2 − p 0
Xd=1 Yp (cid:16) (cid:17)pY|abq(cid:16) (cid:17) Yp|b (cid:16) (cid:17)
gcd(d,q)=1 p∤q
as required to complete the proof of the lemma. (cid:3)
Rather than Lemma 1, we will actually need a corresponding estimate
in which the summand is replaced by f (n)g(n), for suitable real-valued
a,b
functions g. This is supplied for us by the following result.
Lemma 2. Let I = [t ,t ], for t < t , and let g : I R be any function
1 2 1 2
→
such that g has a continuous derivative on I which changes its sign only
R (I) < times on I. Let α Z such that gcd(α,q) = 1. Then we have
g
∞ ∈
c
f (n)g(n) = 0 g(t)dt+O 2ω(b) (log I ) M (g) ,
a,b I
q · | | ·
n∈I∩Z ZI
n≡αX(mod q) (cid:0) (cid:1)
with c given by (3.3) and M (g) := (1+R (I)) sup g(t).
0 I g · t∈I | |
8 T.D.BROWNINGANDU.DERENTHAL
Proof. Let S denote the sum that is to be estimated, and write
M(t) := f (n),
a,b
n6t
X
n≡α (mod q)
for any t > 0. By partial summation,
t2
S = M(t )g(t ) M(t )g(t ) M(t)g′(t)dt.
2 2 1 1
− −
Zt1
An application of Lemma 1 reveals that M(t) = c t/q+O(2ω(b)log(2+ t )).
0
| |
Hence partial integration yields
c t2
S = 0 g(t)dt+O 2ω(b) (log I ) (g(t ) + g(t ) + g′(t) dt) .
2 1
q · | | · | | | | | |
ZI Zt1
Splitting I into the R (cid:0)intervals where g′ has constant sign therefore c(cid:1)om-
g
pletes the proof of the lemma. (cid:3)
4. The universal torsor
The purpose of this section is to establish a completely explicit bijection
betweentherationalpointsontheopensubsetU ofourA quarticdelPezzo
4
surface S, and the integral points on the universal torsor above S which are
subject to a number of coprimality conditions. In doing so we shall follow
the strategy of the second author’s joint work with Tschinkel [9]e.
Along the way we will introduce new variables η ,...,η and α ,α . It
1 7 1 2
will be convenient to henceforth write
η = (η ,...,η ), η′ = (η ,...,η ), α = (α ,α ). (4.1)
1 5 1 7 1 2
Furthermore, we will make frequent use of the notation
5
η(k1,k2,k3,k4,k5) := ηki, (4.2)
i
i=1
Y
for any (k ,...,k ) Q5.
1 5
∈
In order to derive the bijection alluded to above, we must begin by col-
lecting together some useful information about the geometric structure of
S, as defined by equations (1.1). By computing the Segre symbol of S, the
definition of which can be found in Hodge and Pedoe [13], we see that S
contains exactly one singularity. This has type A and is easily determined
4
as p = (0 : 0 : 0 : 0 : 1). By the classification of singular quartic del Pezzo
surfaces found in Coray and Tsfasman [6, Proposition 6.1], S contains ex-
actly three lines. Let us call these lines E′′, E′′ and E′′, where E′′ and E′′
5 6 7 5 6
intersect in the singularity p, and E′′ intersects E′′ outside p. We easily de-
7 6
termine these lines as E′′ = x = x = x = 0 , E′′ = x = x = x = 0
5 { 0 2 3 } 6 { 0 1 3 }
and E′′ = x = x = x = 0 .
7 { 1 3 4 }
The projection x (x : x : x ) from E′′ is a birational map φ : S 99K
7→ 1 3 4 7
P2, which maps
U := S (E′′ E′′ E′′)= (x :... : x ) S x = 0
\ 5 ∪ 6 ∪ 7 { 0 4 ∈ | 3 6 }
isomorphically to
(α : η : α ) P2 η = 0,α η +α2 = 0 P2.
{ 2 5 1 ∈ | 5 6 1 5 2 6 } ⊂
MANIN’S CONJECTURE FOR A QUARTIC DEL PEZZO SURFACE 9
The inverse map is ψ :P2 99K S given by
ψ : (α : η : α ) (η3 :α η :α η2 :η η :α η ), (4.3)
2 5 1 7→ 5 2 7 2 5 5 7 1 7
where η = (α η +α2).
7 − 1 5 2
By [6, Proposition 6.1, Diagram 12], blowing up the singularity p leads
to a minimal desingularisation π : S S containing four ( 2)-curves
0
→ −
E ,...,E (the four exceptional divisors obtained by blowing up p) and
1 4
three ( 1)-curves E ,E ,E (the stricet transforms of the lines E′′,E′′,E′′
− 5 6 7 5 6 7
on S). The configuration of these ( 1)- and ( 2)-curves on S is described
− −
by Figure 1, where the number of edges between two curves is the intersec-
tion number, and self intersection numbers are given as uppereindices. The
divisors A ,A will be introduced momentarily.
1 2
[1] [−1]
A E
1 CCCCCCCCCCCCCCCC 5 EEEEEEEE
[−1] [−1] [−2] [−2] [−2]
E E E E E
7 6 4 3 2
{{{{{{{{ yyyyyyyy
[0] [−2]
A E
2 1
Figure 1. Configuration of curves on S.
The surface S is a blow-up π : S P2 in five points. While there
e
→
are several ways to construct S as such a blow-up of P2, we describe a
map π that is ceompatible with the emap φ : S 99K P2 in the sense that
φ π : S S 99K P2 coincidees with π where it is defined. Such a map
0
◦ →
π : S P2 is obtained by contracting E ,E ,E ,E ,E on S in this order.
6 4 3 2 1
→
We choosee thesame coordinates (α :η : α ) on P2 as before. Thenπ maps
2 5 1
E ,eE ,E ,E ,E to (0 : 0 : 1). Furthermore, E is the sterict transform
1 2 3 4 6 7
of E′ = η = (α η +α2) = 0 P2 and E is the strict transform of
7 { 7 − 1 5 2 } ⊂ 5
E′ = η = 0 P2 under π.
5 { 5 } ⊂
To describe which points on P2 we must blow up in order to recover S,
we introduce A′ = α = 0 P2 and A′ = α = 0 P2. We note that
1 { 1 } ⊂ 2 { 2 } ⊂
its strict transforms A ,A under π on S intersect E ,...,E as describeed
1 2 1 7
by Figure 1, where A ,A ,E meet in one point which maps under π to
1 2 7 0
(1 : 0 : 0 : 0 : 0) S. Given E′,E′,Ae′,A′ P2 as above, we may now
∈ 5 7 1 2 ⊂
perform the following sequence of five blow-ups to obtain S:
blow up the intersection of E ,E ,A to obtain E ;
5 7 2 1
•
blow up the intersection of E ,E ,E to obtain E e;
1 5 7 2
•
blow up the intersection of E ,E to obtain E ;
2 7 3
•
blow up the intersection of E ,E to obtain E ;
3 7 4
•
blow up the intersection of E ,E to obtain E .
4 7 6
•
Here we have renamed E′ to E and A′ to A , and we have used the same
i i j j
names for a divisor and its strict transform in each blow-up in the sequence.
We proceed to establish the claim made in 1.
§
10 T.D.BROWNINGANDU.DERENTHAL
Lemma 3. The surface S is not an equivariant compactification of G2.
a
Proof. ToestablishthelemmaweassumeforacontradictionthatS isofthis
typeandapplytheworkofHassettandTschinkel[12]. IfS isanequivariant
compactification of G2 then the map φ :S 99K P2 has to be G2-equivariant,
a a
resultinginanactionofG2onP2whichleavesE′ = η = (α η +α2) = 0
a 7 { 7 − 1 5 2 }
invariant. However, we can check that the two distinct G2-structures on
a
P2 (see [12, Proposition 3.2]) do not leave any irreducible quadric curve
invariant. (cid:3)
We are now ready to derive the promised bijection between U(Q) and
integral points on the universal torsor lying above S. The map ψ given by
(4.3) induces a bijection
ψ :(α ,α ,η ,η ) (η3,α η ,α η2,ηeη ,α η )
0 1 2 5 7 7→ 5 2 7 2 5 5 7 1 7
between
(α,η ,η ) Z2 Z Z α η +α2+η = 0,gcd(α ,α ,η )= 1
{ 5 7 ∈ × >0× 6=0 | 1 5 2 7 1 2 5 }
and
U(Q) = (x :... :x ) S(Q) x = 0 S(Q).
0 4 3
{ ∈ | 6 } ⊂
Note that
max ψ (α ,α ,η ,η )
06i64 0 1 2 5 7 i
H(ψ (α ,α ,η ,η )) = | | .
0 1 2 5 7 gcd( ψ (α ,α ,η ,η ) 0 6 i6 4 )
0 1 2 5 7 i
{ | }
Motivated by the sequence of blow-ups above, we introduce new variables
η := gcd(α ,η ,η ), η := gcd(η ,η ,η ), η := gcd(η ,η ),
1 2 5 7 2 1 5 7 3 2 7
η := gcd(η ,η ), η := gcd(η ,η ),
4 3 7 6 4 7
and in each step transform and rename the previous variables accordingly.
Observe that this gives a bijection
(η′,α) (η(2,4,3,2,3)η ,η(1,1,1,1,0)η η α ,η(2,3,2,1,2)α ,η(1,2,2,2,1)η2η ,η α ),
7→ 6 6 7 2 2 6 7 7 1
which we call Ψ, between
η α +η α2+η η2η3η = 0
:= (η′,α) Z6 Z Z2 5 1 1 2 3 4 6 7
T ( ∈ >0× 6=0× (cid:12)coprimality conditions hold)
(cid:12)
andU(Q). Thecoprimalityconditionsa(cid:12)redescribedbytheextendedDynkin
(cid:12)
diagram of E ,...,E ,A ,A in Figure 1, following the rule that any of the
1 7 1 2
variables η ,α are coprime if and only if there is no line connecting the
i j
divisors E ,A in the Dynkin diagram. Once taken in conjunction with the
i j
equation
T(η′,α) = η α +η α2+η η2η3η = 0,
5 1 1 2 3 4 6 7
thatissatisfiedbytheelementsof ,itiseasilycheckedthatthecoprimality
T
conditions can be rewritten as
gcd(α ,η η ) = 1 (4.4)
1 2 6
gcd(α ,η η η )= 1 (4.5)
2 2 3 4
gcd(η ,η η η η ) = 1 (4.6)
6 1 2 3 5
gcd(η ,η η η η η )= 1 (4.7)
7 1 2 3 4 5
gcd(η ,η η η )= 1, gcd(η ,η )= 1, gcd(η ,η η ) = 1. (4.8)
1 3 4 5 2 4 5 3 4