C OMPOSITIO M ATHEMATICA
K LAUS H ULEK
C ONSTANTIN K AHN S TEVEN H. W EINTRAUB
Singularities of the moduli spaces of certain abelian surfaces
Compositio Mathematica, tome 79, n
o2 (1991), p. 231-253
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Singularities of the moduli spaces of certain Abelian surfaces
KLAUS
HULEK1,
CONSTANTINKAHN1
and STEVEN H.WEINTRAUB2
© 1991 Kluwer Academic Publishers. Printed in the
1Institut für
Mathematik, Universitât Hannover, Welfengarten 1, W-3000 Hannover 1, F.R.G.;2Department
of Mathematics, Louisiana State University, Baton Rouge, LA 70803, U.S.A.Received 22 February 1990; accepted 22 October 1990
Introduction
The space of
principally polarized
abelian surfaces has been studiedextensively.
Gottschling [2, 3],
see also[ 1 ],
has determined thesingularities
of this space.They
occur in codimension 2 and 3. The moduli spaces of abelian surfaces withnon-principal polarization
have been studied much less. It turns out that thereagain
the situation is verycomplicated. However,
in this situation we have anatural notion of level structure. Our main
objects
of interest in this paper arethe moduli spaces
.91 l,p
of(1, p)-polarized
abelian surfaces with level structure,i.e.,
oftriples (A, H, a)
where A is an abelian surface defined over thecomplex numbers, H
is apolarization
oftype (1, p)
onA,
and a is anaccording
levelstructure on A.
(For
definitions see, e.g.,[4].)
Since
singularities
in moduli spaces of abelian surfaces arise from surfaces with non-trivialautomorphism
groups, and since the presence of a level structure breaks many of thesesymmetries,
one canhope
to be able toclassify
the
singularities
of the spacesA1,p.
Our aim is to show that this is indeed thecase. In
fact,
we determine thesingularities
of a suitable toroidal com-pactification
of.91 l,p.
For
simplicity
we shallalways
assume that p is aprime
and thatp ~
2. In order to describe our results we introduce some notation which we will usethroughout
the paper. Letbe the
Siegel
upperhalf space of degree
2.Then,
thesymplectic
group defined withrespect
to the standardsymplectic
formJ,
acts
properly
Ydiscontinuously
on2 Namely, g
9 =C B) c- Sp(4, Q), where .4,
B, C,
D denote 2 x 2matrices,
mapsZ~J2
to(AZ
+B)(CZ
+D)-1.
We alsonote that
if g
issymplectic
and of thisform,
then its inverse can beexpressed
asIn
Sp(4, Q)
we consider the arithmeticsubgroup
of
symplectic
transformations which preserve(1, p)-polarizations
and levelstructures.
By taking
thequotient Of Y2
withrespect
to the left actionof rl,p
weobtain
the moduli space
of (1, p)-polarized
abeliansurfaces
with level structure.d l,p
hasat most
quotient singularities.
We denoteby A*1,p
a suitable toroidal com-pactification
of.91 l,p
which we shallbriefly
describe at thebeginning
of section 2.See
chapter
1 of ourforthcoming
book[5]
for details about this com-pactification
which is intended to be ananalogue
for theIgusa compactification
of the moduli space of
principally polarized
abelian surfaces.Denote
by r a b)~SL(2, Z)|a
= d -1(p),
b ~ c =0
theprin-
cipal
congruencesubgroup of
level p inSL(2, Z)
which acts onJ1
={z~C| Im z
>01.
Thequotient X°(p)
=F,(p)BY,
is called(open)
modularcurve
of
level p. Its naturalcompactification
to a smooth curveby adding points
at the
(p2 -1)/2
cusps is denotedby X(p)
and also called modular curveof
level p.A non-isolated 3-dimensional
singularity
is said to be of transversal type ’X’along C,
where C is a smooth curve, if itlocally
isisomorphic
to aproduct
of Cand an isolated surface
singularity
oftype
’X’. The surfacesingularities
ofinterest to us are the rational double
point AI
which isisomorphic
toC’/l ± 12},
and the rational
triple point C3,1
which isisomorphic
toC2/{03C1k12|k
=0, 1, 2},
p =
e203C0i/3.
Thesingularity types Ai
resp.C3,1
occur at the vertices of the affinecones over the rational normal curves of
degree
2 inP2
resp. ofdegree
3 inP3.
An isolated 3-dimensional
cyclic quotient singularity
is oftype ’1 n(q1, q2, q3)’
if it is determined
by
the action of thediagonal
matrixdiag(03B6q1, (Q2, 03B6q3), 03B6 = e203C0i/n,
as a
generator
ofZn .
Note that acyclic quotient singularity
oftype 1 2(1, 1, 1)
occurs at the vertex of the cone over the Veronese surface in
P s.
See[8, §1]
fordetails.
We summarize our results in
THEOREM
(2.15, 3.4). sli,p
contains twodisjoint
curvesC*1
andCi isomorphic
to
X(p)
such thatA*1,p is singular
with transversalA1-type along Ci,
and withtransversal
C3,1-type along C*2.
Thecomplement A*1,p - (CT u C*2)
containsonly
isolated
cyclic quotient singularities
which ail lie on theboundary A*1,p - A1,p, namely (p2 -1)/2 singularities of each of the
types1 2(1, 1, 1)
and1(1,2, 1).
The first section is devoted to
characterizing
thesingularities
in the moduli spaceA1,p
in casep
5. Indoing
so werely
on workof Gottschling ([2, 3])
andUeno
([9]).
In section 2 westudy
the toroidalcompactification A*1,p of A1,p
inorder to determine its
singularities
on theboundary sli,p - si l,p.
In the finalsection we treat the case p = 3 which is
slightly
différent fromp
5.1.
Singularities
in the moduli spaceA
1,pWe first determine the
isotropy subgroups
contained in03931,p corresponding
tofixed
points
ing 2
up toconjugacy,
thenstudy
theirrespective
fixedvarieties,
and
finally
characterize thecorresponding quotient singularities
in.91
1,p .DEFINITION 1.1. The
following
four matrices are elements of finite order in03931,p:
PROPOSITION 1.2.
Every
non-trivial elementof finite
order inrl,p, p 5,
isconjugate
with respect toSp(4, Z)
toexactly
oneof the following eight
matrices:(a)
Involutions: S and T.(b)
Elementsof
order 4: U andU-1.
(c)
Elementsof
order 6: V andV-1.
(d)
Elementsof
order 3 :V2
andV - 2.
Furthermore, U2 = V3
= S.Proof. Gottschling
has determined theconjugacy
classes of all elements of finite order inSp(4, Z) ([2];
see[9, 1, §2; II, Appendix]
forrepresentatives
of all56
conjugacy classes).
Sincerl,p
is asubgroup
ofSp(4, Z)
weonly
need tocharacterize which of these classes contain elements of
r 1,p .
Two necessary conditions aregiven by
LEMMA 1.3.
Suppose
that M ESp(4, Z)
isconjugate
to an elementof 03931,p
and letXM(À)
=det(M - 03BB14)
be the characteristicpolynomial of
M.Then,
(1) Xm(Â)
is divisibleby (À - 1)2
modulo p;(2) Xm(Â)
is divisibleby
À - 1 modulop2.
Proof.
We may assumeM ~ 03931,p.
With thediagonal
matrix F =diag(l, 1, 1, p),
we obtainfrom which both statements of the lemma are easy consequences. D We continue with the
proof
ofProposition
1.2. For the convenience of the reader wereproduce part
of the classification of matrices of finite order inSp(4, Z) according
to their characteristicpolynomials
from Ueno[9, II,
p.198]:
Applying 1.3(1)
withp
5yields
that the classes1(2), 11(2), 11(3), 11(4), III(1), 111(3), 111(6)
and all of the classesIV(k)
with thepossible exception only
ofIV(4)
in case p = 5 do not contain elements of
rl,p. Moreover,
classIV(4)
can beexcluded even if p = 5
by 1.3(2).
Of the
remaining classes, 1(1)
is theidentity. II(1)
contains two involutions which areconjugate
to S and T of(a).
For the elementconjugate
to S this isobvious,
for the otherrepresentative given by
Ueno we haveFinally, 111(2), 111(4),
and111(5) correspond
to(d), (c),
and(b), respectively.
Inthese cases Ueno’s
representatives
are those of theproposition.
DPROPOSITION 1.4. Let
Z ~ J2. Then, the isotropy subgroup {g
e03931,p | g(Z)
=Z}
is either trivial orconjugate
with respect toSp(4, Z) to
oneof the four cyclic
groupsgenerated by S, T, U,
and MProof. Suppose
thatG ~ 03931,p
is a non-trivialisotropy
group. It is shown in[2,
Proof of Lemma2]
that withrespect
toGL(4, C)
the group G isconjugate
toa group of
matrices (X 0) with unitary X E U(2). We may thus identify
G with
a
subgroup
G’ ofU(2). Moreover,
sinceby
1.2 everyelement ~
1 of G has + 1 as aneigenvalue
ofmultiplicity 2,
all elements of G’ arepseudo-reflections, i.e.,
oneof their
eigenvalues
is + 1.We claim that a finite
subgroup
ofU(2)
which containsonly pseudo-
reflections is
cyclic.
LetA,
B EU(2)
be two non-trivialpseudo-reflections
suchthat AB is also a
pseudo-reflection.
We may assume A= (1 0) and
B =(bij)
with
eigenvalues
1 and03B2.
Since AB is apseudo-reflection
anddet(AB)
=03B103B2,
wefind
tr(AB)
=b11
+03B103B222
= 1 +03B103B2.
Wtihtr(B)
=b11
+b2 2
= 1+ 03B2
thisimplies (a - 1)(03B2 - b22)
=0,
henceb22
=03B2
andb11
=1,
and hence B= (1 0)
because B is
unitary.
It follows that A and Bgenerate
acyclic
group which isenough
to prove the claim.Consequently,
G isSp(4, Z)-conjugate
to one of thecyclic
groupsgenerated by
the elements listed in 1.2.
Finally,
observe that if G containsgV2g-1, g
eSp(4, Z),
then also
gVg-1
is inG,
as Yand y2
have the samefixed-point
set onJ2.
Wehave to show that
g Yg -1 E r
1,p which follows from V =2 314
+2 3V2 - 1 3V4
since itis
easily
seen that2 3A + 2 3B - te
eSp(4, Z)
lies in03931,p
ifA, B,
C are inFi p. (Here
we have used
3 f
p. The statement,however,
remains true for p =3,
cf. theproof
of
3.3.)
DDEFINITION 1.5.
(1)
Denote the fixed varietiescorresponding
to the four non-trival
isotropy
groups ofProposition
1.4by
(2)
Therespective images
of the fixed lociK1, Yf 2, b1, and b2
under thenatural
projection
fromJ2
onto.9Il,p
will be calledHl, H2, Ci,
andC2.
REMARKS 1.6.
(1)
Thepoints
ofH1 c .9Il,p correspond
to(1, p)-polarized
abelian surfaces A which are determined
by period
matrices of the form( /1 0 ïi 0B and
which are henceproducts of elliptic
curves A ~Et!
xE’03C43
whereE03C4 = C/(Z
+Z03C4)
andE’03C4
=C/(pZ
+Zr).
These surfaces carryproduct polariza-
tions which are trivial on the first factor and of
degree p
on the second factor.(2)
Thepoints of H2 ~ A1,p correspond
to(1, p)-polarized bielliptic
abeliansurfaces, i.e.,
suitable covers of Jacobians ofgenus-2
curves which admit anelliptic
involution. For aprecise
definition see[6].
(3)
The surfacesJe;,
resp.Hi,
i =1, 2,
areexamples
of Humbertsurfaces
in thesense of
[1]
or[5].
(4) Any
abelian surface which admits a non-trivial involutioncorresponds
toa
point
inHl u H2 ([6, Proposition 2.3]).
Sinceby
1.2 every non-trivialisotropy
group in
03931,p
contains an involution this shows that.9Il,p
isnon-singular
outside
H1 ~ H2. Proposition
1.4implies H1
nH2
=0.
(5)
The curvesCl
andC2
inH1 parametrize product
surfaces in the sense of(1)
of the formEi
xE’03C4
resp.Ep
xEz.
Since the second factor is endowed with alevel-p
structure,Cl
andC2
areisomorphic
to the(non-compact)
modular curveX°(p)
=03931(p)BJ1. Proposition
1.4implies Cl
nC2
=0.
All elements in
03931
,pconjugate
to either U or V lead in.9Il,p
toimages
of theirrespective
fixed loci whichparametrize
abelian surfacesadmitting
non-involutory automorphisms.
Thefollowing proposition shows, however,
that no other loci arise in this way than the curvesC1 and C2. Namely,
everySp(4, Z)-
translate of
Wi
cJ2
which is the fixed locus of an element inrl,p
ismapped
onto
Ci
in.9Il,p,
i =1,
2.PROPOSITION 1.7.
If
X E03931,p
isconjugate
to U(resp. V)
with respect toSp(4, Z),
then it is alsoconjugate
to U(resp. V)
with respect tor 1 ,p.
Proof. Suppose
that X =gUg-1 Er 1,p
with g =(gij)
ESp(4, Z).
We construct~ 03931,p
such that X= U-1 by letting g
=gh
andchoosing h ~ Sp(4, Z) appropriately.
SinceU2 =
S it follows fromgUg-1 E rl,p
thatgSg-1 Er 1,p
andthis
implies
Because of
det(g)
=1,
aproperty
ofsymplectic matrices,
at least one of theremaining
three 2 x 2 minorslying
on the first and third columnsof g
must be~ 0(p).
But thennecessarily
g21 ~ g23 ~OM’
Fromdet(g)
= 1 it then follows that at least one of g22 and g24 is invertible modulop2.
We can findrelatively prime integers a
and c such that ag22 + cg24 =1(p2)
and hence there exists amatrix
(a b) ~ SL(2, Z) such that (g22,g24). (a b) ~ (1, 0)(p2). Consequently,
where we let
It follows
that g
=gh
lies in03931,p. (The remaining
congruence conditions can be deduced from-1
=14 using
the factthat g
issymplectic.)
Since h commuteswith U we also
have U-1 = gUg-1 =
X. We have thus proven the assertion about U. For Y one argues inexactly
the same way. 1:1We state the main result of this section
using
thepreceding
definitions:THEOREM 1.8. Assume that
p
5.Then,
the modulispace A1,p
issingular
withtransversal
Al-
resp.C3,,-type along
the smooth curvesCI
andC2.
Thecomple-
ment
A1,p-(C1 ~ C2)
isnon-singular.
Proof.
All that remains to beinvestigated
is how theisotropy
groups ofProposition
1.4 act onIf 2
withrespect
to local coordinates around their fixedpoints.
Ingeneral,
ifZo
=(03C41 T2 )
is apoint
inIf 2 having
a non-trivialisotropy
group we will introduce local coordinates(x,
y,z)
in aneighbourhood
of 0~C3 by Z = (03C41+x 03C42+y). For 03C42+y 03C43+z) arbitrary fixed points of S, T, U,
and V
taken from the fixed loci
YCI, Je 2’ b1, and W2 (cf.
Definition1.5)
we findby straightforward computation
Hence,
withrespect
to local coordinates andneglecting higher
order terms,i.e., looking
at thetangent
space instead of aneighbourhood
of the fixedpoint
and thus
linearizing
theaction,
we obtainThe two involutions
locally
actby
reflections-for T this iseasily
seenby checking
the characteristicpolynomial
of the matrixdescribing
the linearizedaction-,
henceby
a well known theorem ofChevalley
thequotients by
theiractions are smooth. The two
cyclic
groupsgenerated by
U and V both contain thecyclic
group of order 2generated by
S.Dividing
out this reflection group weget cyclic
groups of order 2 resp. 3 which in the(X, Y)-plane give
rise toquotient singularities of type A
resp.C3,1
while in the Z-directionalong
the curverc
b1resp.
rc 2
the action is trivial. Hence the claim. D2.
Singularities
on theboundary
ofA*1,p
In this section we are concerned with the
singularities
on theboundary
of aparticular
toroidalcompactification di,p
of the moduli spaceA1,p.
In ourconstruction of
A*1,p
we mimic thedescription
of theIgusa compactification
inthe
principally polarized
case which Namikawa hasgiven
in[7].
A brief outline of it follows.However,
since ourquestion
is local in nature, not much will be said about theglobal
structure ofA*1,p,
nor will there beproofs
since a detailedexposition
will begiven
in[5].
We refer to[7]
forgeneralities
on toroidalcompactification.
The
starting point
for thecompactification
of .91 1,p is the Titsbuilding
of r 1,p agraph
made up of therB p-conjugacy
classesof parabolic subgroups of r 1,p,
orequivalently
therl,p-cosets
of non-trivialisotropic subspaces of Q4
withrespect
to the
symplectic
formJ,
whereedges
are drawn for inclusion relations.PROPOSITION 2.1
([5]). (1)
Therl,p-equivalence
classesof isotropic
lines inQ4
arerepresented by
the line10 generated by (0, 0, 1, 0)
and the(p2 -1)/2 lines 1(a,b) generated by (0, alp, 0, b)
where a and b arerelatively prime integers representing (a, b)
E(Zp
xZp - {(0, 0)}) ±
1.(2)
The03931,p-equivalence
classesof isotropic planes
inQ4
arerepresented by
thep + 1 planes h[a:b] spanned by 10
and1(a,b),
where[a : b] ~ P1(Zp).
(3)
The Titsbuilding of 03931,p
is thefollowing graph
Then, A*1,p
is obtainedfrom .91 l,p by adding boundary
components atinfinity
which are indexed
by
the vertices of the Titsbuilding.
There is a one-to-onecorrespondence
betweenboundary components
of codimension k andrl,p-
classes of
isotropic subspaces
of rank k inQ4
withadjacency
ofboundary components corresponding
toedges
in the Titsbuilding.
Inparticular, (with (a, b)
and[a : b]
as in2.1 ):
where the
Dl ’s
are open, irreduciblesurfaces,
and theEh’s
areconnected, compact
curves which are reducible.By D,
we denote the Zariski closureof D03BFl
inA*1,p.
Thecomponent Dlo
is called centralboundary component
while the othercomponents Dl(a,b)
are referred to asperipheral boundary components.
Let r be an
isotropic subspace
inQ4. Then,
theboundary component
associated with r can-in thepresent
context-be described as follows: LetPr c I-’ 1,p
be theparabolic subgroup stabilizing
r. InP,
there is anintrinsically
defined normal
subgroup P’r ~ P,
such that N =P’rBJ2
is a toroidal"neigh-
bourhood
of infinity."
Thepartial compactification
ofA1,p
in the "direction r" is definedby
the choice of a suitable toroidalembedding N
of N such that the action ofPr - Pr/Pr
on N extends toN. Then, P"rB(N - N)
is theboundary component belonging
to r, andP"rBN
will become an openneighbourhood
of itin
A*1,p. (Of
course,compatibility
conditions need to be satisfied for the variouspartial compactifications
toglue together.)
Concretely,
in our situationP’ is always conjugate
to a latticeconsisting
ofmatrices
(12 B)~03931,p, B = tB which
acton J2 by
translations Z~Z+ B,
so that the
quotient
maper:J2 ~ P’rBy2
isconveniently expressed by
expon- ential functions. N =P’rBJ2
islocally
aroundinfinity isomorphic
to eitherJ1
x C x C* if r is anisotropic line,
or to(C*)3
if r is aplane.
In the first case wesimply
choose the trivialpartial compactification J1
x C x C while in the second case we need to be more careful-see 2.11 below. In either case we obtaina smooth space
N
on whichPr
actsproperly discontinuously,
so in order to findthe
singularities
ofA*1,p
on theboundary component
associated to r it is sufficient to consider the fixedpoints
of theP"r-action
onN lying
onN -
N.We also note that under the
larger
group039303BF1,p
ofsymplectic
transformationspreserving only (1, p)-polarizations
all linesl(a,b)
areequivalent,
and allplanes h[a:b]
areequivalent.
Since wenaturally
choose thepartial compactifications belonging
to thesesubspaces
to be identifiedaccordingly,
we may restrict ourselves tolooking
atA*1,p locally
aroundDi, D4o.1)’
andEh[0.1].
In
Propositions 2.2, 2.5,
and 2.11 below we summarize the technical details of the construction of thesecomponents.
Thecomputations
involved arealways
straightforward
but sometimeslengthy.
PROPOSITION 2.2
([5]).
Thestabilizing subgroup of 10
inhl,p
iswhere entries ’*’ are determined
by
the conditionsof symplecticity.
It contains the latticeP’lo = {(12 B)|B
=q 0 q~Z}. Hence, the corresponding
toroidal
neighbourhood of infinity
is the image of g 2
under the map
An open
neighbourhood of Dîo
inA*1,p
is then obtained as thequotient of
aneighbourhood of {0}
x Cx g
1by
the induced actionof Pl’ 0
=P,.IP’,o
onC x C
J1.
I nparticular,
we mayidentify P"l0
and its action on C x CJ
1 asfollows:
REMARK 2.3. It follows that
D03BFl0
isisomorphic
to the open Kummer modularsurface K’(p)
which is defined to be thequotient
of the Shioda modular surfaceS°(p) by
the involutionacting simultaneously
on all fibersof S°(p) by
the natural involution x~-x ofelliptic
curves.K°(p)
is a smooth surface ifp
2.PROPOSITION 2.4.
A*1,p is non-singular locally
aroundD03BFl0.
Proof. Suppose
that we have(tl,
i2,!3)ECxCxgl
andThe invariance of T3
implies
that(1/p)03C43
is a fixedpoint
of(a b)
e03931(p) acting on Y 1.
It follows that(a b) = 1
since03931(p)
hasonly
trivial fixedpoints
on.
If e =
+ 1,
then invariance of r2implies m
= n =0,
andhence 9
= 1. Nowassume that 03B5 = -1.
Then,
from the second coordinate we obtain -CI =1 2(m03C43
+pn).
The fixedpoints arising
in this way arejust
those of the"Kummer involution"
(cf. 2.3)
and itsPl’-conjugates, i.e.,
the 2-divisionpoints
inthe
i2-plane
withrespect
to the latticepZ
+ZT3.
Theelement g
actsby
which
locally
around !2 =1 2(m03C43
+pn)
isequivalent
to a reflection.Hence,
thequotient
space is smooth. DPROPOSITION 2.5
([5]).
Thestabilizing subgroup of 1(0,1)
inrl,p is
where entries ’*’ are
determined by
the conditionsof symplecticity.
1 t contains thelattice
P’l(0,1) = {(12 B)|B = (0 0), q~Z}. Hence,
thecorresponding
tor-oidal
neighbourhood of infinity
is theimage Of J2
under the mapAn open
neighbourhood of D03BFl(0,1)
ind!,p
is then obtained as thequotient of
aneighbourhood of J1
x C x{0} by
the induced actionof PÍ;O.l)
=Pl(0,1)/P’l(0,1)
onY,
x C x C. Inparticular,
we mayidentify Pl’ (0’,)
and its action onY,
x C x C asfollows:
REMARK 2.6. It follows that
D03BFl(0,1)
isisomorphic
to the open Kummer modular surfaceK’(1)
which contains foursingular points.
DEFINITION 2.7. For i =
1, 2,
denoteby Ci
the Zariski closure of the curveCi
inA*1,p. (Cf. 1.5(2)
for the definition ofCi.)
PROPOSITION 2.8. There are
precisely four singular points of A*1,p
onD03BFl(0,1),
namely (1)
Two non-isolatedsingular points Q,
andQ2, represented by
thepoints (i, 0, 0)
and(p, 0, 0),
p =e21tiI3, o f J1
x C x C in thesetting of 2.5, respectively.
Fori =
1, 2,
thepoint Qi
is thepoint of
intersectionof Ci
withD03BFl(0,1)
whereCi
issmooth at
Qi.
Thesingularities of A*1,p
around thesepoints
areof
transversalA1-
typealong Ci
andof
transversalC3,1-type along C2.
(2)
Two isolatedcyclic quotient singularities Q’1 and Q2
which arerepresented by (i, p 2(1 - i), 0)
resp.( p, 23(l - p), 0) in J1
x C xC,
and which areof type 1 2(1, 1, 1)
resp.
1 3(1, 2, 1). (Cf.
theintroduction.)
Proof. Suppose
that we have(il,
i2,t3)E!/1
x C x C andFrom the first coordinate we read off that g0 =
a b E SL(2, Z) acting on fil 1
leaves il
fixed. We then either have go =+ 12
which acttrivially
onJ1,
or03C41 = i g0 = ±(0
10’ or ! 1 = P and go E :t 1
1-1), ±(-1 0
The case
of go = ±12
can be dealt with inexactly
the same way as in theproof
of 2.4. Since it is easy to see that go and
go 1 give
rise to the samesingularities,
forthe
remaining
cases wejust
have to look into thefollowing
threepossibilities:
a
-cl =i,
g0 =0
-1). Then,
T2 =m
+n)(1 - i)
- n and modulo theP"l(0,1)-action
thesepoints belong
to the two orbitsrepresented by
i2 = 0 and03C42 = p 2(1 - i).
Let i2
=0, i.e.,
m = n = 0. Thiselement g
leaves all thepoints
fixed in{i}
x{0}
x C which contains theimage
ofb1.
Withrespect
to local co- ordinates introducedby (i
+x, y, z)
around(i, 0, 0)
the actionof g
isg:
(x,
y,z)
~( -
x, -iy, z)
+ ..- up to first order. Hence the claim about transver- salA1-type along Ci locally
aroundQl.
Now let i2
= p 2(1 i ), i.e.,
m = 1 and n = 0. In local coordinates defined around(i, p 2(1 - i), 0) by (i
+x, p 2(1 - i )
+y, z)
the actionof g
is(x,
y,z) H
(-x, p 2(1 - i)x - iy, - z)
+ - - - which after achange
of base isequivalent
to(, , )~(-, i, -).
Afterdividing
out the reflection inducedby g2
we seethat we
get
acyclic quotient singularity
oftype 1 2(1, 1, 1)
atQ’1.
(b)
i = p, go =1 1 . Then, 12 = p(m - n) -
pnp, hence moduloPl’ (0.1)
allthese
points
areéquivalent tO’r2
=0, corresponding
to m = n = 0.Again
we finda
point-wise
fixed curve{03C1}
x{0}
x Ccontaining
theimage of b2.
Withcoordinates defined
by (p
+ x, y,z)
around(p, 0, 0)
weget (x,
y,z)
~(p2x, -
py,z)
+ ..- for the action of g.Hence,
theresulting singularities
are of transversal
C3,1-type along Ci.
(c)
i = p, g0 =(-1
1-1)
.Then,
2= p 3(m
+n)(1 - P)
pn, and moduleP"l(0,1)
thesepoints belong
to twoorbits, namely
thoserepresented by
12 = 0 and i21 fll - p). (Note
that the orbitsof p 3(1 - p)
and2p/3(1 - p)
coincide. Thiscan be seen
by applying
the Kummer involution03C42 ~ -03C42.)
In the case i2 = 0 thecorresponding element g
is the square of the one considered in(b)
above.Hence,
let 12= p 3(1- p),
and define local coordinates around(p, p 3(1 - p), 0) by (p
+x, p 3(1 - p)
+ y,z). Then,
g:(x,
y,z)
~(03C1x, -p 303C1(1 - p)x
+03C12y, pz)
+ ... andthis defines a
cyclic quotient singularity
oftype 3(1, 2, 1)
atQ2
as claimed. DNow that we know the structure of
di,p
around the open codimension-1boundary components
it remains tostudy
aneighbourhood
of the codimension- 2boundary component Eh[0.1].
Unlike in the twopreceding
cases wecannot just
use a trivial
embedding
of the toroidalneighbourhood.
Beforegiving
the detailswe state a few facts about torus
embeddings
and define the fandetermining
theparticular
torusembedding
which we aregoing
to use.Let T xé
(C*)’
be analgebraic
torus of rank r, and denoteby
M =Hom(T, C*)
and N
= Hom(C*, T)
itsrespective
groups of characters and1-parameter subgroups.
M and N are lattices of rank r,naturally
dual to eachother,
withassociated real vector spaces
MR
=M ~Z
R andNR
=N ~Z
R. If 03C3 ~NR
is acone defined
by
rationalhyperplanes,
then defineX03C3
=Spec C[03C3~
nM]
whereC[u’ n M]
is thesemi-group ring
definedby
U’ ={x~MR|x·y
0 for ally~03C3},
the cone dual to 6. Moregenerally,
let E bea fan (also
called a rationalpartial polyhedral decomposition)
inNR, i.e.,
a collection of cones which do not contain linearsubspaces
such that with u c- 1 all faces of 6 are inE,
and such that 03C31 n u 2 is a common face for any two 03C31, (J 2 E E. If 6’ is a face of 6 EE,
a naturalopen inclusion
X03C3’
4X03C3
is inducedby
03C3’~ ~ 03C3~.( T
=Xjoj X03C3
is aspecial
case.)
The torusembedding X,
associated to L is defined to be the scheme obtained frompatching together
allX03C3,
u EE, using
the identificationsX03C3’ X 6 coming
from faces a’ c 6.DEFINITION 2.9
(Cf. [7, §7]).
Denoteby Symm(2)
the 3-dimensional real vector spaceof symmetric
2 x 2matrices,
andby Symm + (2)
the cone ofpositive
definite matrices therein.
Then,
the 2nd Voronoidecomposition Y-
ofSymm+(2)
isdefined to be the fan
consisting
of allGL(2, Z)-translates
of the conetogether
with allrespective faces,
whereGL(2, Z)
acts onSymm(2) by g(M)
=tg-1Mg-1
forg E GL(2, Z)
andM E Symm(2).
REMARK 2.10
([7, (6.14)]).
Let N be the latticeof integral
matrices inSymm(2).
Then,
the 3-dimensional torusembedding X03A3
associated with the 2nd Voronoidecomposition 1
is smooth.Furthermore,
the action ofGL(2, Z)
on theembedded torus T = N Q C* induced from the
GL(2, Z)-action
on Nnaturally
extends to an action on
X03A3. (This
is so becauseg(03C3)~03A3
holds for every g eGL(2, Z)
and a eE.)
PROPOSITION 2.11
([5]).
Thestabilizing subgroup of h[0:11
inrl,p
iswhere
The
corresponding
toroidalneighbourhood of infinity
is theimage of f/ 2
underThe group
P"h[0:1]
=Ph[0.1]/P’h[0:1]
can beidentified
withby sending
the 2 x 2 matrix A in thedescription of Ph[0.1]
above toFAF-1
whereF =
1 0).
With respect to coordinates(t1,
t2,t3)
onT,
G then acts on Tby
This action
corresponds
to the actionof
G cGL(2, Z) given
in 2.9if
the latticeof integral
matrices inSymm(2)
isidentified
withHom(C*, T) by sending
el e2e2
e3to the
1-parameter subgroup
03BB~(03BBe1, 03BBe2, 03BBe3).
An open
neighbourhood of Eh[0:1]
indi,p
is then obtained as thequotient of
aneighbourhood of X03A3 -
T inX03A3 by
the induced actionof G,
where TX03A3
denotes the torusembedding
determinedby
the 2nd Voronoidecomposition
E.PROPOSITION 2.12.
1 p is non-singular locally
aroundEh[0:1].
Proof. (I)
Thedescription
for the action of G on Tgiven
in 2.11 allows todefine an action of
GL(2, Z)
onT,
andby
definition of X this action extends toX03A3.
In order to prove smoothness ofW*, 1 p
aroundEh[0:1]
it suffices to show thatevery non-trivial
isotropy
group in G of apoint
inXE -
T isgenerated by pseudo-reflections.
We will prove thestronger
assertion that everyg ~
±12 in GL(2, Z)
which isconjugate
to an element of G and has a fixedpoint
onXE -
Tacts
locally
like a reflection.Let u, c- 1 be defined as in 2.9. We first observe that it is sufficient to consider
only
fixedpoints
in the affinepiece X03C30
cX03A3
because everypoint
x~ X03A3
lies insome
Xg(03C30),
g EGL(2, Z),
and there is a naturalisomorphism
g:X (10 Xg(03C30) by
which fixed
points
inXg(03C30) correspond
to fixedpoints (with respect
toconjugate
group
elements)
inX03C30.
SinceX03C30
isisomorphic
toC3
with(C*)3 being
theimage
of
T,
we have to look for fixedpoints lying
on the three axes C x{0}
x{0}, {0}
x C x{0},
and{0}
x{0}
x C.However,
since thestabilizing subgroup
of thecone 6o in
GL(2, Z) permutes
thegenerators
of Uo[7, (8.7)],
and hence alsopermutes
the three axes inX03C30, by
a similarargument
it suffices to consideronly
one
of them,
e.g.,{0}
x{0}
x C. Ourproblem
nowsplits
up into twoparts:
Fixedpoints
in101
x{0}
x C*representing generic points
ofEh[0:1],
and(0, 0, 0)
as afixed
point, representing
so-calleddeepest points
ofEh[0:1].
Before
treating
the two casesseparately,
we make two observations which will become useful.Firstly,
note that if x~ X03C30
is a fixedpoint of g
EGL(2, Z),
thenthere exists an open
neighbourhood W
of x inX03C30
such thatg(W)
cX03C30. (A
direct consequence of
continuity.)
In order to make use of this fact we need tounderstand where
points
ofX03C30
aremapped
toby
elementsg E GL(2, Z).
Conveniently using
the coordinate functions tl, t2, t3 on T forgenerators
of the lattice M =Hom(T, C*)
we find that the dual cone03C3~0
isgenerated by
thecharacters tlt2,
t2t3,
andt2
1 in M. If we use them torepresent
coordinate functionsTi, T2, T3
onX03C30 ~ C3,
then the actualembedding
of the torus T inX03C30
isexpressed by
For g = a b ~ GL(2, Z)
weformally
define~g = l03BFg03BFl-1
as a rationalmorphism
fromXao
to itself. Wherever ~g is a map it describes thepart
of theaction
of g
that takesplace
insideX ao. Concretely,
The second observation we want to make is the
following
lemma which is a trivial consequence of the definition of G in 2.11:LEMMA 2.14.
If g E GL(2, Z) is conjugate
to an elementof G,
then its characteristicpair (tr(g), det(g))
is either congruent to(0, -1)
or to(2, 1)
modulo p.(II)
Fixedpoints
on{0}
x{0}
x C*. Assume that thepoint (Ti, T2, T3)
withT1 = T, =
0 andT3 e
0 is fixedby g
=t b). Using
the fact thatis
definedin an open
neighbourhood
of the fixedpoint,
we obtain fromT,
= 0 and(2.13)
that bd =
0,
hence b = 0 or d = 0.Similarly, T2
= 0implies
ac =0,
so a = 0 orc = 0. Of these four cases,
obviously
those where a = b = 0 or c = d = 0 cannot lead to invertible matrices. The cases b = c = 0 resp. a = d = 0together
withdet(g) =
+ 1 lead to thefollowing eight possibilities:
Finally,
2.14 shows thatif g
EGL(2, Z)
isconjugate
to an element of G and has a fixedpoint
on{0}
x{0}
xC*, then g
must be one of thefollowing:
We shall describe the action of these matrices on
X03C30
inpart (IV)
below.(III)
The"deepest point" (0, 0, 0)~X03C30. If(Ti, T2, T2)
withT1 = T2 = T3 = 0 is
fixed
by g
=(a b), then arguing
as inII
we obtain fromT -
0 with2.13
that