C OMPOSITIO M ATHEMATICA
K IERAN G. O’G RADY
On the Kodaira dimension of moduli spaces of abelian surfaces
Compositio Mathematica, tome 72, n
o2 (1989), p. 121-163
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On the Kodaira dimension of moduli spaces of abelian surfaces
KIERAN G. O’GRADY
Department of Mathematics, Columbia University, New York, N. Y., USA and
Dipartimento di Matematica, Seconda Università di Roma, Italy
Received 25 July 1988; accepted in revised form 15 February 1989 Compositio
(Ç) 1989 Kluwer Academic Publishers. Printed in the Netherlands.
It is known
(e.g. [Î])
that the moduli space ofprincipally polarized
abeliansurfaces is rational. On the other hand it follows from
general
results of Mumford[M-1]
that the moduli spaceof p.p.a.s.’s
with a full level-n structure is ofgeneral
type for n
big.
In this paper we prove that moduli spaces ofp.p.a.s.’s
with anintermediate
level-p
structure are ofgeneral
type for pbig.
Moreprecisely
letA2(p) be the
moduli space ofcouples (S, H)
where S is a p.p.a.s. and H cS[p]
a rank two
subspace
of thep-torsion points, non-isotropic
for the Weilpairing (p
is a
prime).
Our main theorem asserts thatA2(p)
is ofgeneral
typeif p 17.
Themotivation for this work came from
studying
moduli spaces of K3 surfaces. LetF2d
be the moduli space of K3 surfaces with aprimitive polarization of degree
2d.For every
n, k
there exists a finitesurjective
mapfn,k: F2n2k ~ :F2k (see
theappendix).
Let’s fixk, say k
= 1. The moduli space:F2
is unirational hence03BA(F2) = -
oo but one istempted
tostudy
the mapsf n,1: F2n2
-F2
in order todetermine the Kodaira dimension of
F2n2
for nbig.
Now letd2,d
be the moduli space of abelian surfaces with apolarization
withelementary
divisors{1, dl;
wethink of
A2,d
asanalogous
toF2d (see
theappendix).
There exist mapsn,k: d2,.2k - d2,k analogous
to the mapsfn,k.
If we set k = 1 and n is aprime
p,then the defintion of
gp,1: A2,p2 ~ A2,1 (= A2,
the moduli space ofp.p.a.s.’s)
identifies
A2,p2
with our moduli spaceA2(p)
and the map gp,1 is identified with the natural map fromA2(p)
toA2.
So the Main Theorem isequivalent
to thestatement that
A2,p2
is ofgeneral type
for p 17(Corollary 5.1);
it suggests thatF2p2
is also ofgeneral type
for pbig.
The
plan
of theproof
of the main theorem is thefollowing.
Letd 2 be the
moduli space of
p.p.a.s.’s;
we choose the(toroidal) compactification A2 of A2 isomorphic
ton2l
the moduli space of stable genus two curves. In Section 1 weestablish some relations between divisor classes on
A2.
Let 03C0:A2(p) ~ A2
be themap obtained
by associating
to thecouple (S, H)
the surfaceS,
i.e.by forgetting
the p-structure. We define
A2(p)
to be the natural toroidalcompactification
ofA2(p)
such that 03C0 extends to a finitesurjective
map 03C0:A2(p) ~ A2.
In Section2 we
apply
Hurwitz’s formula to 03C0 in order to get anexpression
for the canonicalclass of
A2(p).
Not allsingularities
ofA2(p)
arecanonical,
i.e. some of them"impose
conditions onadjoints".
In Section 3 we construct apartial desingular-
ization
2(p)
ofA2(p)
all of whosesingularities
are canonical. In Section 4 weshow that
h0(nK2(p)) Q(p)n3
+O(n2 )
for nsufficiently divisible,
whereQ(p)
> 0for p
17. Hence tr.deg. ~~n=0H0(nK2(p))
= 4 for p17; since Â2(p)
iscanonical
d2 (P)
is ofgeneral
type(p 17).
It’s a
pleasure
to thank Joe Harris for crucialhelp during
the initial stage of this work. Thanks also go toHenry
Pinkham forsuggesting
to useFujiki’s algorithm
for
resolving cyclic quotient singularities.
NOTATION: Let S be an abelian
surface,
thenS[n]
will be thesubgroup
ofn-torsion
points.
Let S be a p.p.a.s.
(or
let C be a genus twocurve);
let - 1 EAut(S)
bemultiplication by - 1 (respectively
let t: C ~ C be thehyperelliptic involution),
then
We will refer to Aut’
(S) (Aut’(C))
as the reduced group ofautomorphisms
ofS
(respectively C).
By elliptic
curve we mean a curve of arithmetic genus one with at most onenodal
singularity.
Welet j(E)
be the usualj-invariant of E,
ifj(E)
=0,
E ~C/Z
+7Le1ti/3,
ifj(E)
=1728,
E ~C/Z
+7Li,
ifj(E)
= oo, E issingular.
(g1’...’ gn )
will be the groupgenerated by
g1, ..., gn .U(x,
y,...,z)
will be the affine space with coordinates x, y, ... , z. en denotes aprimitive
nth root ofunity.
Let M be the moduli space of a class
of varieties,
letf:V ~
T be afamily
of suchvarieties,
we will denoteby m (sometims
mU ormp)
the induced map from T to M.In
particular
if V is one suchvariety m(V)
E M will be the modulipoint
of V.Section 1. Divisors on
rol2
Let
rol2
be the moduli space ofDeligne-Mumford
stable curves of arithmetic genus two.DEFINITION 1.1.
(i)
Let03940
crol2
be the divisorparametrizing
curves with one(at least) nondisconnecting
node.(ii)
Let03941 B’2
be the divisorparametrizing
curves with onedisconnecting
node.
(iii)
Let02 ~ M2
be the divisor whosegeneric point
is the moduli of a doublecover of an
elliptic
curve.REMARK. The
generic
curve whose modulibelongs
to02
isgiven by
It has two involutions whose
quotient
is anelliptic
curve,namely
t1’ t2’ whereDEFINITION 1.2.
Let, by
abuse ofnotation, 03940, 03941, 1 A2
EPic(M2) Q9 Q
be theclasses of the reduced divisors
03940, 03941, 03942.
REMARK. The
singularities
ofrol2
arequotient singularities
hence every Weil divisor isQ-Cartier,
soDo, 03941, A2
are indeed elements ofPic(012)
~ Q. Ourclasses
0394i’s
aregiven by
the reducedDi’s; they
differ from Munford’s[M-2]
classes
[0394i]Q.
In fact the relation should beDEFINITION 1.3. Let
f:q ~
T be afamily
of stable genus two curves. TheHodge
bundle(over T)
isThe
Hodge
bundle can be viewed as an element of the functorial Picard group ofrol2.
Due to curves withextra-automorphisms
it does not corne from a line bundleon
rol2. A sufficiently
divisible power of the functorialHodge
bundle(a
commonmultiple
of the ordersof automorphism
groups of stable genus two curves willdo)
is the
pull-back
of a line bundle onrol2,
hence we cangive
DEFINITION 1.4. Let A e
Pic(M2)~Q
be theHodge
bundle.DEFINITION 1.5. Let
{Et} (t
e T ~P1)
be a Lefschetzpencil
on a smooth cubicsurface
in ¡p3.
Let F be a fixedgeneric elliptic
curve,i.e. j (F) 1= 0, 1728,
00. LetCt
=Et ~
F be obtainedby gluing Et
and Falong
the zeroes of the group laws(notice
that thepencil {Et}
has three sections so we can choose one as the curve ofzeroes of the
Et’s) .
Letf:q ~
T be theresulting family
of stable genus two curves.LEMMA 1.1: Let m: T ~
rol2 be the
moduli map associated tothe family f:y ~ T, (i) deg m*(03940) = 12,
(ii) deg m*(03941) = -2, (iii) deg m*(03942) = 24, (iv) deg m*(03BB) = 1.
Proof (i)
There are 12singular
fibers in thepencil {Et},
since it is a Lefschetzpencil
the curvem(T)
crol2
is transverse toOo
at eachpoint
of intersection.Hence
deg m*(03940)
= 12.(ii)
Let 9: é -T,03C8:
àK - T be two families ofelliptic
curves. Letg:q ~
T bethe
family
ofcurves {Ct
=E, u F, 1
obtainedby gluing
the zeroes, and let03C3:T 03B5, 03C4:T F be
the sectionsgiven by
the zeroes. Let m : T ~R2
be theassociated moduli map,
by
definitionm(T)
c03941. Following [H-M],
page51,
we have thatIn our case
N03C3(T)/03B5 ~ UT( -1)
andN03C4(T)/F ~ UT
hencedeg m*(03941)
= - 2.(iii)
It is not difficult to check thatm(t)
E02
if andonly
ifEt ~
F. Sincedeg m*(03940)
= 12 there are 12 such values of t.Let to
be such avalue,
let U be theuniversal deformation space
of Cto
=Et0 ~ F,
let m: T ~ U be the map associated to thefamily f: W -+
T and let mU :U ~ M2
be the moduli map.Let Ai ( U)
cU, 02 ( U)
c U be the divisors such thatmU(03941(U))
=03941, mU(03942(U))
=03942;
it is easy to check thatthey
are transverse. Let C = E ~ F be a curve(in
the universalfamily
overU) lying
overAi ( U);
let x =j(E),
y= j (F), they
are local coordinateson
03941(U).
We have that02(U)
n03941(U) = {P c-,àl (U) x(P)
=y(P)j
and(T) =
{P
EA 1 (U) y(P) = j(F)},
hencem(T)
is transverse toA2 (U) -
SincemU:U ~ D2
is ramified with index 2
along 03942(U)
we get thatdeg m*(03942)
= 2·( #{t
ET|
m(t)
EA2
= 24.(iv)
LetAT
be theHodge
bundle of thepencil {Et}, i.e., AT
= ~*03C903B5/T, then03BB’T ~ AT.
An easycomputation gives
thatdeg AT
= 1 hencedeg ÂT
= 1.DEFINITION 1.6. Let E be a fixed
elliptic
curvewith j (E) 1= 0, 1728,
oo. LetS = E x E and let 9:
~
X be the blow up of S at(P, P)
where P E E is the zero ofthe group law. Let
, ~
be the strict transforms of thediagonal A
and of03A3 = {P}
x Erespectively.
Let03C02~: ~ E
be thecomposition
of ç andprojection
on the secondfactor; à and Ï
are sections of thefamily
ofelliptic
curves 03C02~. Let W be obtained
from S by gluing à and 1
in the obvious way sothat we get a
family g:q ~
E of genus two stable curves with one nondiscon-necting
node each. The fiberof g
overQ ~
P is obtained from Eby gluing Q
andP;
the fiber over P is the union of E and thesingular elliptic
curve.LEMMA 1.2: Let m: E ~
«2 be
the moduli map associated tothe family g:q ~ E,
thenProof. (i) By
definitionm(E)
c03940.
We haveSince Â-Â = Ë’Ë = -1
we get thatdeg m*(D0) = -2.
(ii) Obviously m-1(03941)
={P}.
Let U be the universal deformation space ofCP=g-1(P)
and let03941(U)
be the divisorparametrizing
curves with onedisconnecting
node. Let m : E ~ U be the map associated to thefamily g:q ~ E,
is one-to-one and theimage
is fixedby
the action of the extraautomorphism
ofCp.
Hencem(E)
is transverse to the divisor fixedby
thisaction,
i.e.03941(U).
Since the moduli map mu :U ~ M2
is ramified with index 2along 03941(U)
we get thatdeg m*(03941)
= 2.(iii)
It is easy to check thathence
#m-1(03942) =
3. An argument similar to theprevious
onegives
thatdeg m* (A2)
= 6.(iv)
We have the exact sequencewhere R is the residue map. Hence
COROLLARY 1.1.
{03940,03941} is a
basisof Pic(M2) ~ Q.
Proof Igusa [I] proved
thatm2 ~ U(x, y, z)/(g)
whereg*(x, y, z) = (es x, e25y, e35z),
hencePic(m2)~Q~{0}.
Sincem2=2B(03940~03941), Pic(m2) ~ Q
isgenerated by Ao
and03941.
Lemmas 1.1 and 1.2 show thatAo
and03941
areindependent,
hencethey
form a basis.Proof. By
theprevious corollary
we know that A =xAo
+y03941
for somex, y E Q.
Using
Lemmas 1.1 and 1.2 we get x = y =10.
COROLLARY 1.3.
A2
=3Ao
+603941.
Proof.
Same asprevious corollary.
LEMMA 1.3.
Km2 = - 11 503940 - 1eà
Proof.
In[H-M]
a formula isgiven
for the canonical classof R.,
the modulispace of stable genus g curves,
with g
4. The same kind of formula holds forKm2
with an extra contribution from02
since thepoints
inA2
represent curveswith an extra
automorphism.
The formula one gets isTaking
into account Corollaries 1.2 and 1.3 we get thatAnother way
of proceeding
is thefollowing.
We know thatKm2
=x0°
+y03941
forsome x, y E Q.
Igusa’s description
of9X2
via invariants ofbinary
sexticsactually
extends to a
description
ofR2 B03941.
Thus one can checkdirectly
that x= - 11 5.
One can then obtain y
= - 16 5 by applying adjunction
to0° .
Let
d 2
be the moduli space ofprincipally polarized
abelian surfaces.By associating
to a genus two curve its Jacobian we get a map Jac:m2 ~ d 2
whichextends to an
isomorphism
Jac:M2 B0° d2.
DEFINITION 1.7. Let
A2 ~ A2
be thecompactification
ofA2 given by Jac-1: A2
’+012 (i.e. IF12 = rol2).
IMPORTANT REMARK. The
compactification FI2
is a toroidal compact-ification of
S2[n].
We will
identify rol2
andj7f2
via theisomorphism
Jac:rol2 ~ A2.
Inparticular
we will denote
by 0°, 03941, 02
the divisor classesJac(03940), Jac(03941), Jac(03942)
EPic(A2)
0 Q. Notice that03941 c FI2
is the closure of the locus of moduli ofp.p.a.s.’s (S, e)
with anelliptic
curve E c S such that E ·0398 = 1.Similarly A2 r- -FI2
is the closure of the locus of moduli of
p.p.a.s.’s (S, 0)
with anelliptic
curveE c S such that E.e = 2.
Section 2. The canonical divisor class on
A2(p)
We now come to the
object
of ourstudy.
DEFINITION 2.1. Let
A2(p)
be the coarse moduli space ofcouples (S, H)
whereS is a p.p.a.s. and H c
S[p]
is a rank twosubspace non-isotropic
for the Weilpairing,
where p is aprime.
Let L be a lattice of rank four and let E be an
alternating
bilinear form onL with
elementary divisors {1,1}.
Let E denote also the extensionof E to L 0 C,
then
H(v, w) = 1 E(v, w)
is a Hermitian form on L (g) C.Siegel’s
upper half space can be realized asH2~{V~L~C|dimV=2,E|V~0,H|V>0}
i.e. as a
classifying
space forweight
oneHodge
structures. Let p be aprime,
letE p
be the
IF p-valued alternating
form that E induces onL p
= L~ Fp
and letXp
cL p
be a fixed rank two
subspace non-isotropic
forEp.
Now let S be a p.p.a.s. andH c
S[p]
anon-isotropic subspace;
the Weilpairing
identifiesH1(S, IF p)
withHl (S, Fp)
hence we can think of H asliving
inH1(S, Fp).
Letf : H1(S, Z) ~
L beany
isomorphism
such thatf * E
is thepolarization
on S and such thatf(H)
=E p;
then
f(H1,0(S)) ~ H2.
It is clear thatby
this constructionIW2 (P)
can be realized as0393pBH2,
whereAs usual
m(S, H)Ed 2(P) (or mp(S, H))
will be the modulipoint of (S, H).
Let C bea smooth genus two curve, we will use
(C, H) (H ~ Jac(C)[p])
as an alternative notation for(Jac(C), H).
DEFINITION 2.2. Let n:
A2(p) ~ A2
be definedby 7r(m(S, H))
=m(S),
i.e.by forgetting
the p-structure on S.REMARKS. Notice that there is an involution 1:
A2(p) ~ A2(p) commuting
with n: let x =
m(S, H)
theni(x)
=m(S, H) (orthogonality
is with respect to the Weilpairing).
The map n can also be defined as the map induced from the inclusion
rp Sp(4, Z).
LEMMA 2.1. Let 7c:
A2(p) ~ d 2’
thendeg n
=p4
+p2.
Proof.
Letm(S) E d 2
be ageneric point
ofd 2’
i.e. let theautomorphism
group of S begenerated by multiplication by -1.
The fiber03C0-1(m(S))
is in one-to-onecorrespondence
with the set ofisomorphism
classes ofcouples (S, H).
Sincemultiplication by - 1
fixes everysubspace of S[p]
thedegree
of n isequal
to thenumber
of subspaces
H cL p
such thatEplH
isnon-degenerate.
The GrassmannianGr(2, Lp)
ofplanes
inL p
is realizedby
the Pluckerembedding
as thevariety
ofrational
points
of a smoothquadric hypersurface
inP(A 2L.).
Theisotropic
subspaces correspond
topoints
on ahyperplane section,
which is smooth ifp > 2. Hence if p > 2 deg
One can check that the formula still holds if p = 2.
PROPOSITION 2.1. Let D c
A2 be
an irreducible componentof
the branch divisorof
n, then Dparametrizes surfaces
with extraautomorphisms.
Proof.
Letm(S)
be ageneric point
ofD,
i.e. letAut’(S)
be contained in the reducedautomorphism
group of all surfaces T such thatm(T)~D.
Let U be theuniversal deformation space of
S,
let m : U ~d 2
be the moduli map. The groupAut’(S)
acts on U andm(U) ~ Aut’(S)B
U. Letm(S, H)
Ed 2(P)
be apoint
in theramification divisor
lying
over D. The deformation space of(S, H)
isisomorphic
to U. Let
Aut’(S, H ) Aut’(S)
be thesubgroup fixing
H(this
makes sense becausemultplication by -1
fixesH).
Let mp: U ~A2(p)
be the moduli map, thenmp(U)) ~ Aut’(S, H)BU.
The map 03C0:mp(U) ~ m( U )
is induced from the inclusionAut’(S, H) Aut’(S).
The inclusion of groups must be proper because 03C0 is ramified henceAut’(S)
cannot be trivial.Q.E.D.
The
preceding
discussion also proves thefollowing.
PROPOSITION 2.2. Let D c
d 2 be
an irreduciblecomponent of
the branchdivisor of ’Tt
and letm(S)
be ageneric point of
D. Theramification
indexof
thecomponent
of 03C0-1(D) through m(S, H)
isequal to
It is easy to check that the divisors on
d 2 parametrizing p.p.a.s.’s
with extraautomorphisms
areexactly 03941 and 02.
DEFINITION 2.3. Let
àl
cA2(p)
be the locus of moduli ofcouples (S, H)
with S a reducible p.p.a.s.
(i.e. S zé
E xF)
and H =E[p]
or H=F[p].
Obviously 03C0(1)
=03941 and 1
is a two sheeted cover of03941.
DEFINITION 2.4. Let
Ri
~d 2(P)
be the(reduced)
divisor suchthat 03C0-1(03941) = A, U R1 .
LEMMA 2.2.
If p
> 2 the map 7r:A2(p) ~ d 2 is unramified along Ã1
and hasramification
index 2along R1. If
p =2, n
isunramified along
allof 03C0-1(03941).
Proof.
Letm(S)
be ageneric point of 03941. Let g
EAut’(S)
act asmultiplication by
-1 on E and as the
identity
onF,
thenAut’(S) = (g) zé Z/(2). If p > 2 E[ p]
andF[p]
are theonly non-isotropic subspaces
fixedby g,
henceAut’(S, E[p] ) ~ Aut’(S,F[p]) ~ Z/(2)
andAut’(S,H) ~ id)
if H ~E[p], F[p].
Thereforeby Proposition 2.2, 03C0
is unramifiedalong 1
and has ramification index onealong R1. If p
=2, since g
acts as theidentity
onS[2], Aut’(S, H) ~ (g)
for allH,
hence03C0 is unramified over
03941.
DEFINITION 2.5. Let
m(S)ELB2’
i.e. S contains anelliptic
curve E such thatE·0398 =
2,
where 0 c S is the theta divisor. Let a : E S be theinclusion,
leta.: E[p]
4S[p]
be the restriction of a top-torsion points.
If p > 203B1(E[p])
isnon-isotropic.
For p > 2 letA2
cA2(p) be
the locus of moduli ofcouples (S, H)
where
m(S)c-A2
and H =03B1(E[p])
or H =03B1(E[p]).
DEFINITION 2.6. For p > 2 let
R2 c d 2(P)
be the(reduced)
divisor definedby
LEMMA 2.3.
If
p > 2 the map03C0:A2(p) ~ W2
isunramified along à2
and hasramification
index 2along R2- If
p = 2 n hasramification
index 2along
allof 03C0-1(03942)
=R2.
Proof.
Letm(S)~03942
begeneric,
i.e. letAut’(S) ~ Z/(2).
The surface S isisomorphic
to E xF/G,
whereG
= {(x, qJ(x» xeE[2]
and 9:E[2] ~ F [2]
is asymplectic isomorphism.
Let S’ = E x
F,
let 0’ c S’ be the reducibleprincipal polarization
and letf:
S’ ~ S be thequotient
map, thenf*(0398) ~
20’(0
is theprincipal polarization
on
S). If p > 2
the mapf: S’[p] ~ S[p]
is anisomorphism of groups.
LetW,
W’be the Weil
pairings
onS,
S’respectively,
thenW(f(x), f(y))
=2 W’(x, y)
henceH c
S[p]
isnon-isotropic
if andonly
iff-1(H)
cS’[p]
isnon-isotropic.
Thereduced group
Aut’(S)
isgenerated by
theautomorphism
induced from the extraautomorphism
of S’. Hence we are reduced to the case of theprevious
lemma andwe
get
that 7r is unramifiedalong A2
and has ramification index 2along R2.
In thecase p = 2 one checks that there are no
non-isotropic subspaces of S[2]
fixedby
the extra
automorphism
hence 7r is ramified with index 2along
all of03C0-1(03942).
COROLLARY 2.2. Let Te:
A2(p) ~ A2
thenSince n is induced from the inclusion
rp Sp(4, Z)
the rationalpolyhedral decompositions defining
the toroidalcompactification d2
cdl
also definea
compactification çî2(p) (-- j72(p)
such that n extends to a finitesurjective
mapn:
A2(p) ~ W2.
We will prove thatA2(p)
is ofgeneral
type for p 17by studying
n-canonical forms on the
compactification -W2(p).
The set of codimension 1
boundary components
ofH2
is in one-to-onecorrespondence
with the set of one-dimensionalsubspaces
of L ~ Q. If w =[v]
is such a
subspace
we can think of v as thevanishing cocycle. Any
two suchsubspaces [v]
and[w]
areSp(4, Z)-equivalent (this
isequivalent
toAo being irreducible).
The smaller grouprp
does not acttransitively
onP(L
0Q),
in factwe have
Claim. There are three
equivalence
classes for the action ofFp
onP(L):
where vp = v (8) 1 is the reduction of v modulo p.
It is clear that
(a), (b), (c)
are notequivalent
underrp .
It is also easy to check thatrp
actstransitively
on each of the sets(a), (b), (c).
DEFINITION 2.7.
(a)
Letào
cFI2(p)
be the divisorcorresponding
to theeqivalence
class(a).
(03B2)
Letào ce j/-2(p)
be the divisorcorresponding
to theequivalence
class(b).
(y)
LetRo
cA2(p)
be the divisorcorresponding
to theequivalence
class(c).
REMARK. The involution on
W2(p) (p. 128)
extends to an involution:A2(p) ~ A2(p) commuting
with n. It is clear that(0)
=Xo i(R0)
=Ro.
LEMMA 2.4. The map
03C0:A2(p) ~ A2 is unramified along ào
andAo,
it hasramification
index palong Ro.
Before
proving
the Lemma wegive
adescription
of the fibers of 7r overAo.
LetC be a stable genus two curve with one
(at least) non-disconnecting
node. LetU be the universal deformation space of
C,
let0°(U)
c U be the divisorparametrizing
curves with one(at least) non-disconnecting node,
it’s a divisorwith normal
crossings.
Let 9: V - U be the cover unbranched outside0°(U)
andwith ramification of
order p
over eachcomponent
of0°(U).
Let W be thepull
back to V of the universal curve over U and let C’ be a fixed smooth reference
curve
(in
thefamily q)
with no extraautomorphisms.
The Picard-Lefschetztransformation(s)
actstrivially
onp-torsion points
ofJac(C’).
The fiber03C0-1(m(C’))
is in one-to-onecorrespondence
with the set ofisomorphism
classesof
couples (C’, H).
We can associate apoint
of03C0-1(m(C))
to everycouple (C, H)
where C is our
singular
curve and H cJac(C’)[p]
is anon-isotropic subspace
of the fixed smooth curve. If we choose another reference fiber C" there is a well defined
isomorphism
between thesubspaces
ofJac(C’)[p]
and thesubspaces
ofJac(C")[p]
becausemonodromy
actstrivially
onp-torsion points of Jac(C’).
Letm: U ~
m2
be the moduli map, letY°
c Y be the open set on which m:V ~rol2 is
unramified
(notice
that C’ maps to apoint
ofY°).
Let G be the group of deck transformations ofm~:V0 ~ mqJ(YO)
and let M be the group of deck transfor- mationsof 9:
V° -qJ(YO)
we have an exact sequence 1 - M ~G ~ Aut’(C) ~
1.The group G acts on the set of
non-isotropic subspaces
ofJac(C’)[p] because,
as we haveremarked,
there is a well definedisomorphism
between
p-torsion points
of any two smooth fibers of y.DEFINITION 2.8. Let H c
Jac(C’)
benon-isotropic,
we defineAut’(C, H)
Gto be the
subgroup fixing
H.Let m p:
V ~FI2 (p)
be the moduli map, thenIn
practice
in order to constructmp(V)
we start with a smooth fiber C’ in theuniversal
family
over U and we choose anon-isotropic
H cJac(C’)[p].
Then welet V’ be the cover of U ramified with index
(p - 1) only
over the components ofDo(Y) corresponding
to Picard-Lefschetz transformations which do not fixH,
and weproceed
as before.Proof of
Lemma 2.4. Letm(C)
EDo
begeneric,
i.e.Aut’(C)
is trivial. Let U be the universal deformation space ofC, 03940(F)
c V is smooth. Let C’ be a fixed reference smooth curve in the universalfamily 16
and letvp~Jac(C’)[p]
be thevanishing cycle:
(i)
Apoint
in0 ~03C0-1(m(C)) corresponds
to H ceJac(C’)[p]
such that H 1 vp, hence H is fixedby
the Picard-Lefschetz transformation.Using
the notation wejust
introduced we have thatAut’(C)~{1},
M ~Z/(p)
hence G ~Z/(p).
SinceH is fixed
by monodromy Aut’(C, H) ~
G ~7LI(p).
Thereforemp(v)
=U,
m( U )
= U and n:mp(V) ~ m(U) is just
theidentity,
so x is indeed unramifiednear
m(C)).
(ii)
Apoint
inAo ~ 03C0-1(m(C)) corresponds
to H cJac(C’)[p]
such thatH~vp,
so
again H
is fixedby
the Picard-Lefschetz transformation. The same argumentas in case
(i)
shows that is unramifiedalong 30.
(iii)
Apoint
inRo
n03C0-1(m(C)) corresponds
to H cJac(C’)[p]
which is notorthogonal to vp
and does not contain vp, hence it is not fixedby
thePicard-Lefschetz transformation. Therefore
Aut’(C,H) ~ {1}
somp(V) ~ V;
since V is a
p-sheeted
cover of V branched overDo(U)
andm(V) ~ V
we see thatn has ramification of order p
along Ro.
COROLLARY 2.3. Let 7t:
A2(p) ~ A2,
thenPROPOSITION 2.4. Let n :
A2(p) ~ A2,
thenProof.
Let the notation be asbefore,
som(C)
EAo
is ageneric point.
The fiber’Tt-1(m(C»
nA.
is in one-to-onecorrespondence
with the set ofnon-isotropic subspaces
H cJac(C’)[p] orthogonal
to thevanishing cycle
vp. So we have tocount the number
of projective
lines inP(vp)
which arenon-isotropic
for the Weilpairing. Obviously,
such a line cannot contain[vp],
and since thepairing
isnon-degenerate
this condition is sufficient for a line to benon-isotropic.
Hencedeg 03C0|0
= #{lines in P2(Fp)
notcontaining
a fixedpoint}
=p2.
A similar countgives deg 03C0|0 = p2;
notice that the involution i onQ-/2 (p) commuting
withrc
interchanges A. and Xo,
therefore we must havedeg 03C0|0
=deg 03C0|0.
Thedegree
of 03C0 restricted to
Ro
isreadily
obtained fromdeg 03C0
=p4
+p2 and 03C0*(03940) =
Proof.
Weapply
Hurwitz’s formula to the finitesurjective morphism
03C0c:A2(p) ~ .d2. Taking
into account Lemmas2.2, 2.3,
2.4 weget
formulas(*)
and(**).
THEOREM 2.1. Let n :
A2(p) ~ A2. If
p > 2 thenApplying Corollary
1.3 and Lemma 1.3 weget
the first formula. When p = 2 weget
which
together
withCorollary
1.3 and Lemma 1.3gives
the second formula.The formula for
03C0*(KA2(2))
agrees with the fact thatdl(2)
is rational. Whenp = 3 we get
Let C be an irreducible genus two curve: it can be realized as the double cover
of P1
branched over sixpoints (some
of whichmight
bemultiple). By
consider-ing pencils
ofsixtuples in pl
we can construct a curve T cm2
such thatm(C)~0393,0393~03941 = 0, 03940·0393>0,
hencethrough
ageneric point
ofroll
therepasses a curve r such that r
03C0*(KA2(3))
0. It follows that the linear systemInKd2(3)1 I is
empty for all n >0,
i.e. the Kodaira dimension ofdl(3)
is - 00. Ifp 5 then
03C0*(KA2(p))
is a linear combination withpositive
coefficientsof Ao
and03941,
thereforeh0(n03C0*KA2(p) = cn3 + O(n2)
for apositive
c(n
divisibleenough).
This suggests that
FI2 (p) might
be ofgeneral
typefor p big,
but it is not sufficientto prove
it;
in fact we will need to furtherstudy A2(p)
to prove that it is ofgeneral
type for p 17.THEOREM 2.2.
If p
3 thenProof.
The formula is obtained from(*)
ofProposition
2.5together
withCorollary 1.3,
Lemma 1.3 and the defnitionsof 2, 1, 0, 2.
Section 3. A
partial desingularization
ofJÎ2(P)
We recall that the Kodaira dimension of a
variety
X, denotedby 03BA(X),
is definedas follows: let X ~ X be a
compactification
of X and letX
be adesingularization ofX,thenK(X)
=tr. deg.(R) - 1
where R is the canonicalring R
=~~n=0H0(nK).
One
always
has that03BA(X)
dim X(possibly 03BA(X) = - ~)
and ifK(X)
= dim Xthen X is said to be of
general type.
The Kodaira dimenson is a birationalinvariant;
sinceK(P’) = -
ooif 03BA(X)
0 then X is notrational;
furthermore one sees that if03BA(X)
0 then X cannot be unirational. From now on we assumep5.
We recall that a germ
(X, P)
of a normalalgebraic singularity
is said to havea canonical
singularity
at P if(i)
there exists aninteger
r > 0 such thatrKx
is Cartier(ii)
for aresolution ~:~X (equivalently
for anyresolution)
with excep- tional set E =~iEi, rK = ~*(rKX) + 03A3iaiEi
withai
0 for all i.If X has
only
canonicalsingularities and 1
is a resolution of XH0(nK) ~ HO(nKx)
hence we need not passto X
in order to determine03BA(X).
We will
apply (when possible)
thefollowing
Shepherd-Barron, Reid,
Tai criterion[H-M]:
Let a finite group G actlinearly
on acomplex
vector space V.Let g
E G beconjugate
towhere (
is aprimitive
mth root ofunity
and0 ai
m. If03A3di=1ai
m forall g
and (
then(GBV,0)
is canonical at 0.In our case the situation is the
following.
Letm(C, H)~A2(p)
be asingular point.
The cotangent space to the deformation space of C iscanonically isomorphic
toH0(03A91C
0wc).
The groupAut’(C)
acts onH0(03A91C
003C9C)
anda
neighborhood
ofM(C)~m2
isisomorphic
toAut’(C)BH0(03A91C
003C9C).
If C is smooth or has
only
onedisconnecting
node then aneighborhood
ofm(C, H)
EA2(p)
isisomorphic
toAut’(C, H)BH0(03A91C
0coc) -
SinceAut’(C, H) Aut’(C)
we see thatm(C, H)
can besingular only
ifAut’(C)
is non-trivial. If C has one(at least) non-disconnecting
node then aneighborhood
ofm(C, H)
isisomorphic
toAut’(C, H)) E
where V is a cover of U(the
deformation space ofC)
branched over
03940(U).
The groupAut’(C, H)
in this case is contained inG,
an extensionof Aut’(C) by
themonodromy
group M. Henceif Aut’(C)
is trivial thenAut’(C, H)
M. It is easy to check that ifm(C) E 0°
andAut’(C)
is trivial then C hasexactly
onenon-disconnecting
node so M ~7LI(p), Aut’(C, H) ~ 7LI(p)
or{1}
and hencem(C, H)
is smooth. Therefore weagain
conclude thatm(C, H)
canbe
singular only
ifAut’(C)
is nontrivial.Igusa [I]
listed all smooth genus two curves withextra-automorphisms,
we caneasily
add a list of all theremaining
stable curves with extra
automorphisms.
( 1 )
C = EuF,
whereE,
F areelliptic
curvesand j(E), j(F) ~ 0, 1728,
i.e.m(C)
is a
generic point
of03941. Aut’(C) = g> ~ Z/(2),
where91E
=(multiplication by -1 ), g|F
=identity.
(2)
C = EuE, j(E) gé 0,1728
som(C)~03941
n02. Aut’(C) = g, h> ~ Z/(2)
EBZ/(2),g|E
=(-1),g|F = (id),
hinterchanges
the two components(3)
C = EuF, j(E) = 1728, j(F)
~0,1728. Aut’(C) = g> ~ Z/(4), g|E
=(mul- tiplication by ~-1), 91F
=(id).
(4)
C = E u F,j(E)
=0, j(F) ~ 0,
1728.Aut’(C) = (g) zé Z/(6), 9 1 E
=(multi- plication by e6 ), 91F
=(id).
(5)
C =E u E, j(E)
= 1728.Aut’(C)
actsnaturally
on the two components of C so it fits into the exact sequence0~N~Aut’(C)~Z/(2)~0
and N zéZ/(4) ? Z/(2).
(6)
C = EuF, j(E)
=0, j(F)
= 1728.Aut’(C) = g> ~ Z/(12), 91E
=(multipli-
cation
by e6), g|F
=(multiplication by ~-1).
(7)
C =E u E, j(E)
= 0.Aut’(C)
actsnaturally
on the twocomponents
ofC,
it fits into the exact sequence 0 ~ N ~Aut’(C) ~ Z/(2) ~
0 where N ééZ/(6) ? Z/(3).
(8)
C =E/P ~ Q,
where P -Q
is a 2-torsionpoint (unless j(E)
= 1728 andP, Q
are chosen so thatthey give
case(9) below). Aut’(C) = g> ~ Z/(2);
if we letP be the
origin then g
is induced frommultiplication by -1.
We havem(C)
eA2
n0° .
(9)
C =E/P ~ Q,
where E isgiven by y2
=x4
+1, so j(E)
=1728,
andP =