The rationality of the moduli spaces of Coble surfaces and of nodal Enriques surfaces

0.15 MB·English

by  Igor Dolgachev

THE RATIONALITY OF THE MODULI SPACES OF COBLE
SURFACES AND OF NODAL ENRIQUES SURFACES
2
1 IGORDOLGACHEVANDSHIGEYUKIKONDO¯
0
2
ABSTRACT. WeprovetherationalityofthecoarsemodulispacesofCoblesur-
n
facesandofnodalEnriquessurfacesoverthefieldofcomplexnumbers.
a
J

0
3

] 1. INTRODUCTION
G
A ThepurposeofthisnoteistoprovetherationalityofthemodulispacesofCoble
surfaces and of nodal Enriques surfaces over the field of complex numbers. A
.
h
Coble surface is a rational surface obtained by blowing up 10 nodes of a rational
t
a plane curve of degree 6, and an Enriques surface is called nodal if it contains a
m
smooth rational curve. The moduli space of nodal Enriques surfaces is a codi-
[
mension one subvariety inthe 10-dimensional moduli space of Enriques surfaces.

1 WhentheK3-coverofanEnriquessurfacedegenerates admittinganordinarydou-
v blepoint fixedunder aninvolution, thequotient bytheinvolution isarational sur-
3
face obtained from a Coble surface by blowing down the proper transform of the
9
0 planesextic. Inthisway,themodulispaceofCoblesurfacescanbeidentifiedwith
6 a codimension one component of the boundary of the moduli space of Enriques
.
1 surfaces.
0 The idea of the proof is similar to the one used by the second author for the
2
proof of rationality of the moduli space of Enriques surfaces [13]. The K3 sur-
1
: face birationally isomorphic to the double cover of the projective plane branched
v
along the union of a cuspidal plane quintic and its cuspidal tangent contains the
i
X lattice D8 ⊕U in its Picard group. It is shown that the moduli space of K3 cov-
r ersofEnriques surfacesandthemodulispaceofK3surfaces admitting thislattice
a
in its Picard group are birationally isomorphic quotients of a bounded symmetric
domainoftype IV.Asimilar ideaisused here. Weprovethat themoduli space of
K3 covers of nodal Enriques surfaces (resp. the K3-covers of Coble surfaces) is
birationallyisomorphictothemodulispaceofK3surfacesbirationallyisomorphic
tothe double coverof theprojective plane branched along theunion ofacuspidal
planequinticanditscuspidal tangentlinewherethequintichasanadditional dou-
blepoint(resp. thecuspidal tangentlinetouches thecurveatanonsingular point).
It is easily proved, by using [15], [22], that the corresponding moduli spaces of
planequintics arerationalvarieties.
ResearchofthesecondauthorispartiallysupportedbyGrant-in-AidforScientificResearch(S),
No22224001,(S),No19104001.
1
2 IGORDOLGACHEVANDSHIGEYUKIKONDO¯
Note that general Enriques and Coble surface are examples of the quotients of
a K3 surface by a non-symplectic involution which acts identically on the Picard
group. Recently Ma [14] showed the rationality of the moduli spaces of such K3
surfaces in many cases. The case of the K3-covers of Coble surfaces is one of
eightexceptional caseswherehismethodsdidnotwork.
TheideathatthemodulispacesofEnriquessurfaces (resp. nodalEnriques sur-
faces, Coble surfaces) should be related to the moduli space of cuspidal quintics
(resp. their special codimension one subvarieties) originates from some (still hy-
pothetical) purely geometric constructions ofthe firstauthor whichmayrelate the
corresponding moduli spaces. We discuss these constructions in the last two sec-
tionsofthepaper.
2. PRELIMINARIES
AlatticeisafreeabeliangroupLoffiniterankequippedwithanon-degenerate
symmetric integral bilinear form L × L → Z whose value on a pair (x,y) will
be denoted by x · y. For x ∈ L ⊗ Q, we call x2 = x · x the norm of x. For a
lattice L and arational number m, wedenote by L(m) the free Z-module L with
theQ-valued bilinear formobtained from thebilinear form ofLbymultiplication
by m. The signature of a lattice is the signature of the real vector space L ⊗ R
equippedwiththesymmetricbilinearformextendedfromoneonLbylinearity. A
latticeiscalledevenifx·x ∈ 2Zforallx ∈ L.
WedenotebyU theevenunimodularlatticeofsignature(1,1),andbyA , D
m n
or E the even negative definite lattice defined by the Cartan matrix of type
k
A , D or E respectively. For an integer m, we denote by hmi the lattice
m n k
ofrank1generatedbyavectorwithnormm. WedenotebyL⊕M theorthogonal
directsumoflatticesLandM,andbyL⊕mtheorthogonaldirectsumofm-copies
ofL. Foranyintegerk wedenotebyM thesetofx ∈ M withnormk.
k
WedenotebyLK3thelatticeE8⊕2⊕U⊕3. Itisisomorphictothe2-cohomology
groupH2(X,Z)ofaK3surfaceequippedwiththestructureofalatticedefinedby
thecup-product. WewillrefertoLK3astheK3-lattice. ThelatticeE = E8⊕U is
calledtheEnriqueslattice. Itisisomorphic tothelatticeNum(S) = Pic(S)/(K )
S
ofnumericalequivalence divisorclassesonanEnriquessurfaceS.
LetLbeanevenlattice and letL∗ = Hom(L,Z)identified withasubgroup of
L⊗Qwiththeextended symmetric bilinear form. Wedenote byA thequotient
L
L∗/Landdefinemaps
q :A → Q/2Z, b :A ×A → Q/Z
L L L L L
byq (x+L)= x·xmod2Zandb (x+L,y+L)= x·ymodZ. Wecallq the
L L L
discriminant quadratic form ofL and b the discriminant bilinear form. Alattice
L
iscalled2-elementary ifitsdiscriminant groupisa2-elementary abeliangroup.
LetO(L)betheorthogonal groupofL,thatis,thegroupofisomorphismsofL
preserving thebilinearform. SimilarlyO(A )denotesthegroupofisomorphisms
L
ofA preserving q . Thereisanatural map
L L
(2.1) φ: O(L)→ O(A )
L
COBLESURFACESANDNODALENRIQUESSURFACES 3
whosekernelisdenoted byO(L)∗.
3. THE MODULI SPACES OF ENRIQUES, NODAL ENRIQUES AND COBLE
SURFACES
First we recall the moduli space of lattice polarized K3 surfaces. For any even
lattice M of signature (1,r −1) primitively embeddable into the K3-lattice LK3,
onecanconstructthecoarsemodulispaceMK3,M (resp.MaK3,M)ofisomorphism
classes of lattice M polarized (resp. amply polarized) K3 surfaces X, i.e. iso-
morphism classes of pairs (X,j), where j : M ֒→ Pic(X) is a primitive lattice
embedding such that the image contains a nef and big (resp. ample) divisor class
(see[10]).1 LetN = M⊥ . Thentheperioddomainisgivenby
LK3
(3.1) D(N)= {[ω] ∈ P(N ⊗C) : ω·ω = 0, ω·ω¯ > 0}
which is a disjoint union of two copies of the 20−r-dimensional bounded sym-
metricdomainoftypeIV.Themodulispaceisconstructed asaquotient
MK3,M = D(N)/O(N)∗, MaK3,M = (cid:0)D(N)\H−2(cid:1)/O(N)∗,
whereO(N)∗ = Ker(O(N) → O(A )),and,foranyd ∈ Z,
N
(3.2) H−2d = [ {[ω] ∈ D(N) : ω·δ = 0}.
δ∈N−2d
We call H−2d the (−2d)-Heegner divisor. Suppose that N and M satisfy the
condition
(3.3) ThenaturalmapsO(N)→ O(A ), O(M) → O(A )aresurjective.
N M
Then
MK3,M/O(AM)∼= D(N)/O(N), MaK3,M/O(AM)= (cid:0)D(N)\H−2(cid:1)/O(N),
are coarse moduli spaces of K3 surfaces wich admit a primitive embedding of M
intoPic(X).
The period point of a marked K3 surface belongs to H−2 if and only if there
exists a primitive embedding of M ⊕h−2i in Pic(X). The image of a generator
of h−2i will be an effective divisor class R with self-intersection −2 such that
R·h = 0 for every divisor class from the image of M. This shows that X does
notadmitanyamplepolarization contained intheimageofM inPic(X). Inother
words,anynefandamplepolarization ofX originated fromM willblowdownR
toadoublerational point.
Now we consider an Enriques surface S. Let π : X → S be its K3-cover and
letσ be thefixed point free involution of X. Thenπ∗(Pic(S)) = π∗(Num(S)) ∼=
E(2). We take E(2) as M and denote by N the orthogonal complement of M in
LK3. Then
(3.4) N ∼= U ⊕E(2).
1Thereissomeadditionaltechnicalrequirementfortheembeddingwhichwerefertoloc.cit.
4 IGORDOLGACHEVANDSHIGEYUKIKONDO¯
Notethatσ∗|M = 1 andσ∗N| = −1 . Itisknownthatanyperiodpoint [ω]of
M N
theK3-coverX isnotcontained inH−2 (e.g. [17]). Thequotient D(N)/O(N)is
a normal quasi-projective variety of dimension 10, and (cid:0)D(N)\H−2(cid:1)/O(N) is
themodulispaceM ofEnriquessurfaces.
En
Next we consider nodal Enriques surfaces, i.e. Enriques surfaces containing
a smooth rational curve ((−2)-curve, for short). Let C be a (−2)-curve on an
EnriquessurfaceS. Thenπ−1(C)splitsintothedisjointsumC1∪C2oftwo(−2)-
curves. The divisor class δ = [C1 −C2] with δ2 = −4 belongs to π∗(Pic(S)⊥).
If we consider all (−2)-curves on S, the corresponding (−4)-vectors δ generate
a negative definite lattice R(2) in U ⊕ E(2) where R is a root lattice. The root
lattice R is a part of the notion of root invariant for Enriques surfaces (see [20]).
Since any period point [ω] of the K3-cover X is orthogonal to an algebraic cycle,
we obtain that the period [ω] belongs to H−4. Thus we define the moduli space
Mnod ofnodalEnriquessurfaces by
En
(3.5) MnEond = (cid:0)H−4\H−2(cid:1)/O(U ⊕E(2)),
where H−4,H−2 are Heegner divisors in the period domain of Enriques surfaces.
It is known that such (−4)-vector δ in U ⊕ E(2) is unique up to the orthogonal
groupO(U⊕E(2)),andtheorthogonalcomplementδ⊥ inU⊕E(2)isisomorphic
to
(3.6) N = U ⊕h4i⊕E8(2)
(see [17]). The orthogonal complement M = N⊥ of N in LK3 contains E(2)⊕
h−4i as a sublattice of index 2, where E(2) = π∗(Pic(S)) and h−4i is generated
by δ. Then the quotient D(N)/O(N) is a 9-dimensional quasi-projective variety.
We call a nodal Enriques surface S is general if R ∼= h−2i, that is, for any two
(−2)-curves C,C′ on S, [C1 −C2] = [C1′ −C2′]. Note that, for a general nodal
Enriquessurface,thedecompositionE(2)⊕h−4iisunique,thatis,itisindependent
onachoiceof(−2)-curves. Hencewehavethefollowing.
Proposition3.1. LetN = U⊕h4i⊕E8(2). ThenthemodulispaceMnodofnodal
En
Enriquessurfacesisbirationally isomorphic toD(N)/O(N).
Finally we consider Coble surfaces. A Coble surface is a smooth rational pro-
jective surface S such that |−K | = ∅ but |−2K | =6 ∅ (see [12]). A classical
S S
exampleofsuchasurfaceistheblow-upoftheprojectiveplaneatthetennodesof
anirreducibleplanecurveC ofdegree6. Thesetsof10pointsintheplanerealized
asthe nodes ofarational sextics areexamples ofspecial sets ofpoints insense of
A. Coble [6] (they were called Cremona special in [3]). They were first studied
byA.Coblein[5]. Inthisnote wewillrestrict ourselves only withtheseclassical
examples.
Denote by M the moduli space of Coble surfaces constructed as a locally
Co
closed subvariety of the GIT-quotient of the variety of 10-tuples of points in P2
modulo the group PGL(3). By taking the double cover of P2 branched along the
planesexticwith10nodes, themodulispaceM canbedescribed asanopenset
Co
COBLESURFACESANDNODALENRIQUESSURFACES 5
of an arithmetic quotient of a 9-dimensional bounded symmetric domain of type
IV.Webrieflyrecallthis.
DenotebyX thedoublecoveroftheCoblesurfaceS branchedalongtheproper
transform of the plane sextic C. Then X is a K3 surface containing the divisors
E0, E1,...,E10, where E0 is the pullback of a line on P2 and E1,...,E10 are
the inverse images of the exceptional curves over the nodes p1,...,p10 of C. It
is easily seen that the corresponding divisor classes e0,e1,...,e10 generate the
sublattice M of Pic(X) isomorphic to M = h2i ⊕h−2i⊕10. Note that M is a
X
2-elementary lattice of signature (1,10) with AM ∼= (Z/2Z)11. The orthogonal
complement of M in H2(X,Z), denoted by N , is a 2-elementary lattice of
X X
signature (2,9) with q = −q (see [18], Corollary 1.6.2). The isomorphism
NX M
classofsuchlatticeisuniquely determined by−q . ThusN isisomorphicto
M X
(3.7) N = h2i⊕E(2)
(seeloc.cit.,Theorem3.6.2). WeremarkthatM ∼= E(2)⊕h−2i.
LetD(N)beasin(3.1),whereN isthelattice(3.7). ThequotientD(N)/O(N)
is a normal quasi-projective variety of dimension 9. The Torelli type theorem for
algebraic K3surfaces, duetoPiatetskii-Shapiro andShafarevich [21],impliesthe
following(formoredetails, see[16]):
Proposition 3.2. Let N = h2i ⊕ E(2). Then the moduli space M of Coble
Co
surfacesisisomorphic toanopensubsetofD(N)/O(N).
Note that N = h2i ⊕ E(2) is isomorphic to the orthogonal complement of a
(−2)-vector in U ⊕ E(2). This implies that the quotient of the (−2)-Heegner
divisorH−2 intheperioddomainofEnriques surfacesbythearithmetic subgroup
O(U⊕E(2))isbirationallyisomorphictothemodulispaceM ofCoblesurfaces.
Co
4. PLANE QUINTICS WITH A CUSP
LetC be aplane quintic curve witha cusp p. LetL be the tangent line of C at
thecusp. Weconsiderthefollowingtwocases.
Case1: ThelineListangent toC atasmoothpointq ofC.
Case2: C hasanordinary nodeq.
Let M be the moduli space of cuspidal quintics, that is, the GIT-quotient
cusp
oftheprojective space ofplane cuspidal curves ofdegree 5by thegroup PGL(3).
The second author proved earlier that M is a rational variety birationally iso-
cusp
morphic to the moduli space M (see [13]). The proof establishes a birational
En
isomorphism between M and the moduli space of K3 surfaces birationally iso-
En
morphic to the double covers of P2 branched along a cuspidal quintic. Here we
willfollow thesamestrategy replacing M withitscodimension 1subvarieties
cusp
M′ (resp. M′′ )corresponding toquintics fromCase1(resp. Case2).
cusp cusp
Theorem4.1. M′ andM′′ arerationalvarieties ofdimension9.
5,cusp 5,cusp
Proof. WestartwithCase1.
Let C be a quintic curve from Case 1. By a linear transformation, we may
choosecoordinates (x0 : x1 : x2)toassumethatp = (1 : 0 : 0)isthecusp, V(t1)
6 IGORDOLGACHEVANDSHIGEYUKIKONDO¯
is the cuspidal tangent line which touches C at the point q = (0 : 0 : 1). Since p
isacuspofC withcuspidaltangentV(x1),thecurveC isgivenbyanequation of
theform
3 2 2
ax0x1+x0A1(x1,x2)+x0A2(x1,x2)+A3(x1,x2) = 0, a 6= 0
where A1,A2,A3 are homogeneous polynomials of degrees 3, 4, and 5, respec-
2
tively. Plugginginx1 = 0,weobtainthebinaryformx0A1(0,x2)+x0A2(0,x2)+
A3(0,x2)invariables x0,x2. Itmusthave azero at(0 : 1)ofmultiplicity 2. This
impliesthatA2 = x1A′2 andA3 = x1A′3 forsomepolynomials A′2,A′3 ofdegrees
3and4,respectively. ThustheequationofC canberewrittenintheform
F = ax30x21+x20A1(x1,x2)+x0x1A′2(x1,x2)+x1A′4(x1,x2) = 0, a 6= 0.
LetV be the linear subspace of S5(C3)∗ consisting ofquintic ternary forms F
as above (with a may be equal to zero). The subgroup G of GL(4) which leaves
invariant V consistsoflineartransformations
x0 → ax0+bx2, x1 → cx1, x2 → dx1+ex2.
ThenM′5,cusp is birational to the quotient P(V)/G. Itfollows that the dimension
of M′ is equal to 9. Note that G is a solvable algebraic group of dimension
5,cusp
5 acting linearly on the linear space V of dimension 14. The assertion of the
rationality nowfollowsfollowsfromaresultofMiyata[15]andVinberg[22].
The Case 2 can be argued in the same way. We may assume that the node
does not lie on the cuspidal tangent line. First transform C to a curve such that
p = (1 :0 :0)isacuspwiththecuspidaltangentlineV(x1)andq = (0 : 1: 0)is
anode. Arguingasabove,wefindthatC canbegivenbyanequation
3 2 2 2
F = ax0x1+x0A1(x1,x2)+x0x2A2(x1,x2)+x2A3(x1,x2) = 0, a 6= 0.
Let V′ be a linear subspace of S5(C3)∗ consisting of quintic ternary forms F as
above (with a may be equal to zero). Its dimension is equal to 13. The subgroup
G′ ofGL(4)leavingV′ invariant consists ofprojective transformations
x0 7→ ax0+bx2, x1 7→ cx1, x2 → dx2.
It is a solvable algebraic group of dimension 4 acting linearly on V. The variety
M′′ isbirationaltothequotientvarietyP(V′)/G′. Itfollowsthatthedimension
5,cusp
of M′′ is 9. Invoking the same result of Miyata and Vinberg, we obtain that
5,cusp
M′′ isrational. (cid:3)
5,cusp
5. K3SURFACES ASSOCIATED WITH A PLANE QUINTIC WITH A CUSP
In this section, we shall show that M′ (resp. M′′ ) is isomorphic to
5,cusp 5,cusp
an open subset of an arithmetic quotient of a 9-dimensional bounded symmetric
domainoftypeIV.
Firstweconsider Case1.
Let C be a cuspidal quintic as in Case 1 and let L be the cuspidal tangent.
Consider the plane sextic curve C +L. Let p be the cusp and q be the a smooth
tangency point of L with C. Let X¯ be the double cover of P2 branched along
COBLESURFACESANDNODALENRIQUESSURFACES 7
C + L. Then X¯ has a rational double point of type E7 over p locally isomor-
2 2 3
phic to V(z + y(y + x )) and a rational double point of type A3 over q lo-
2 2
cally isomorphic to V(z + x(x + y )). Denote by X the minimal resolution
of X¯ and by τ the covering transformation. Then X is a K3 surface contain-
ing11smoothrationalcurvesE1,...,E11 withintersection graphpicturedbelow.
E2 E3 E4 E5 E6 E7 E8 E9 E10
• • • • • • • • •
E1 E11
• •
WeseethatE1,...,E7formtheintersectiongraphoftypeE7,E8istheinverse
imageofLandE9,E10,E11 formtheintersection graphoftypeA3. Thecovering
transformation τ preserves each of E1,...,E9 and changes E10 and E11. Note
thatthelinearsystem
|E1+E3+2(E4 +···+E9)+E10+E11|
definesanelliptic fibration withasingular fiberoftype D˜9,andE2 isasection of
this fibration. This implies that these 11 curves generate a lattice M of Pic(X)
X
isomorphic to M = U ⊕D9. Here U is generated by the class of a fiber and the
section E2 and D9 is generated by E1,E4,...,E11. Since the discriminant of M
isequalto4,andtherearenoevenunimodularlatticeswithsignature(1,10), M
X
isprimitiveinH2(X,Z)andM = Pic(X)forgeneralX.
X
Let N be the orthogonal complement of M in H2(X,Z). Then N has
X X X
signature (2,9). Itfollowsfrom [18],Corollary 1.6.2thatqNX ∼= −qM. Notethat
AM ∼= Z/4Z. Alsoitfollowsfromloc. cit.,Theorem 1.14.2thattheisomorphism
classofN isuniquely determined byq . ThusN isisomorphic to
X M X
(5.1) N = h4i⊕U ⊕E8
ObviouslyO(MX)∼= Z/2Z. Wehavethefollowinglemmawhichiseasytoprove.
Lemma 5.1. The group O(A ), and hence O(A ), is generated by the cover-
SX TX
ing involution τ. In particular, the natural maps (2.1) O(M ) → O(A ) and
X MX
O(N ) → O(A )aresurjective.
X NX
LetD(N)beasin(3.1)withthelatticeN from(5.1). ThequotientD(N)/O(N)
isanormalquasi-projective varietyofdimension 9.
WefixaprimitiveembeddingofM intotheK3-latticeLK3withN = M⊥. We
also fix a basis {e } of M which has the same incidence relation as {E }. It fol-
i i
lowsfromLemma5.1thatthereexistsanisometryfromH2(X,Z)toLK3sending
theclasses ofE toe . Thisdefines aM-lattice polarization onthecorresponding
i i
K3 surface (see [10]) satisfying condition (3.3). Note that the M-marking deter-
minestheactionoftheinvolution ofτ onH2(X,Z): τ∗ actstriviallyone1,...,e9,
changes e10 and e11, and acts on N as −1. Conversely let (X,j),(X′,j′) be two
M-amplypolarized K3surfaces whoseperiods coincide inD(N)/O(N). Wede-
note by τ (resp. τ′) the involution of X (resp. X′). It follows from Lemma 5.1
thatthereexistsaHodgeisometryφ: H2(X,Z) → H2(X′,Z)preservingtheM-
markings. The Torelli type theorem for algebraic K3 surfaces implies that there
8 IGORDOLGACHEVANDSHIGEYUKIKONDO¯
exists an isomorphism ϕ : X → X′ with ϕ∗ = φ. Moreover ϕ ◦ τ = τ′ ◦ ϕ.
Henceϕinduces anisomorphism betweenthecorresponding planequintics. Thus
wehavethefollowingtheorem.
Theorem5.2. LetN = h4i⊕U⊕E8. ThenthemodulispaceM′5,cuspisisomor-
phictoanopensubsetofD(N)/O(N).
NextwestudyCase2.
ConsideragaintheplanesexticcurveC+L. LetLintersectC atthecusppand
twodistinctpointsq1,q2. LetX¯ bethedoublecoverofP2 branched alongC +L.
Then X¯ has a rational double point of type E7 over p and three rational double
point of type A1 over q1,q2,q. Denote by X the minimal resolution of X¯ and
by τ the covering transformation. Then X is a K3 surface containing 12 smooth
rationalcurvesE1,...,E12 whoseintersection graphispicturedbelow.
E12 E11 E2 E3 E4 E5 E6 E7 E8 E9
• • • • • • • • • •
E1 E10
• •
Here E9,E10 or E12 corresponds to the exceptional curve over q1,q2 or q, re-
spectively, and E1,...,E7 correspond to the exceptional curves over p, E8 is the
inverseimageofLandE11istheinverseimageofthelinepassingthroughpandq.
Thecoveringtransformation τ preserves eachofE1,...,E12. Notethatthelinear
system
|2(E4 +···+E8)+E1+E3+E9+E10|
definesanellipticfibrationwithasingularfiberoftypeD˜8 andoftypeA˜1,andE2
isasection ofthisfibration. Thisimplies that these12 curves generate thesublat-
ticeMX ofPic(X)isomorphictoM = U ⊕D8⊕A1. HereU isgenerated bythe
classoffiberandthesectionE2,thesublatticeD8isgeneratedbyE1,E4,...,E10,
and the sublattice A1 is generated by E12. Since the fixed locus of τ consists of
a smooth curve of genus 4 and four smooth rational curves E2,E4,E6,E8, the
invariant sublattice of H2(X,Z) under the action of τ∗ coincides with M ([19],
X
Theorem 4.2.2). In particular, M is primitive in H2(X,Z) and M = Pic(X)
X X
forgeneric X.
Let N be the orthogonal complement of M in H2(X,Z). Then M has
X X X
signature (2,9). N isisomorphic to
X
(5.2) N = h2i⊕U(2)⊕E8,
ItiseasytoseethatO(AMX) ∼=Z/2Z. ThisimpliesthefollowingLemma.
Lemma 5.3. The group O(A ) is generated by the isometry of S acting triv-
MX X
ially on E3,...,E12 and switching E1 and E2. The group O(ANX) is generated
bythecovering involution τ. Inparticular, thenaturalmapsO(M ) → O(A )
X MX
andO(N ) → O(A )aresurjective.
X NX
LetD(N)beasin(3.1)withN definedin(5.2). Usingthesameargumentasin
Case1,weprovethefollowingtheorem.
COBLESURFACESANDNODALENRIQUESSURFACES 9
Theorem 5.4. Let N = h2i ⊕ U(2) ⊕ E8. Then the moduli space M′5′,cusp is
isomorphictoanopensubsetofD(N)/O(N).
Remark5.5. Ma[14]provedtherationality ofthemodulispaceofK3surfaces in
thecase2byusinganother modelofthequotientX/τ.
6. PROOF OF THE RATIONALITY
Inthissection, weprovetherationality ofM andMnod.
Co En
Theorem6.1. LetN (resp. N′)bethelatticefrom(3.7)(resp. (5.1)). Then
D(N)/O(N) ∼= D(N′)/O(N′).
Proof. We have N(1/2) = h1i ⊕ U ⊕ E8. The odd unimodular lattice N(1/2)
contains the unique even sublattice of index 2 isomorphic to N′. Thus we can
consider N′ as a sublattice of N(1/2). Then any isometry of N(1/2) preserves
N′ andhence O(N(1/2)) ⊂ O(N′). Conversely, consider N′∗ = h41i⊕U ⊕E8.
The discriminant group AN′ is a finite cyclic group of order 4 and contains the
unique subgroup N(1/2)/N′ of order 2. This implies that any isometry of N′
can be extended to the one of N(1/2) and hence O(N′) ⊂ O(N(1/2)). Thus
we have O(N(1/2)) = O(N′). Now consider the bounded symmetric domain
D(N(1/2)) = D(N′). Then
D(N(1/2))/O(N(1/2)) = D(N′)/O(N′).
Obviously
D(N(1/2))/O(N(1/2)) ∼= D(N)/O(N).
Thereforewehaveprovedtheassertion. (cid:3)
Theorem6.2. LetN (resp. N′)bethelatticefrom(3.6)(resp. (5.2)). Then
D(N)/O(N) ∼= D(N′)/O(N′).
Proof. Consider
N′(1/2) = h1i⊕U ⊕E8(1/2), (N′(1/2))∗ = h1i⊕U ⊕E8(2).
ThenN isthe evensublattice of (N′(1/2))∗. Hence O(N′(1/2)) ⊂ O(N). Con-
versely,consider N∗ = h41i⊕U ⊕E8(1/2)andthediscriminant quadratic form
q :N∗/N → Q/2Z.
N
We remark that N′(1/2) is characterized as being the maximal submodule K of
N∗ suchthatq (K/N) ⊂ Z/2Z. Anyisometry ofN canbeextended totheone,
N
denoted by φ, of N∗. Then the above remark implies that φ preserves N′(1/2).
HenceO(N)⊂ O(N′(1/2))). Therefore
D(N)/O(N) = D(N′(1/2))/O(N′(1/2)) ∼= D(N′)/O(N′).
(cid:3)
Combining Propositions 3.2, 3.1 and Theorems 5.2, 5.4, 6.1, 6.2, we have the
followingtheorem.
10 IGORDOLGACHEVANDSHIGEYUKIKONDO¯
Theorem6.3. Therearebirational isomorphisms
(6.1) Υ : MEn ∼= M5,cusp,
(6.2) Υ′ : MCo ∼= M′5,cusp,
(6.3) Υ′′ : MnEond ∼= M′5′,cusp.
ByTheorem4.1,weobtainthefollowingmaintheorem.
Theorem6.4. M andMnod arerationalvarieties.
Co En
Remark 6.5. The K3-cover X of a general nodal Enriques surface is isomorphic
to a minimal nonsingular model of Cayley quartic symmetroid, the locus Y of
singularquadricsinageneral3-dimensionallinearsystemLofquadricsinP3(see
[7]). ThesurfaceY has10nodescorrespondingtoreduciblequadricsinL. Theset
of10pointsinP3 realizedasthetwnnodesofaCayleyquarticsymmetroidisone
ofspecialsetsofpointsinP3inthesenseofA.Coble(see[6]). Thereisabeautiful
relationshipbetweenCayleyquarticsymmetroidsandrationalsextics(seeloc.cit.).
The variety of such sets modulo projective equivalence is birationally isomorphic
totheGIT-quotientoftheGrassmannian G3(L)ofwebsofquadricsinP3 modulo
PGL(4). ItisbirationallyisomorphictosomefinitecoverofMnod. Therationality
En
ofG3(L)/PGL(4)isadifficultproblem.
7. A GEOMETRIC CONSTRUCTION: ENRIQUES SURFACES
Let M (2) be the moduli space of degree 2 polarized Enriques surfaces, i.e.
En
2
thecoarsemodulispaceofpairs(X,h),wherehisanefdivisorclasswithh = 2.
2
It is known that h = F1 + F2, where F1,F2 are nef divisors with Fi = 0 and
F1 ·F2 = 1, or h = 2F1 + R, where F1 is as above and R is (−2)-curve with
F1 ·R = 0 (see [9], Corollary 4.5.1). We call h non-degenerate if h is as in the
firstcase,anddegenerate otherwise.
Foranyhasabove,thelinearsystem|2h|definesadegree2mapφ : X → P4
h
whoseimageisaquarticdelPezzosurface. Ifhisnon-degenerate,Dhas4ordinary
double points, otherwise, it has 2 ordinary double points, and one rational double
pointoftypeA3. WecallDa4-nodalquarticdelPezzosurfaceinthefirstcaseand
adegenerate4-nodalquarticdelPezzosurfaceinthesecondcase(see[9],Chapter
0, §4). The set of fixed points of the deck transformation σ of the double cover
φ consists ofasmooth curve W and 4isolated points. Theimage W ofW onD
h
is a curve of arithmetic genus 5 from the linear system |OD(2)|. It does not pass
throughthesingular pointsofD. Themap(φ ) : W → W isthenormalization
h |W
map.
LetMEn(2)ndegbetheGIT-quotient|OD(2)|//Aut(D),whereDisanon-degenerate
quartic del Pezzo surface, and let M (2)deg be the same when D is a degenerate
En
quarticdelPezzosurface. Thefirstvariety(resp. secondone)isaprojectivevariety
ofdimension 10(resp. 9). Thedisjointunion
M (2) = M (2)ndeg ∪M (2)deg
En En En