A simply connected numerical Campedelli surface with an involution

0.32 MB·English

by  Heesang Park

A SIMPLY CONNECTED NUMERICAL CAMPEDELLI SURFACE
WITH AN INVOLUTION
HEESANGPARK,DONGSOOSHIN,ANDGIANCARLOURZU´A
2
1 ABSTRACT. Weconstructasimplyconnectedminimalcomplexsurfaceofgeneraltype
0 with pg=0andK2=2whichhasaninvolutionsuchthattheminimalresolutionofthe
2 quotientbytheinvolutionisasimplyconnectedminimalcomplexsurfaceofgeneraltype
n withpg=0andK2=1.Inordertoconstructtheexample,wecombineadoublecovering
a andQ-Gorensteindeformation. Especially,wedevelopamethodforprovingunobstruct-
J ednessfordeformationsofasingularsurfacebygeneralizingaresultofBurnsandWahl
whichcharacterizesthespaceoffirstorderdeformationsofasingularsurfacewithonly
4
rationaldoublepoints.WedescribethestablemodelinthesenseofKolla´randShepherd-

Barronofthesingularsurfacesusedforconstructingtheexample.Wecountthedimension
]
G oftheinvariantpartofthedeformationspaceoftheexampleundertheinducedZ/2Z-
action.
A
.
h
t
a
1. INTRODUCTION
m
[ Oneofthefundamentalproblemsintheclassificationofcomplexsurfacesistofinda
new family of complex surfaces of general type with pg =0. In this paper we construct
3
newsimplyconnectednumericalCampedellisurfaceswithaninvolution,i.e. simplycon-
v
nected minimal complex surfaces of general type with p =0 and K2 =2, that have an
7 g
9 automorphismoforder2.
7 Therehasbeenagrowinginterestforcomplexsurfacesofgeneraltypewithp =0hav-
g
0 inganinvolution; cf.J.Keum-Y.Lee[12], Calabri-Ciliberto-MendesLopes[4], Calabri-
.
8 Mendes Lopes-Pardini [5], Y. Lee-Y. Shin [17], Rito [23]. A classification of numerical
0 Godeauxsurfaces(i.e. minimalcomplexsurfacesofgeneraltypewith p =0andK2=1)
g
1
with an involution is given in Calabri-Ciliberto-Mendes Lopes [4]. It is known that the
1
quotientsurfaceofanumericalGodeauxsurfacebyitsinvolutioniseitherrationalorbira-
:
v tionaltoanEnriquessurface,andthebicanonicalmapofthenumericalGodeauxsurface
i
X factorsthroughthequotientmap.
However,thesituationismoreinvolvedinthecaseofnumericalCampedellisurfaces,
r
a becausethebicanonicalmapmaynotfactorthroughthequotientmap;cf.Calabri-Mendes
Lopes-Pardini [5]. In particular it can happen that the quotient is of general type. More
precisely,letX beanumericalCampedellisurfacewithaninvolutionσ.Ifσ hasonlyfixed
points and no fixed divisors, then the minimal resolution S of the quotientY =X/σ is a
numericalGodeauxsurfaceandσ hasonlyfourfixedpoints;cf.Barlow[2].Conversely,if
Sisofgeneraltype,thenσ hasonlyfourfixedpointsandnofixeddivisors;Calabri-Mendes
Lopes-Pardini[5].
There are some examples of numerical Campedelli surfaces X with an involution σ
having only four fixed points. Barlow [1] constructed examples with π (X)=Z/2Z⊕
1
Date:December31,2011.
2000MathematicsSubjectClassification. 14J29,14J10,14J17,53D05.
Keywordsandphrases. surfaceofgeneraltype,involution,Q-Gorensteinsmoothing.
1
2 H.PARK,D.SHIN,ANDG.URZU´A
Z/4Z,Z/8Z. Barlow [2] also constructed examples with π (X)=Z/5Z whose minimal
1
resolutionofthequotientbytheinvolutionisthefirstexampleofasimplyconnected nu-
merical Godeaux surface. Also all Catanese’s surfaces [6] have such an involution and
π =Z/5Z. Recently Calabri, Mendes Lopes, and Pardini [5] constructed a numerical
1
CampedellisurfacewithtorsionZ/3Z⊕Z/3Zandtwoinvolutions.
ItisknownthattheordersofthealgebraicfundamentalgroupsofnumericalCampedelli
surfaces are at most 9 and the dihedral groups D and D cannot be realized. Recently,
3 4
the existence question for numerical Campedelli surfaces with |πalg|≤9 was settled by
1
the construction of examples with πalg =Z/4Z; Frapporti [10] and H. Park-J. Park-D.
1
Shin [22]. Hence it would be an interesting problem to construct numerical Campedelli
surfaceshaving aninvolutionwith πalg=G foreach givengroupG with |G|≤9. Espe-
1
cially we are concerned with the simply connected case because the fundamental groups
ofalltheknownexampleswithaninvolutionhavelargeorder: |G|≥5. Furthermorethe
firstexampleofsimplyconnectednumericalCampedellisurfacesisveryrecent(Y.Lee-J.
Park[15]),butwehavenoinformationabouttheexistenceofaninvolutionintheirexam-
ple. Themaintheoremofthispaperis:
Theorem (Corollary 3.6). There are simply connected minimal complex surfaces X of
generaltypewith p (X)=0andK2 =2whichhaveaninvolutionσ suchthattheminimal
g X
resolution S of the quotientY =X/σ is a simply connected minimal complex surface of
generaltypewith p (S)=0andK2=1.
g S
WealsoshowthattheminimalresolutionSofthequotientY =X/σ hasalocaldefor-
mationspaceofdimension4correspondingtodeformationsS ofS suchthatitsgeneral
fiberS istheminimalresolutionofaquotientX/σ ofanumericalCampedellisurfaceX
t t t t
byaninvolutionσ;Theorem5.2. Inaddition,weshowthattheresolutionSshouldbeal-
t
wayssimplyconnectedifthedoublecoverX isalreadysimplyconnected;Proposition3.7.
Conversely Barlow [2] showed that if the resolution S is a simply connected numerical
Godeauxsurfacethenthepossibleorderofthealgebraicfundamentalgroupofthedouble
coverX is1,3,5,7,or9. Asfarasweknow,theexampleinBarlow[2]wastheonlyone
whosequotientissimplyconnected. Ithasπ (X)=Z/5Zasmentionedearlier. Herewe
1
findanexamplewithπ (X)=1. Henceitwouldbeanintriguingprobleminthiscontext
1
toconstructanexamplewithπ (X)=Z/3Z.
1
Inordertoconstructtheexamples,wecombineadoublecoveringandaQ-Gorenstein
smoothing method developed in Y. Lee-J. Park [15]. First we build singular surfaces by
blowing up points and then contracting curves over a specific rational elliptic surface.
Thesesingularsurfacesdifferbycontractingcertain(−2)-curves. Ifwecontractallofthe
(−2)-curves, we obtain a stable surfaceY(cid:48) in the sense of Kolla´r–Shepherd-Barron [14],
andweprovethatthespaceofQ-GorensteindeformationsofY(cid:48) issmoothand8dimen-
sional; Proposition2.2. A(Q-Gorenstein)smoothingofY(cid:48) inthisspaceproducessimply
connectednumericalGodeauxsurfaces. Inparticular,thesmoothingofY(cid:48) givestheexis-
tenceofatwodimensionalfamilyofsimplyconnectednumericalGodeauxsurfaceswith
six(−2)-curves;Corollary2.3. Wealsoprovethatafourdimensionalfamilyinthisspace
producessimplyconnectednumericalGodeauxsurfaceswitha2-divisibledivisorconsist-
ingoffourdisjoint(−2)-curves;Theorem2.4andTheorem5.2.ThesenumericalGodeaux
surfacesareusedtoconstructthenumericalCampedellisurfaceswithaninvolution. The
desirednumericalCampedellisurfacesareobtainedbytakingdoublecoveringsofthenu-
mericalGodeauxsurfacesbranchedalongthefourdisjoint(−2)-curves;Theorem3.4.
ANUMERICALCAMPEDELLISURFACEWITHANINVOLUTION 3
OntheotherhandwecanalsoobtaintheCampedellifamilyexplicitlyfromasingular
stablesurfaceX(cid:48). Itcomesfromblowinguppointsandcontractingcurvesoveracertain
rational elliptic surface; Proposition 3.1. The Q-Gorenstein space of deformations of X(cid:48)
issmoothand6dimensional;Proposition3.3. InbothGodeauxandCampedellicaseswe
computeH2(T)=0toshownolocal-to-globalobstructiontodeformthem;Theorem2.1
andTheorem3.2. Thisinvolvesanewtechnique(Theorem4.4)whichgeneralizesaresult
ofBurns-Wahl[3]describingthespaceoffirstorderdeformationsofasingularcomplex
surfacewithonlyrationaldoublepoints.
Notations. A cyclic quotient singularity (germ at (0,0) of) C2/G, where G=(cid:104)(x,y)(cid:55)→
(ζx,ζqy)(cid:105) with ζ a m-th primitive root of 1, 1<q<m, and (q,m)=1, is denoted by
1(1,q). A means 1(1,m−1). A(−1)-curve(or(−2)-curve)inasmoothsurfaceisan
m m m
embedded CP1 with self-intersection −1 (respectively, −2). Throughout this paper we
use the same letter to denote a curve and its proper transform under a birational map. A
singularityofclassT isaquotientsingularitywhichadmitsaQ-Gorensteinoneparameter
smoothing. Theyare eitherrationaldouble pointsor 1 (1,dna−1)with1<a<n and
dn2
(n,a)=1;seeKolla´r–Shepherd-Barron[14,§3]. ForanormalvarietyX itstangentsheaf
TX isHomOX(Ω1X,OX). ThedimensionofHiishi.
Acknowledgements. TheauthorswouldliketothankProfessorYongnamLeeforhelpful
discussionduringthework,carefulreadingofthedraftversion,andmanyvaluablecom-
ments. The authors also wish to thank Professor Jenia Tevelev for indicating a mistake
in an earlier version of this paper. Heesang park was supported by Basic Science Re-
search Program through the National Research Foundation of Korea (NRF) grant funded
bytheKoreanGovernment(2011-0012111). DongsooShinwassupportedbyBasicSci-
enceResearchProgramthroughtheNationalResearchFoundationofKorea(NRF)grant
fundedbytheKoreanGovernment(2010-0002678). GiancarloUrzu´awassupportedbya
FONDECYTIniciograntfundedbytheChileanGovernment(11110047).
2. NUMERICALGODEAUXSURFACESWITHA2-DIVISIBLEDIVISOR
InthissectionweconstructafamilyofsimplyconnectednumericalGodeauxsurfaces
havinga2-divisibledivisorconsistingoffourdisjoint(−2)-curvesbysmoothingasingu-
lar surfaceY(cid:101); Theorem 2.4. This is the key to construct numerical Campedelli surfaces
withaninvolution. Inaddition, wedescribetheexplicitstablemodelofthesingularsur-
faceY(cid:101). Infact,weconstructarationalnormalprojectivesurfaceY(cid:48) withfoursingularities
A3, A3, 812(1,8·5−1), 712(1,7·4−1) and KY(cid:48) ample. HenceY(cid:48) is a stable surface (cf.
Kolla´r-Shepherd-Barron[14],Hacking[11]). WewillprovethattheversalQ-Gorenstein
deformationspaceDefQG(Y(cid:48))(cf.Hacking[11,§3])issmoothand8dimensional,andthat
theQ-GorensteinsmoothingsofY(cid:48) aresimplyconnectednumericalGodeauxsurfaces. In
particular,thisshowsthattherearesimplyconnectednumericalGodeauxsurfaceswhose
canonicalmodelhaspreciselytwoA singularities;Corollary2.3. Furthermoreafourdi-
3
mensionalfamilyinDefQG(Y(cid:48))producestheabovesimplyconnectednumericalGodeaux
surfaces with a 2-divisible divisor consisting of four disjoint (−2)-curves; Theorem 2.4
andTheorem5.2.
2.1. ArationalellipticsurfaceE(1). WestartwitharationalellipticsurfaceE(1)with
an I -singular fiber, an I -singular fiber, and two nodal singular fibers. In fact we will
8 2
usethesamerationalellipticsurfaceE(1)inthepapersH.Park-J.Park-D.Shin[20,21].
4 H.PARK,D.SHIN,ANDG.URZU´A
However,weneedtosketchtheconstructionofE(1)toshowtherelevantcurvesthatwill
beusedtobuildthesingularsurfacesY(cid:101)andY(cid:48).
LetL , L , L , and(cid:96)belinesinCP2 andletcbeasmoothconicinCP2 givenbythe
1 2 3
followingequations. TheyintersectasinFigure1.
√ √
L :2y−3z=0, L :y+ 3x=0, L :y− 3x=0
1 2 3
(cid:96):x=0, c:x2+(y−2z)2−z2=0.
ℓ
p
b
c s
b L3
L1
qb L2 rb
FIGURE1. Apencilofcubiccurves
Weconsiderthepencilofcubics
√ √
{λ(y− 3x)(y+ 3x)(2y−3z)+µx(x2+(y−2z)2−z2)|[λ :µ]∈CP1}
generatedbythetwocubiccurvesL +L +L and(cid:96)+c. Thispencilhasfourbasepoints
1 2 3 √
p,q,r,s,andfoursingularmemberscorrespondingto[λ :µ]=[1:0],[0:1],[2:3 3],[2:
√
−3 3]. Thelattertwosingularmembersarenodalcurves, denotedbyF andF respec-
√ √ 1 2
tively. Theyhavenodesat[− 3:0:1]and[ 3:0:1],respectively.
In order to obtain a rational elliptic surface E(1) from the pencil, we resolve all base
points (including infinitely near base-points) of the pencil by blowing-up 9 times as fol-
lows. Wefirstblowupatthepoints p,q,r,s. Lete ,e ,e ,e betheexceptionaldivisors
1 2 3 4
overp,q,r,s,respectively.Weblowupagainatthethreepointse ∩L ,e ∩L ,e ∩(cid:96).Let
1 3 2 2 3
e ,e ,e betheexceptionaldivisorsovertheintersectionpoints,respectively. Wefinally
5 6 7
blowupateachintersectionpointse ∩cande ∩c. Lete ande betheexceptionaldi-
5 6 8 9
visorsovertheblown-uppoints. WethengetarationalellipticsurfaceE(1)=CP2(cid:93)9CP2
overCP1;seeFigure2.
The four exceptional curves e , e , e , e are sections of the elliptic fibration E(1),
4 7 8 9
which correspond to the four base points s, r, p, q, respectively. The elliptic fibration
E(1)hasoneI8-singularfiber∑8i=1Bi containingallLi (i=1,2,3): B2=L2,B3=L3,and
B =L ;cf.Figure2. WewillusefrequentlythesumB=B +B +B +B ,whichwill
5 1 1 2 3 4
beshowntobe2-divisible. ThesurfaceE(1)hasalsooneI -singularfiberconsistingof(cid:96)
2
andc,andithastwomorenodalsingularfibersF andF .
1 2
ThereisaspecialbisectiononE(1). LetMbethelineinCP2passingthroughthepoint
√ √
r=[0:0:1]andthetwonodes[− 3:0:1]and[ 3:0:1]ofF andF . SinceMmeets
1 2
every member in the pencil at three points but it passes through only one base point, the
proper transform of M is a bisection of the elliptic fibration E(1)→CP1; cf. Figure 2.
NotethatM2=0inE(1).
ANUMERICALCAMPEDELLISURFACEWITHANINVOLUTION 5
e
4
e
9
F F ℓ
B 1 2
8
B B
4 3 c
B
7 M
B
5
B B
1 2
B
6
e
7
e
8
FIGURE2. ArationalellipticsurfaceE(1)
A 2-divisible divisor on E(1). Let h∈Pic(E(1)) be the class of the pull-back of a line
in CP2. We denote again by e ∈Pic(E(1)) the class of the pull-back of the exceptional
i
divisore. WehavethefollowinglinearequivalencesofdivisorsinE(1):
i
B ∼e −e , B ∼h−e −e −e , B ∼h−e −e −e ,
1 2 6 2 2 3 6 3 1 3 5
B ∼e −e , B ∼h−e −e −e , B ∼e −e ,
4 1 5 5 1 2 4 6 6 9
B ∼e −e , B ∼e −e , F ∼3h−e −···−e ,
7 3 7 8 5 8 1 1 9
F ∼3h−e −···−e , (cid:96)∼h−e −e −e .
2 1 9 3 4 7
Let L:=h−e −e −e . Note that the divisor B=B +B +B +B is 2-divisible
3 5 6 1 2 3 4
becauseoftherelation
B=B +B +B +B ∼2(h−e −e −e )=2L. (2.1)
1 2 3 4 3 5 6
2.2. ArationalsurfaceZ=E(1)(cid:93)7CP2. IntheconstructionofZ,weuseonlyonesection
S:=e . WefirstblowupatthetwonodesofthenodalsingularfibersF andF sothatwe
4 1 2
obtainablown-uprationalellipticsurfaceW =E(1)(cid:93)2CP2;Figure3.LetE andE bethe
1 2
exceptionalcurvesoverthenodesofF andF , respectively. Wefurtherblowupateach
1 2
threemarkedpoints•andblowuptwiceatthemarkedpoint(cid:74)inFigure3. Wethenget
Z=E(1)(cid:93)7CP2 asinFigure4. ThereexisttwolinearchainsoftheCP1 inZ whosedual
graphsaregivenby:
−2 −3 −5 −3 −2 −6 −2 −3
C = ◦ − ◦ − ◦ − ◦, C = ◦ − ◦ − ◦ − ◦,
8,5 7,4
whereC consistsof(cid:96),S,F ,E ,andC containsF ,E ,M.
8,5 1 1 7,4 2 2
2.3. Numerical Godeaux surfaces. We construct rational singular surfaces which pro-
duceunderQ-Gorensteinsmoothingssimplyconnectedsurfacesofgeneraltypewithp =
g
0andK2=1.
We first contract the two chainsC andC of CP1’s from the surface Z so that we
8,5 7,4
have a normal projective surfaceY(cid:101) with two singularities p1, p2 of class T: 812(1,8·5−
1),712(1,7·4−1). Denotethecontractionmorphismbyα(cid:101): Z→Y(cid:101). LetY bethesurface
obtained by contracting the four (−2)-curves B1,...,B4 inY(cid:101). We denote the contraction
morphism by α: Y(cid:101) →Y. Then Y is also a normal projective surface with singularities
p1, p2 fromY(cid:101), and four A1’s (ordinary double points), denoted by q1,...,q4. We finally
6 H.PARK,D.SHIN,ANDG.URZU´A
S
b ×2
F F ℓ
B 1 2
8
B B
4 3 E1 E2 c
B
7 M
B b
5
B B
1 2
B6 b b
FIGURE3. Ablown-uprationalellipticsurfaceW =E(1)(cid:93)2CP2
−3
−2−1
−5 −6 −2
B
8
B B
4 3 −3 −2
−3
−1
−1 −1
B B
1 2
B
6
FIGURE4. ArationalsurfaceZ=E(1)(cid:93)7CP2
contractC ,C ,B +B +B andB +B +B inZtoobtainY(cid:48). Ithasthesingularities
8,5 7,4 4 8 3 1 6 2
1(1,8·5−1), 1(1,7·4−1),andtwoA = 1(1,3)’s. Letα(cid:48): Z→Y(cid:48)bethecontraction.
82 72 3 4
In Section 4 we will prove that the obstruction spaces to local-to-global deformations
ofthesingularsurfacesY(cid:101),Y andY(cid:48)vanish. Thatis:
Theorem2.1. H2(Y(cid:101),TY(cid:101))=0,H2(Y,TY)=0,andH2(Y(cid:48),TY(cid:48))=0.
ThesingularsurfaceY(cid:48)isthestablemodelofthesingularsurfacesY(cid:101)andY:
Proposition 2.2. The surfaceY(cid:48) has KY2(cid:48) =1, pg(Y(cid:48))=0, and KY(cid:48) is ample. The space
DefQG(Y(cid:48)) is smooth and 8 dimensional. A Q-Gorenstein smoothing of Y(cid:48) is a simply
connectedcanonicalsurfaceofgeneraltypewith p =0andK2=1.
g
Proof. ForasurfaceY(cid:48)withonlysingularitiesoftypeT wehave
K2 =K2+ ∑ s − ∑ µ ,
Y(cid:48) Z p p
p∈Sing(Y(cid:48)) p∈Sing(Y(cid:48))
wheres isthenumberofexceptionalcurvesover pandµ istheMilnornumberof p. In
p p
ourcase,K2 =−7+4+4=1. Wehave p (Y(cid:48))=q(Y(cid:48))=0becauseoftherationalityof
Y(cid:48) g
thesingularities.
ANUMERICALCAMPEDELLISURFACEWITHANINVOLUTION 7
Wenowcomputeα(cid:48)∗(KY(cid:48))inaQ-numericallyeffectiveway. LetF bethegeneralfiber
oftheellipticfibrationinZ. LetE ,E ,E betheexceptionalcurvesoverM∩E ,F ∩E ,
3 4 5 1 1 1
F ∩E respectively. Let E , E be the exceptional curves over F ∩S with E2 =−1.
2 2 6 7 2 7
Then,KZ∼−F+∑7i=1Ei+E3+E4+E5+E7. WealsohaveF∼F1+2E1+2E3+3E4∼
F +2E +3E +E +E . WritingF≡ 1F+1F inK andaddingthediscrepanciesfrom
2 2 5 6 7 2 2 Z
p and p ,weobtain
1 2
3 5 5 5
α(cid:48)∗(KY(cid:48))≡ F1+ F2+ E1+ E2+E3
8 14 8 7
1 1 13 3 4 3 6
+ E + E + E + E + M+ (cid:96)+ S.
4 5 6 7
2 2 14 2 7 8 8
We now intersect α(cid:48)∗(KY(cid:48)) with all the curves in its support, which are not contracted
byα(cid:48), tocheckthatKY(cid:48) isnef. Moreover, ifΓ.α(cid:48)∗(KY(cid:48))=0foracurveΓnotcontracted
byα(cid:48),thenΓisacomponentofafiberintheellipticfibrationwhichdoesnotintersectany
curve in the support. This is because F , E , E , and E belong to the support, and they
1 1 3 4
are the components of a fiber. One easily checks that Γ does not exist, proving that KY(cid:48)
is ample. Therefore any Q-Gorenstein smoothing ofY(cid:48) over a (small) disk will produce
canonicalsurfaces;cf.Kolla´r-Mori[13,p.34].
TocomputethefundamentalgroupofaQ-Gorensteinsmoothingweusetherecipein
Y.Lee-J.Park[15]. WefollowtheargumentasinY.Lee-J.Park[15,p.493]. Considerthe
normalcirclesaroundMandE . WecancomparethemthroughthetransversalsphereE .
1 3
Sincetheordersofthecirclesare49and64,whicharecoprime,weobtainthatbothend
upbeingtrivial.
The smoothness of DefQG(Y(cid:48)) follows from Theorem 2.1 and Hacking [11, §3]. To
computethedimension,weobservethatifY(cid:48)→∆isaQ-GorensteinsmoothingofY(cid:48)=
0
Y(cid:48) and TY(cid:48)|∆ is the dual of Ω1Y(cid:48)|∆, then TY(cid:48)|∆ restricts to Yt(cid:48) as TYt(cid:48) (tangent bundle of
Yt(cid:48))whent(cid:54)=0,andTY(cid:48)|∆|Y(cid:48)⊂TY(cid:48) withcokernelsupportedatthesingularpointsofY0(cid:48);
cf.Wahl[25]. Thentheflatn0essofT0Y(cid:48)|∆ andsemicontinuityincohomologyplusthefact
that H2(Y(cid:48),TY(cid:48))=0 gives H2(Yt(cid:48),TYt(cid:48))=0 for anyt. But then, since Yt(cid:48) is of general
type,Hirzebruch-Riemann-RochTheoremsays
H1(Yt(cid:48),TYt(cid:48))=10χ(Yt(cid:48),OYt(cid:48))−2KY2t(cid:48) =10−2=8.
Thisprovestheclaim. (cid:3)
Corollary2.3. Thereisatwodimensionalfamilyofsimplyconnectedcanonicalnumerical
GodeauxsurfaceswithtwoA singularities.
3
Proof. Weconsiderthesequence
0→H1(Y(cid:48),TY(cid:48))→TQ1G,Y(cid:48) →H0(Y(cid:48),TQ1G,Y(cid:48))→0
attheendofHacking[11,§3]. WejustprovedthatT1 is8dimensional,andweknow
QG,Y(cid:48)
thatH0(Y(cid:48),T1 )is8dimensional,sinceeachA gives3dimensionsandeach p gives1
QG,Y(cid:48) 3 i
dimension. ThereforeH1(Y(cid:48),TY(cid:48))=0. Toproducetheclaimedfamilyweneedtosmooth
upatthesametime p and p . (cid:3)
1 2
A simply connected numerical Godeaux surfaces with a 2-divisible divisor consisting
of four disjoint (−2)-curves is obtained from a Q-Gorenstein smoothing of the singular
surfaceY(cid:101):
8 H.PARK,D.SHIN,ANDG.URZU´A
Theorem2.4. (a) ThereisaQ-GorensteinsmoothingY(cid:102)→∆overadisk∆withcentral
fiberY(cid:102)0=Y(cid:101)andaneffectivedivisorB⊂Y(cid:102)suchthattherestrictiontoafiberY(cid:101)t over
t∈∆
Bt :=B∩Y(cid:101)t =B1,t+B2,t+B3,t+B4,t
is2-divisibleinY(cid:101)t consistingoffourdisjoint(−2)-curvesandB0=B.
(b) ThereisaQ-GorensteindeformationY →∆ofY withcentralfiberY =Y suchthat
0
afiberY overt (cid:54)=0hasfourordinarydoublepointsasitsonlysingularitiesandthe
t
minimalresolutionofYt isthecorrespondingfiberY(cid:101)t ofY(cid:102)→∆.
Proof. WeapplyasimilarmethodinY.Lee-J.Park[16]. Sinceanylocaldeformationsof
thesingularitiesofY canbeglobalizedbyTheorem2.1,thereareQ-Gorensteindeforma-
tionsofY overadisk∆whichkeepallfourordinarydoublepointsandsmoothup p and
1
p . LetY →∆besuchdeformation, withY asitscentralfiber, andY (t (cid:54)=0)anormal
2 t
projectivesurfacewithfourA sasitsonlysingularities. Weresolvesimultaneouslythese
1
four singularities in each fiber Yt. We then get a family Y(cid:102)→∆ that is a Q-Gorenstein
smoothingofthecentralfiberY(cid:101),whichshowsthataQ-GorensteinsmoothingofY canbe
liftedtoaQ-Gorensteinsmoothingofthepair(Y,B),i.e. the2-divisibledivisorBonY(cid:101) is
extendedtoaneffectivedivisorB⊂Y(cid:102).
We finally show that the effective divisor Bt is 2-divisible inY(cid:101)t for t (cid:54)=0. According
to Manetti [18, Lemma 2], the natural restriction map rt :Pic(Y(cid:102))→Pic(Y(cid:101)t) is injective
foreveryt ∈∆andbijectivefor0∈∆. Hereweareusingthat pg(Y(cid:101))=q(Y(cid:101))=0. Since
the divisor B is nonsingular, B is also nonsingular. Since B∼2L in (2.1), it follows
t
that Bt ∼2Lt, where L is extended to a line bundle L ⊂Y(cid:102)and Lt is the corresponding
restriction. (cid:3)
3. NUMERICALCAMPEDELLISURFACESWITHANINVOLUTION
ThemainpurposeofthissectionistoconstructsimplyconnectednumericalCampedelli
surfaceswithaninvolution. Alongtheway,wewillintroducearationalnormalprojective
surfaceX(cid:48)with6singularities(twoA1,two 812(1,8·5−1),andtwo 712(1,7·4−1))andKX(cid:48)
ample.AcertainfourdimensionalQ-GorensteindeformationofX(cid:48)willproducenumerical
Campedellisurfaceswithaninvolution.
Recall that the rational surface Z has a 2-divisible divisor B = B +B +B +B ;
1 2 3 4
cf. (2.1). LetV be the double cover of Z branched along the divisor B, where the dou-
blecoverisgivenbythedataB∼2L,L=h−e −e −e . Wedenotethedoublecovering
3 5 6
byψ :V →Z. ThesurfaceV hastwoC ’sandtwoC ’s.
8,5 7,4
OntheotherhandthesurfaceV canbeobtainedfromacertainrationalellipticsurface
by blowing-ups, as we now explain. The morphism ψ: V →Z blows down to a double
coverψ(cid:48):V(cid:48)→E(1)branchedalongB. Theramificationdivisorψ(cid:48)−1(B)consistsoffour
disjoint (−1)-curves R ,...,R . We blow down them fromV(cid:48) to obtain a surface E(1)(cid:48);
1 4
cf. Figure 5. In Figure 5 the pull-back of the B are the B(cid:48), of the curve (cid:96) is (cid:96) +(cid:96) , of
i i 1 2
thesectionS isS +S , andofthedoublesectionM isM +M . EachI inE(1)(cid:48) isthe
1 2 1 2 2
pull-backofeachI inE(1).
1
NotethatthesurfaceE(1)(cid:48) hasanellipticfibrationstructurewithtwoI -singularfibers
4
and two I -singular fibers. In fact, the surface E(1)(cid:48) can be obtained from the pencil of
2
cubicsinCP2
{λx(y−z)(x−2z)+µy(x−z)(y−2z)|[λ :µ]∈CP1}
ANUMERICALCAMPEDELLISURFACEWITHANINVOLUTION 9
S
1
B′ ℓ
8 1
M
1
B′ B′
5 7 M2
B6′ ℓ2
S
2
FIGURE5. RelevantcurvesintheellipticrationalsurfaceE(1)(cid:48)
wheretheI -singularfiberscomefromx(y−z)(x−2z)=0andy(x−z)(y−2z)=0,andthe
4
I -singularfiberscomefrom(x+y−2z)(xy−yz−xz)=0and(x−y)(xy−xz+2z2−yz)=
2 √
0. The two double sections M and M are defined by the lines x = (1+ −1)z and
√ 1 2
x=(1− −1)z. Insummary:
Proposition3.1. ThesurfaceV(cid:48)istheblow-upatfournodesofoneI -singularfiberofthe
4
rationalellipticfibrationE(1)(cid:48)→CP1. HencethesurfaceV canbeobtainedfromV(cid:48) by
blowing-upintheobviousway.
LetX(cid:101)bethedoublecoverofthesingularsurfaceY(cid:101)branchedalongthedivisorB. Note
that the surface X(cid:101) is a normal projective surface with four singularities of class T whose
resolutiongraphsconsistoftwoC8,5’sandtwoC7,4’s. TheramificationdivisorinX(cid:101) con-
sistsofthefourdisjoint(−1)-curvesR1,...,R4. Letφ(cid:101):X(cid:101)→Y(cid:101)bethedoublecovering. On
theotherhandthesurfaceX(cid:101)canbeobtainedfromtherationalsurfaceV bycontractingthe
twoC8,5’sandtwoC7,4’s. Letβ(cid:101):V →X(cid:101)bethecontractionmorphism.
Let X be the surface obtained by blowing down the four (−1)-curves R ,...,R from
1 4
X(cid:101). Wedenotetheblowing-downmorphismbyβ :X(cid:101)→X. Thenthereisadoublecovering
φ :X →Y branchedalongthefourordinarydoublepointsq ,...,q . Finally,letX(cid:48) bethe
1 4
contractionofthe(−2)-curvesB(cid:48) andB(cid:48) inX. Letβ(cid:48):V →X(cid:48)bethecontraction. Tosum
8 6
up,wehavethefollowingcommutativediagram:
E(1)(cid:48) (cid:111)(cid:111) V(cid:48) (cid:111)(cid:111) V β(cid:101) (cid:47)(cid:47)X(cid:101) β (cid:47)(cid:47)X (cid:47)(cid:47)X(cid:48)
ψ(cid:48) ψ φ(cid:101) φ
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
E(1)(cid:111)(cid:111) Z α(cid:101) (cid:47)(cid:47)Y(cid:101) α (cid:47)(cid:47)Y (cid:47)(cid:47)Y(cid:48)
WewillshowinSection4theobstructionspacestolocal-to-globaldeformationsofthe
singularsurfacesX(cid:101),X andX(cid:48)vanish:
Theorem3.2. H2(X(cid:101),TX(cid:101))=0,H2(X,TX)=0,andH2(X(cid:48),TX(cid:48))=0.
ThesingularsurfaceX(cid:48)isthestablemodelofX(cid:101)andX:
Proposition 3.3. The surface X(cid:48) has KX2(cid:48) =2, pg(X(cid:48))=0, and KX(cid:48) ample. The space
DefQG(X(cid:48)) is smooth and 6 dimensional. A Q-Gorenstein smoothing of X(cid:48) is a simply
connectedcanonicalsurfaceofgeneraltypewith p =0andK2=2.
g
10 H.PARK,D.SHIN,ANDG.URZU´A
Proof. The proof goes as the one forY(cid:48) in Proposition 2.2, using the explicit model we
have forV by blowing-up E(1)(cid:48) in Proposition 3.1. One can check that an intersection
computationasinProposition2.2verifiesamplenessforKX(cid:48). (cid:3)
TheproofofthenextmainresultfollowseasilyfromTheorem2.4.
Theorem3.4. ThereexistQ-GorensteinsmoothingsX(cid:102)→∆ofX(cid:101) andX →∆ofX that
are compatible with the Q-Gorenstein deformations of Y(cid:102)→∆ ofY(cid:101) and Y →∆ ofY in
Theorem2.4,respectively;thatis,thedoublecoveringsφ(cid:101):X(cid:101)→Y(cid:101) andφ :X →Y extend
tothedoublecoveringsφ(cid:101)t:X(cid:101)t→Y(cid:101)t andφt:Xt→Yt betweenthefibersoftheQ-Gorenstein
deformations.
Remark3.5. ByTheorem3.2,theobstructionH2(X,T )tolocal-to-globaldeformations
X
of the singular surface X vanishes. The point of the above theorem is that there is a Q-
GorensteinsmoothingofthecoverXthatiscompatiblewiththeQ-Gorensteindeformation
ofthebaseY.
Corollary3.6. AgeneralfiberX oftheQ-GorensteinsmoothingX →∆ofX isasim-
t
plyconnectednumericalCampedellisurfacewithaninvolutionσ suchthattheminimal
t
resolutionofthequotientY =X/σ isasimplyconnectednumericalGodeauxsurface.
t t t
3.1. The fundamental group of the quotient by an involution. Let X be a minimal
complex surface of general type with p =0 and K2 =2. Suppose that the group Z/2Z
g
acts on X with just 4 fixed points. LetY =X/(Z/2Z) be the quotient and let S→Y be
the minimal resolution ofY. Barlow [2, Proposition 1.3] proved that if πalg(S)=1 then
1
|πalg(X)|=1,3,5,7,9. Conversely:
1
Proposition3.7. IfX issimplyconnected,thensoisS.
Proof. Let f :X →Y be the quotient map. Then f is a double covering branched along
the four ordinary double points ofY. Let T be the blow-up of X at the four ramification
points. DenotethefourexceptionaldivisorsbyE ,...,E . Thenthereisadoublecovering
1 4
g:T →S branched along the four (−2)-curves R ,...,R which is an extension of the
1 4
covering f :X →Y. LetX =T\∪4 E andY =S\∪4 R. Theng| :X →Y isan
0 i=1 i 0 i=1 i X0 0 0
e´tale double covering. Since π (X )=π (X)=1, we have π (Y )=Z/2Z. Therefore
1 0 1 1 0
π (S)istrivialorZ/2ZbyvanKampentheorem.
1
Supposethatπ (S)=Z/2Z. Thenthereisane´taledoublecoveringh:U →Ssuchthat
1
U issimplyconnectedandconnected. ThenthefiberproductofT andU overSisane´tale
doublecoverofT. Letφ :T× U →U bethedoublecovering,whichisbranchedalong
Y
theeight(−2)-curvesh∗(∑4i=1Ri). Thatis,wehavethefollowingcommutativediagram:
(cid:111)(cid:111) (cid:111)(cid:111) e´tale
X T T× U
Y
2:1
f g φ
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
(cid:111)(cid:111) (cid:111)(cid:111) h
Y S U
Set U0 =U\h∗(∑4i=1Ri). Since h:U0 →Y0 is e´tale and π1(Y0)=Z/2Z, it follows
that π (U )=1. Furthermore, since T is simply connected, the fiber product T × U
1 0 Y
is diffeomorphic to the disjoint union T(cid:116)T. Then φ :X (cid:116)X →U is an e´tale double
0 0 0
coveringandπ (U )=1andU isconnected. ThereforeX (cid:116)X isdiffeomorphictoU (cid:116)
1 0 0 0 0 0
U . Notethat∂X isdiffeomorphictothedisjointunionof3-spheresS3. However∂U is
0 0 0
diffeomorphictothedisjointunionofthelensspaceL(2,1),whichisacontradiction. (cid:3)