Two Irreducible Components of the Moduli Space M can 1,3

Chen, Yifan
Two Irreducible Components of the Moduli Space M can 1,3

Author: Chen, Yifan
Year: 2012
Source: https://epub.uni-bayreuth.de/id/eprint/260/1/Diss.pdf
Two I educible Componen s o
he Moduli Space Mcan
1,3
DISSERTATION
zu E langung
des DOKTORGRADES(DR. RER. NAT.)
de FAKULT¨
AT F¨
UR MATHEMATIK, PHYSIK UND INFORMATIK
de UNIVERSIT¨
AT BAYREUTH
o geleg on
YIFAN CHEN
aus P. R. China
BAYREUTH
Tag de Ein eichung: 13. Janua 2012
Tag des Kolloquiums: 23. Feb ua 2012
Ange e ig mi de Genehmigung de Fakul ¨a ¨u Ma hema ik, Physik
und In o ma ik de Uni e si ¨a Bay eu h.
1. Gu ach e : P o . Do . Fab izio Ca anese
2. Gu ach e : P o . D . Miles Reid, Uni e si y o Wa wick
3. Gu ach e : P o . D . Jin-Xing Cai, Peking Uni e si y
E kl¨a ung
Ich e siche e eidess a lich, dass ich die A bei selbs ¨andig e ass und keine
ande en als die on mi angegebenen Quellen und Hil smi el benu z habe.
Ich bes ¨a ige, dass ich keine ¨uhe e P omo ions e suche gemach habe.
Un e sch i des Au o s
Con en s
Zusammen assung i
Abs ac iii
Acknowledgemen s
In oduc ion i
No a ion and con en ions xi
Figu es xi
I P elimina ies 1
1 Bidouble Co e s o Su aces 1
2 In olu ions on Ra ional Double Poin s 4
3 No mal Cubic Su aces 9
3.1 3A1- ype Cubic Su aces . . . . . . . . . . . . . . . . . . . . . 12
3.2 D4(1)- ype and D4(2)- ype Cubic Su aces . . . . . . . . . . . 13
3.3 4A1- ype Cubic Su ace . . . . . . . . . . . . . . . . . . . . . . 15
II The I educible Componen Con aining
he Ex ended Bu nia Su aces 16
4 Bu nia Su aces and Ex ended Bu nia Su aces 17
5 One Pa ame e Limi s o he Ex ended Bu nia Su aces 20
6 Exclusion o Ce ain Cubic Su aces 27
7D4-gene alized Bu nia Su aces 30
7.1 Con igu a ion o B anch Di iso s . . . . . . . . . . . . . . . . 31
7.2 D4-gene alized Bu nia Su aces . . . . . . . . . . . . . . . . . 34
84A1-gene alized Bu nia Su aces 35
8.1 Con igu a ion o B anch Di iso s . . . . . . . . . . . . . . . . 35
8.2 4A1-gene alized Bu nia Su aces . . . . . . . . . . . . . . . . 38
9 I educible Componen 40
9.1 Con igu a ion o B anch Di iso s on 3A1- ype Cubic Su aces . 40
9.2 Closu e o he Open Subse N EB3............... 42
III De o ma ions o Gene alized Bu nia Su aces 43
10 Key Tools o Calcula e he Cohomology G oups o
he Tangen Shea es 44
11 De o ma ions o he D4-gene alized Bu nia Su aces 47
12 De o ma ions o he 4A1-gene alized Bu nia Su aces 56
IV The I educible Componen con aining
he Keum-Naie-Mendes Lopes-Pa dini Su aces 61
13 Keum-Naie-Mendes Lopes-Pa dini Su aces 62
14 A Sub amily o KNMP Su aces 63
15 Local De o ma ions and I educible Componen 66
Re e ences 75

ZUSAMMENFASSUNG
Zusammen assung
Das Ziel diese Disse a ion is es zwei Familien on Fl¨achen on allgemeinem
Typ mi pg= 0 und K2= 3 zu s udie en. Genaue gesag handel es sich
um die e wei e en Bu nia Fl¨achen mi K2= 3 und die Keum-Naie-Mendes
Lopes-Pa dini Fl¨achen. Wi konzen ie en uns au die lokalen De o ma ionen
diese Fl¨achen und au die Modul ¨aume, die diesen Fl¨achen en sp echen.
Die e wei e en Bu nia Fl¨achen mi K2= 3 wu den zue s on Baue
und Ca anese in [BC10-b] kons uie , wo sie auch Bu nia Fl¨achen mi K2=
3 s udie en ( gl. auch [Bu66] und [Pe 77]). Sie haben gezeig , dass de
en sp echende Modul aum in dem Modul aum on Fl¨achen on allgemeinem
Typ i eduzibel, o en und on de Dimension 4 is , und, dass de Abschluss
dieses Modul aums eine i eduzible Komponen e des Modul ams on Fl¨achen
on allgemeinem Typ is .
Das e s e Ziel diese A bei is , alle Degene a ionen de e wei e en Bu -
nia Fl¨achen mi K2= 3 zu besch eiben. Dazu zeigen wi zue s , dass die
einpa ame ige Degene a ion de kanonischen Modelle diese Fl¨achen eine
endliche, lache (Z/2Z)2-¨
Ube lage ung on no malen singul¨a en kubischen
Fl¨achen is . Danach zeigen wi mi els de Klassi ika ions heo ie de ku-
bischen Fl¨achen und du ch die Un e suchung des Ve zweigungso s diese
¨
Ube lage ungen, dass genau zwei Familien on Degene a ionen exis ie en,
die in [BC10-b] besch ieben wu den. Somi beweisen wi , dass die Ve eini-
gung de R¨aume, besch ieben in [BC10-b], a s¨achlich die ganze i eduzible
Komponen e des Modul ams is .
Da ¨ube hinaus s udie en wi die lokalen De o ma ionen de Degene a-
ionen de e wei e en Bu nia Fl¨achen mi K2= 3. Un e Zuhil enahme
des S uk u sa zes de (Z/2Z)2-¨
Ube lage ungen sind wi in de Lage, die
Dimensionen de Eigen ¨aume de Kohomologieg uppen de Tangen ialga be
zu bes immen. Wi zeigen, dass de Basis aum de Ku anishi Familie eine
Fl¨ache in eine de zwei Familien de Degene a ionen gla is .
Im zwei en Teil de Disse a ion un e suchen wi Keum-Naie Fl¨achen mi
K2= 3 ([Ke88] und [Na94]) und de en De o ma ionen, die on Mendes Lopes
und Pa dini kons uie wu den. Wi nennen wi diese Fl¨achen Keum-Naie-
i
ZUSAMMENFASSUNG
Mendes Lopes-Pa dini Fl¨achen. In [MP04] wu de gezeig , dass de Abschluss
de en sp echenden Teilmenge diese Fl¨achen im Modul aum i eduzibel,
uni uled und de Dimension 6 is .
Wi kons uie en eine Un e amilie diese Fl¨achen. Die Fl¨achen in un-
se e Familie sind endliche lache (Z/2Z)2-¨
Ube lage ungen eine kubischen
Fl¨ache mi ie Kno en. Sie haben einen amplen kanonischen Di iso . Die
bikanonische Abbildung diese Fl¨ache is die Komposi ion de ¨
Ube lage ung
mi de an ikanonischen Einbe ung de kubischen Fl¨ache. Da aus olg , dass
die bikanonische Abbildung diese Fl¨ache eine Komposi ion mi eine In olu-
ion aus de Galoisg uppe de ¨
Ube lage ung is , so dass die Quo ien en l¨ache
diese In olu ion eine En iques Fl¨ache mi A1-Singula i ¨a en is . Diese Eigen-
scha cha ak e isie alle Mendes Lopes-Pa dini Fl¨achen [MP04].
Un e Zuhil enahme des S uk u sa zes de (Z/2Z)2-¨
Ube lage ungen sind
wi in de Lage eine obe e Sch anke ¨u die Dimension de Kohomologieg up-
pen de Tangen ialga be diese Fl¨achen zu geben. Du ch Kombina ion un-
se e E gebnisse und den E gebnissen aus [MP04] zeigen wi , dass ¨u eine
gene ische Fl¨ache Sin unse e Un e amilie h1(S, ΘS) = 6, h2(S, ΘS) = 2 gil ,
und de Basis aum de Ku anishi Familie gla is . Somi zeigen wi , dass
de Abschluss de Teilmenge des Modul aums, die den Keum-Naie-Mendes
Lopes-Pa dini Fl¨achen en sp ich , eine i eduzible Komponen e is .
ii
ABSTRACT
Abs ac
This hesis is de o ed o he s udy o wo amilies o su aces o gene al ype
wi h pg= 0 and K2= 3: ex ended Bu nia su aces wi h K2= 3 and Keum-
Naie-Mendes Lopes-Pa dini su aces. We ocus on he local de o ma ions o
hese su aces and he co esponding subse s in he Gieseke moduli space.
Ex ended Bu nia su aces wi h K2= 3 we e cons uc ed by Baue and
Ca anese [BC10-b] in he cou se o s udying Bu nia su aces wi h K2= 3
(c . [Bu66] and [Pe 77]). They showed ha he co esponding subse in he
moduli space is an i educible open subse o dimension 4,and i s closu e is
an i educible componen o he moduli space.
The i s goal o his hesis is o desc ibe all he degene a ions o he
ex ended Bu nia su aces wi h K2= 3.Fo his, we i s show ha he
one pa ame e limi s o he canonical models o hese su aces a e ini e la
(Z/2Z)2-co e s o no mal singula cubic su aces. Then by applying he
classi ica ion heo y o cubic su aces and by in es iga ing he b anch loci o
such co e s, we show ha he e a e exac ly wo amilies o degene a ions,
which had been desc ibed in [BC10-b]. Thus we p o e ha he union o he
loci desc ibed in [BC10-b] is indeed he ull i educible componen in he
moduli space.
We also s udy he local de o ma ions o he degene a ions o ex ended
Bu nia su aces wi h K2= 3.Using he s uc u e heo em o (Z/2Z)2-
co e s, we a e able o calcula e he dimensions o he eigenspaces o he
cohomology g oups o he angen shea es. We show ha he base o he
Ku anishi amily o a su ace in one o he wo amilies o degene a ions is
smoo h.
Ano he opic o his hesis is o s udy he Keum-Naie su aces wi h
K2= 3 (c . [Ke88] and [Na94]) and hei de o ma ions cons uc ed by Mendes
Lopes and Pa dini [MP04]. We call all hese su aces Keum-Naie-Mendes
Lopes-Pa dini su aces. I is showed in [MP04] ha he closu e o he co e-
sponding subse o such su aces in he moduli space is i educible, uni uled
and o dimension 6.
We cons uc a sub amily o such su aces. The su aces in ou amily
iii
ABSTRACT
a e ini e la (Z/2Z)2-co e s o a 4-nodal cubic su ace. They ha e ample
canonical di iso s. Mo eo e , he bicanonical maps o hese su aces a e he
composi ion o he co e ing mo phisms and he an icanonical embedding o
he 4-nodal cubic su ace. I ollows ha he bicanonical map o such a
su ace is composed wi h an in olu ion in he Galois g oup (∼
=(Z/2Z)2) o
he co e , such ha he quo ien o he su ace by he in olu ion is a nodal
En iques su ace. This is a p ope y cha ac e izing all he Mendes Lopes-
Pa dini su aces [MP04].
Again using he s uc u e heo em o (Z/2Z)2-co e s, we gi e uppe
bounds o he dimensions o he cohomology g oups o he angen shea es
o hese su aces. Combining he esul s in [MP04], we show ha o a gen-
e al su ace Sin ou sub amily, h1(S, ΘS) = 6, h2(S, ΘS) = 2 and he base o
he Ku anishi amily o Sis smoo h. We hus show ha he closu e o he
co esponding subse o he Keum-Naie-Mendes Lopes-Pa dini su aces is an
i educible componen o he moduli space.
i
FIGURES
No a ion and con en ions
•A su ace will mean a p ojec i e, i educible and educed su ace de-
ined o e he complex numbe ield Cunless o he wise speci ied.
•A canonical su ace will mean he canonical model o a minimal smoo h
su ace o gene al ype.
•We will only ea (ex ended) Bu nia su aces wi h K2= 3,so some-
imes we call hem b ie ly (ex ended) Bu nia su aces. The same con-
en ion will be used o Keum-Naie su aces.
•Fo a smoo h su ace Sand a shea Fon S, we will deno e by hk(S, F)
he dimension o he cohomology g oup Hk(S, F).
•Fo a su ace S, we will deno e by ΘS he shea associa ed o he angen
bundle, Ωp
S he shea o holomo phic p- o ms on S, pg(S) := h0(S, Ω2
S)
he geome ic genus, q(S) := h0(S, Ω1
S) he i egula i y o S, χ(S) :=
1 + pg(S)−q(S) he holomo phic Eule -Poinca ´e cha ac e is ic and by
K2
S he sel -in e sec ion numbe o he canonical di iso .
•Deno e by ≡ he linea equi alence o di iso s and by num
≡ he nume -
ical equi alence o di iso s.
•An An-singula i y o a su ace is a singula i y analy ically isomo phic
o x2+y2+zn+1 = 0.An A1-singula i y is also called a node.
•A−m-cu e on a smoo h su ace is an i educible smoo h a ional cu e
wi h sel -in e sec ion numbe −m, whe e mis a non-nega i e in ege .
•The indices i∈ {1,2,3}should be unde s ood as esidue classes modulo
3 h ough he whole hesis.
•Deno e by G={0,g1, g2, g3}a g oup, which is isomo phic o (Z/2Z)2.
And le G∗={1,χ1, χ2, χ3}be he g oup o cha ac e s o G, whe e
χi(gi) = 1 and χi(gi+1) = χi(gi+2) = −1.
Figu es
xi

FIGURES
P1P2
P3
P0
1
P0
2
P0
3
Figu e 1: A plane model o a gene al 3A1- ype cubic su ace.
P1
P2
P3
P0
1
P0
2
P0
3
Figu e 2: A plane model o a special 3A1- ype cubic su ace.
P1P2
P3
Q1
Q2
Q3
P0
1P0
2
P0
3
Figu e 3: Ano he plane model o a gene al 3A1- ype cubic su ace.
xii
FIGURES
P1P2P3
P0
1P0
2P0
3
Figu e 4: A plane model o he D4(1)- ype cubic su ace.
P1P2P3
Q1
Q2
Q3
P0
1P0
2P0
3
Figu e 5: A plane model o he D4(2)- ype cubic su ace.
P1
P2
P3
P0
2P0
3
P0
1
Figu e 6: A plane model o he 4A1- ype cubic su ace.
xiii
FIGURES
P1= (1 : 0 : 0) P2= (1 : 1 : 0) P3= (0 : 1 : 0)
Q1= (0 : 1 : 1)
Q2= (0 : 0 : 1)
Q3= (1 : 0 : −1)
P0
1P0
2P0
3
Figu e 7: Coo dina es o he p oo o P oposi ion 11.4 and he calcula ion o
h0(e
Y , Ω1
e
Y(log Γ3)(2L−2E0
1−E0
2−E0
3)).
P1= (1 : 1 : 0)
P2= (1 : 1 : 1)
P3= (0 : 0 : 1)
P0
2= (0 : 1 : 0) P0
3= (1 : 0 : 0)
P0
1
P0
1= (0 : 1 : 1)
Figu e 8: Coo dina es o he p oo o P oposi ion 12.3 and he calcula ion o
h0(e
Y , Ω1
e
Y(log N1,log N3,log Γ3)(2L−E1−E2−E0
1−E0
3)).
P1= (1 : −1 : 0)
P2= (0 : 1 : 0)
P3= (1 : 0 : 0)
P0
2= (1 : 0 : 1) P0
3= (0 : 1 : 1)
P0
1= (0 : 0 : 1)
Figu e 9: Coo dina es o he p oo o P oposi ion 15.2 and he calcula ion o
h0(e
Y , Ω1
e
Y(log N2,log N3,log Z)(2L−E2−E0
2−E0
3)).
xi
1. BIDOUBLE COVERS OF SURFACES
Pa I
P elimina ies
1 Bidouble Co e s o Su aces
This sec ion gi es a b ie in oduc ion o he heo y o bidouble co e s. Fo
simplici y, we es ic ou sel es o he case o algeb aic su aces. We quo e
he esul s in [Ca 84], [Pa 91] and [Ca 99] wi hou p oo .
De ini ion 1.1 ([Ca 84], [Pa 91, De ini ion 1.1]).Le e
Ybe a no mal su ace.
A bidouble co e o e
Yis a ini e mo phism π:e
S→e
Y , oge he wi h a ai h ul
G-ac ion on e
Ssuch ha πexhibi s e
Yas he quo ien o e
Sby G.
De ini ion 1.2. (1) Assume ha π:e
S→e
Yis a bidouble co e be ween
no mal su aces. We de ine he ami ica ion locus Ro π, o be he
locus o poin s o e
Swhich ha e non i ial s abilize s. The b anch locus
Bo πis he image o Ron e
Y .
(2) Fo i= 1,2,3,de ine a b anch di iso Bico esponding o gi, o be he
image o all he 1-dimensional i educible componen s o R, whose
ine ia g oups a e he subg oup {0, gi}.
He e o a 1-dimensional i educible componen Do R, he ine ia
g oup Ho Dis de ined as ollows: H={g∈G|gx =x o any x∈D}
(c . [Pa 91, De ini ion 1.2]).
Assume ha e
Yis smoo h and e
Sis no mal. Then by [Be , Sec ion 3], πis
la , and he ami ica ion locus o πis o pu e codimension 1 (c . [Za 58]). I
ollows ha he b anch locus is also o pu e codimension 1.The nex heo em
desc ibes he s uc u e o a bidouble co e unde his assump ion.
Theo em 1.1 ([Ca 84, Sec ion 1], [Pa 91, Theo em 2.1], [Ca 99, Theo-
em 2]).Le π:e
S→e
Ybe a bidouble co e o su aces. Assume ha e
Y
is smoo h.
(1) Assume ha e
Sis no mal. Then
π∗(Oe
S)∼
=Oe
Y⊕ Oe
Y(−L1)⊕ Oe
Y(−L2)⊕ Oe
Y(−L3),
1
1. BIDOUBLE COVERS OF SURFACES
whe e Li’s a e di iso s on e
Y , and Gac s on Oe
Y(−Li) ia he cha ac e
χi.Mo eo e , he e a e h ee e ec i e di iso s ∆1,∆2,∆3on e
Ysuch
ha
2Li≡∆i+1 + ∆i+2,(1.1)
Li+ ∆i≡ Li+1 +Li+2,(1.2)
o i= 1,2,3,and ∆iis he b anch di iso co esponding o gi.
(2) Con e sely, gi en h ee di iso s L1,L2,L3and h ee e ec i e di iso s
∆1,∆2,∆3on e
Y , sa is ying (1.1) and (1.2), we can associa e a bidouble
co e π:e
S→e
Yas ollows (c . [BC11, Sec ion 2]):
o each i= 1,2,3,locally le ∆i= di (δi)and le uibe a ib e co-
o dina e o he geome ic line bundle Li,whose shea o holomo phic
sec ions is Oe
Y(Li).Then e
S⊂L1⊕L2⊕L3is gi en by he equa ions:
u1u2=δ3u3, u2
3=δ1δ2,
u2u3=δ1u1, u2
1=δ2δ3,
u3u1=δ2u2, u2
2=δ3δ1.
(1.3)
Acco ding o his heo em, o cons uc a bidouble co e o e a smoo h
su ace e
Y , i su ices o ind di iso s L1,L2,L3and e ec i e di iso s ∆1,∆2,
∆3sa is ying equa ions (1.1) and (1.2).
Rema k 1.1. (1) I we sum up he le hand side and he igh hand side
o (1.2) o all i= 1,2,3,we ob ain L1+L2+L3≡∆1+ ∆2+ ∆3.
(2) In he ollowing sec ions, e
Ywill be a a ional su ace, and hus P ic(e
Y)
has no o sion. Hence he equa ions (1.1) and (1.2) a e equi alen . We
usually jus e e o equa ions (1.1), o jus e e o ∆1,∆2,∆3,such
ha he sum o any wo is e en in P ic(e
Y),wi hou men ioning he Li’s.
Conce ning he cons uc ion o a bidouble co e in Theo em 1.1 (2), he
ollowing p oposi ion gi es a c i e ion o he no mali y ( espec i ely, smoo h-
ness) o e
S.
P oposi ion 1.2 ([Pa 91, P oposi ion 3.1], [Ca 99, Theo em 2]).Le e
Ybe
a smoo h su ace, and le π:e
S→e
Ybe he bidouble co e co esponding o
he da a L1,L2,L3and ∆1,∆2,∆3,sa is ying (1.1) and (1.2). Then
2

2. INVOLUTIONS ON RATIONAL DOUBLE POINTS
(1) e
Sis no mal i and only i he o al b anch di iso ∆ = ∆1+ ∆2+ ∆3
is educed.
(2) e
Sis smoo h i and only i each ∆iis smoo h o i= 1,2,3,and he
o al b anch di iso ∆has only no mal c ossing singula i ies.
In P oposi ion 1.2 (2), i we do no equi e he condi ion “∆ has only
no mal c ossing singula i ies”, hen e
Smigh ha e singula i ies.
Example 1.1. Assume ha ∆iin e sec s ∆i+1 ans e sely a a common
poin P o i= 1,2,3.Then he local equa ions (1.3) o e
Sshows ha π−1(P)
consis s o one poin Q, which is a 1
4(1,1)-singula i y on e
S. See [BC11, Sec-
ion 2] o de ails.
The ollowing heo em shows how o calcula e he in a ian s o e
S om
he co e ing da a ∆i’s and Li’s.
Theo em 1.3 ([Ca 84, Lemma 2.15], [Ca 99, Sec ion 2]).Le e
Ybe a smoo h
su ace, and le π:e
S→e
Ybe he bidouble co e associa ed o he da a
L1,L2,L3and ∆1,∆2,∆3,sa is ying (1.1) and (1.2). Assume ha ∆ =
∆1+ ∆2+ ∆3is educed and has only no mal c ossing singula i ies. Then
(1) π∗(Oe
S(Ke
S)) ∼
=Oe
Y(Ke
Y)⊕⊕3
i=1Oe
Y(Ke
Y+Li).
(2) 2Ke
S≡π∗(2Ke
Y+L1+L2+L3)≡π∗(2Ke
Y+ ∆1+ ∆2+ ∆3),
π∗(Oe
S(2Ke
S)) ∼
=Oe
Y(2Ke
Y+L1+L2+L3)⊕⊕3
i=1Oe
Y(2Ke
Y+Li+Li+1).
Co olla y 1.4. In he si ua ion o Theo em 1.3,
K2
e
S= (2Ke
Y+L1+L2+L3)2,
χ(Oe
S) = 4χ(Oe
Y) + 1
2
3
X
i=1
Li(Li+Ke
Y),
pg(e
S) = pg(e
Y) +
3
X
i=1
h0(e
Y , Ke
Y+Li),
P2(e
S) = h0(e
Y , 2Ke
Y+L1+L2+L3) +
3
X
i=1
h0(e
Y , 2Ke
Y+Li+Li+1).
3
2. INVOLUTIONS ON RATIONAL DOUBLE POINTS
2 In olu ions on Ra ional Double Poin s
The p e ious sec ion conside ed a bidouble co e π:e
S→e
Ywhen e
Yis a
smoo h su ace. In ou applica ions, bo h e
Sand e
Ymigh ha e singula i ies.
We would like o know when he quo ien o a a ional double poin by
aZ/2Z-ac ion o a (Z/2Z)2-ac ion emains a a ional double poin . This
p oblem has been s udied and sol ed in [Ca 87]. We quo e he main esul s
and ollow he no a ion in [Ca 87] o con enience.
Le us i s gi e a lis o a ional double poin s.
Table 1:
Singula i ies (X0, x0) Equa ion
E8z2+x3+y5= 0
E7z2+x(y3+x2) = 0
E6z2+x3+y4= 0
Dn(n≥4) z2+x(y2+xn−2) = 0
Anz2+x2+yn+1 = 0,o u +yn+1 = 0
De ini ion 2.1 ([Ca 87, De ini ion 1.3]).The in olu ion τo a a ional dou-
ble poin (X0, x0) such ha τ∗(z) = −z, τ∗(x) = x, τ∗(y) = yis called he
i ial in olu ion. Any in olu ion σconjuga e o τis also said o be i ial,
and has he p ope y ha X0/σ ∼
=(C2,0).
The nex heo em classi ies all he in olu ions on a ional double poin s.
Theo em 2.1 ([Ca 87, Theo em 2.1]).The only in olu ion ac ing on E7,
E8is he i ial one. The o he a ional double poin s admi he ollowing
non i ial conjugacy classes o in olu ions:
(a) (x, y, z)7→ (x, −y, z) (E6, Dn, A2k+1),
(b) (x, y, z)7→ (x, −y, −z) (E6, Dn, A2k+1),
(c) (u, , y)7→ (−u, , −y) (A2n),
(d) (x, y, z)7→ (−x, y, −z) (An),
(e) (u, , y)7→ (−u, − , −y) (A2k+1).
The ollowing heo ems classi y he quo ien s o a ional double poin s by
in olu ions. We also calcula e he ami ica ion loci o he quo ien maps.
4
2. INVOLUTIONS ON RATIONAL DOUBLE POINTS
Theo em 2.2 ([Ca 87, Theo em 2.2]).The quo ien o a a ional double
poin by a non i ial in olu ion no o ype (c),(e), is again a a ional double
poin acco ding o Table 2.
Table 2:
Singula i ies
(X0, x0)In olu ions Quo ien s
(Y0, y0)Rami ica ion locus
E6:z2+x3+y4= 0 (x, y, z)7→ (x, −y, z)A2z2+x3= 0
E6:z2+x3+y4= 0 (x, y, z)7→ (x, −y, −z)E7(0,0,0)
Dn:z2+x(y2+xn−2) = 0 (x, y, z)7→ (x, −y, z)A1z2+xn−1= 0
Dn:z2+x(y2+xn−2) = 0 (x, y, z)7→ (x, −y, −z)D2n−2(0,0,0)
A2k+1 :z2+x2+y2k+2 = 0 (x, y, z)7→ (x, −y, z)Akz2+x2= 0
A2k+1 :z2+x2+y2k+2 = 0 (x, y, z)7→ (x, −y, −z)Dk+3 (0,0,0)
An:z2+x2+yn+1 = 0 (x, y, z)7→ (−x, y, −z)A2n+1 (0,0,0)
Theo em 2.3 ([Ca 87, Theo em 2.4]).The quo ien Bko he singula i y
A2kby an in olu ion o ype (c) is de ined in C4,wi h coo dina es (u, w, , η)
by he ideal Ik= (ηw − 2, uw + ηk, u +ηk+1).
The ( educed) excep ional di iso Do i s minimal esolu ion Thas no -
mal c ossings, consis s o ksmoo h a ional cu es, and i s Dynkin diag am
is
◦◦ · · · ◦ ◦ ◦
−3
Theo em 2.4 ([Ca 87, Theo em 2.5]).Le Zbe he a ine cone o e he
Ve onese su ace, i.e., he se o symme ic ma ices



x1x2x6
x2x3x4
x6x4x5



o ank ≤1.
Then he quo ien Yk+1 o he singula i y A2k+1 by he in olu ion (e) is
he in e sec ion o Zwi h he hype su ace φ=x6−xk+1
3= 0.In pa icula ,
5
2. INVOLUTIONS ON RATIONAL DOUBLE POINTS
Yk+1 can also be de ined as he singula i y in C5de ined by he ideal
Jk= (x1x3−x2
2, x2x4−xk+2
3, x3x5−x2
4, x1x4−x2xk+1
3,
x2x5−xk+1
3x4, x1x5−x2k+2
3).
The excep ional di iso Din he minimal esolu ion To Yk+1 has no -
mal c ossings, consis s o (k+ 1) smoo h a ional cu es, and he associa ed
Dynkin diag am is
◦ o k= 0
−4
◦◦ · · · ◦ ◦ o k≥1
−3−3
Rema k 2.1. The Y1-singula i y ( espec i ely, B1-singula i y) is he 1
4(1,1)-
singula i y ( espec i ely, he 1
3(1,1)-singula i y), i.e., he cone o e he a io-
nal no mal cu e o deg ee 4 in P4( espec i ely, o deg ee 3 in P3).
Conside he in olu ion o ype (e) on an A1-singula i y:
σ: (X0, x0) : u +y2= 0 →(X0, x0) : u +y2= 0,
(u, , y)7→ (−u, − , −y).
Then by Theo em 2.4, he quo ien Y0:= X0/σ has a Y1-singula i y y0.Le
ρ:X0→X0be he minimal esolu ion o x0and deno e by N he (−2)-cu e.
Since Ncan be iewed as he p ojec i iza ion o he angen cone o X0 o x0,
σcan be li ed o X0and i has Nas ixed locus. We see ha he image o N
on he quo ien X0/σ is a (−4)-cu e. Hence X0/σ is he minimal esolu ion
o (Y0, y0).
Theo em 2.5 ([Ca 87, Theo em 2.7]).Le (X0, x0)be a a ional double poin
and le Hbe a subg oup o Au (X0, x0),which is isomo phic o (Z/2Z)2.Then
His conjuga e o a subg oup lis ed in Table 3.
Rema k 2.2 ([Ca 87, Rema k 2.8]).F om Theo em 2.5 Table 3, we conclude
ha he quo ien o a a ional double poin (X0, x0) by a ai h ul (Z/2Z)2-
ac ion is again a a ional double poin o a smoo h poin . This s a emen
also holds o he case (X0, x0)∼
=(C2,0).This ema k will be e y impo an
in he p oo o Theo em 5.2, Sec ion 5.
6
3. NORMAL CUBIC SURFACES
P oo . C2:y0y2−y2
1= 0 and C3:y0y2(y0−(a+ 1)y1+ay2) = 0 in e sec a
six poin s
Q0= (1 : 1 : 1), Q1= (1 : 0 : 0), Q2= (0 : 0 : 1), Q3= (a2:a: 1),
and he in ini ely nea poin s Q0
k(k= 1,2) co esponding o he angen line
o C2 o Qk(k= 1,2): Q1Q0
1:y2= 0 and Q2Q0
2:y0= 0.
e
Ycan be ob ained by blowing up hese six poin s. Apply he quad a ic
ans o ma ion cen e ed a Q0, Q1, Q2,namely, i s blow up σ1:Y0→P2a
Q0, Q1, Q2, hen blow down he s ic ans o ms o Q0Q1, Q0Q2and Q1Q2 o
h ee poin s P2, P1and P0
3 espec i ely. Deno e he images o Q0
1, Q0
2, Q3by
P0
2, P0
1, P3 espec i ely. Then P1,...,P0
3sa is y he con igu a ion abo e.
Rema k 3.2. F om he equa ion o a 3A1- ype cubic su ace, we see ha i
has one pa ame e a. I a=−1, he e a e h ee lines x0=x3= 0, x2=x3=
0, x0−x2=x3= 0 con aining a smoo h poin (0,1,0,0) o Y. Co espond-
ingly, h ee lines PiP0
i’s pass h ough a common poin in he con igu a ion o
P1,...,P0
3,and h ee (−1)-cu es Γ1,Γ2,Γ3pass h ough a common poin o
e
Y . See Figu e 2.
3.2 D4(1)- ype and D4(2)- ype Cubic Su aces
Resolu ion o he D4(1)- ype and he D4(2)- ype cubic su aces.
Assume ha Yis a D4(1)- ype o a D4(2)- ype cubic su ace. Then e
Y
can be ob ained as he blowup σ:e
Y→P2o six poin s wi h he ollowing
con igu a ion (see Figu e 4 and Figu e 5):
P1, P2, P3a e h ee dis inc collinea poin s on P2,and P0
iis an in ini ely
nea poin lying o e Pi o all i= 1,2,3.
I Yis o D4(1)- ype, we equi e he h ee lines PiP0
i’s o pass h ough
a common poin . I Yis o D4(2)- ype, we equi e he h ee lines PiP0
i’s o
o m a iangle, wi h e ices Q1, Q2, Q3,whe e Qiis he in e sec ion poin
o he lines Pi+1P0
i+1 and Pi+2P0
i+2.
Ra ional cu es on e
Y.In bo h cases, e
Yhas ou (−2)-cu es,
Ni=Ei−E0
i, Z =L−E1−E2−E3,wi h Ni.Z = 1 and Ni.Ni+1 = 0,
13

3. NORMAL CUBIC SURFACES
and six (−1)-cu es
E0
i,Γi:= L−Ei−E0
i, o i= 1,2,3.
Fo each i= 1,2,3,e
Yhas a pencil o a ional cu es Ciin he linea sys em
|2L−Ei+1 −Ei+2 −E0
i+1 −E0
i+2|,
so ha Ci+ Γi≡ −Ke
Y.The only singula elemen in he pencil is:
Γi+1 + Γi+2.
P oo . (1) Assume ha Yis o D4(1)- ype. C2:y2
0= 0 and C3:y3
1+y3
2= 0
in e sec a six poin s:
P1= (0 : 1 : −1), P2= (0 : 1 : −ζ), P3= (0 : 1 : −ζ2),
whe e ζis a p imi i e cubic oo o 1,and he in ini ely nea poin s
P0
i’s co esponding o he lines
P1P0
1:y1+y2= 0, P2P0
2:y1+ζ2y2= 0, P3P0
3:y1+ζy2= 0.
No e ha hese h ee lines in e sec a a common poin (1 : 0 : 0).
(2) Assume ha Yis o D4(2)- ype. C2:y2
0= 0 and C3:y3
1+y3
2+y0y1y2= 0
in e sec a six poin s:
P1= (0 : 1 : −1), P2= (0 : 1 : −ζ), P3= (0 : 1 : −ζ2),
and he in ini ely nea poin s P0
i’s co esponding o he angen lines
o C3 o Pi’s:
P1P0
1:−y0+ 3y1+ 3y2= 0,
P2P0
2:−ζy0+ 3y1+ 3ζ2y2= 0,
P3P0
3:−ζ2y0+ 3y1+ 3ζy2= 0.
No e ha hese h ee lines o m a iangle wi h e ices
Q1= (−3 : 1 : 1), Q2= (−3 : ζ:ζ2), Q3= (−3 : ζ2:ζ).
Then he conclusion ollows om Theo em 3.1 (3).
14
3. NORMAL CUBIC SURFACES
3.3 4A1- ype Cubic Su ace
Resolu ion o he 4A1- ype cubic su ace. Assume ha Yis a 4A1-
ype cubic su ace. Then e
Ycan be ob ained as he blowup σ:e
Y→P2o six
poin s wi h he ollowing con igu a ion (see Figu e 6):
P1, P2, P3a e collinea , and Pi, P0
i+1, P 0
i+2 a e collinea o all i= 1,2,3,
i.e., P1,...,P0
3a e e ices o a comple e quad ila e al.
Ra ional cu es on e
Y.e
Yhas ou disjoin (−2)-cu es,
Ni=L−Ei−E0
i+1 −E0
i+2, Z =L−E1−E2−E3,
and nine (−1)-cu es,
Ei, E0
i,Γi:= L−Ei−E0
i, o i= 1,2,3.
Fo each i= 1,2,3,e
Yhas a pencil o a ional cu es Ciin he linea sys em
|2L−Ei+1 −Ei+2 −E0
i+1 −E0
i+2|,
so ha Ci+ Γi≡ −Ke
Y.The singula elemen s in he pencil a e:
Γi+1 + Γi+2, Ni+1 +Ni+2 + 2E0
i, Z +Ni+ 2Ei.
P oo . C2:y0y2−y2
1= 0 and C3: (y0−y1)(y1−y2)y1= 0 in e sec a six
poin s
Q1= (1 : 0 : 0), Q2= (0 : 0 : 1), Q3= (1 : 1 : 1),
and in ini ely nea poin s Q0
ico esponding o he angen line o C2 o Qi:
Q1Q0
1:y2= 0, Q2Q0
2:y0= 0, Q3Q0
3:y0−2y1+y2= 0.
e
Ycan be ob ained by blowing up hese six poin s. Apply he quad a ic
ans o ma ion cen e ed a Q1, Q2, Q3,namely, i s blow up σ1:Y0→P2a
Q1, Q2, Q3, hen blow down he s ic ans o ms o Q1Q2, Q2Q3and Q3Q1 o
h ee poin s P0
3, P0
1and P0
2 espec i ely. Deno e he images o Q0
1, Q0
2, Q0
3by
P1, P2, P3 espec i ely. Then P1,...,P0
3sa is y he con igu a ion abo e.
15
3. NORMAL CUBIC SURFACES
The geome y o he 4A1- ype cubic su ace. We explain mo e abou
he geome y o he 4A1- ype cubic su ace Y. See he ollowing igu e.
Yhas 4 nodes Q0, Q1, Q2, Q3,which do no lie in a plane. By B´ezou ’s
heo em, any line connec ing wo nodes is con ained in Y. We can iew
Q0, Q1, Q2, Q3as he e ices o a e ahed on. The edges o he e ahed on
co espond o six lines o Y. The (−1)-cu es Eiand E0
ion e
Yco espond o
a pai o opposi e edges o he e ahed on, o i= 1,2,3.
The e a e h ee mo e lines l1, l2, l3o Ywhich do no pass any nodes.
They lie in a plane and o m a iangle. Each one o hem in e sec s exac ly
one o he h ee pai s o opposi e edges. The h ee (−1)-cu es Γ1,Γ2,Γ3on
e
Yco espond o hese h ee lines. F om his we see ha he pencil o cu es
Cion e
Yco espond o he esidual conics cu by planes con aining one o
he li’s.
Q0
Q1
Q2
Q3
l1
l3
l2
Figu e 10: Singula i ies and lines o he 4A1- ype cubic su ace.
16
4. BURNIAT SURFACES AND EXTENDED BURNIAT SURFACES
Pa II
The I educible Componen
Con aining he
Ex ended Bu nia Su aces
4 Bu nia Su aces and
Ex ended Bu nia Su aces
This sec ion gi es an in oduc ion o he cons uc ion o he (ex ended) Bu -
nia su aces wi h K2= 3 and he main esul s on hei moduli spaces
ob ained in [BC10-b].
Assume ha Yis a 3A1- ype cubic su ace and e
Yis i s minimal esolu ion.
Recall he no a ion in oduced in Subsec ion 3.1. Assume ha he
lines PiP0
i’s do no pass h ough a common poin . See Figu e 1.
De ini ion 4.1 ([Pe 77], [BC10-b, De ini ion 1.1 and De ini ion 1.3]).
(1) De ine s ic ly ex ended Bu nia di iso s on e
Yas ollows:
∆1= Γ1+N2+C3,∆2= Γ2+N3+C1,∆3= Γ3+N1+C2,(4.1)
whe e all Ci’s a e i educible smoo h cu es.
(2) I one o wo o he h ee Ci’s become educible in he way
Ci=Ni+Ei+|L−Ei+1 −Ei+2|,
hen we de ine h ee new di iso s by sub ac ing om ∆i+1 he di iso
Ni,and sub ac ing om ∆i−1 he di iso Ni,and adding i o ∆i.
These new di iso s and he s ic ly ex ended Bu nia di iso s a e all
called ex ended Bu nia di iso s.
17
4. BURNIAT SURFACES AND EXTENDED BURNIAT SURFACES
(3) I all h ee Ci’s become educible in he way abo e, hen we ge h ee
new di iso s, called nodal Bu nia di iso s:
D1=|L−E1−E2|+N1+ Γ1+E3,
D2=|L−E2−E3|+N2+ Γ2+E1,
D3=|L−E3−E1|+N3+ Γ3+E2.
(4.2)
De ini ion 4.2 ([BC10-b, De ini ion 1.4]).A (s ic ly) ex ended Bu nia
su ace wi h K2= 3 is he minimal model So a bidouble co e π:e
S→e
Y
associa ed o a (s ic ly) ex ended Bu nia di iso .
A nodal Bu nia su ace wi h K2= 3 is he minimal model So a bidouble
co e π:e
S→e
Yassocia ed o a nodal Bu nia di iso .
Rema k 4.1. (1) By P oposi ion 1.2, e
Sin he de ini ion is a smoo h su -
ace. Howe e , i is no necessa ily minimal. Whene e Niis a connec ed
componen in ∆, π−1Niis a disjoin union o wo (−1)-cu es.
(2) In pa icula , o a s ic ly ex ended Bu nia di iso ∆,all Ni’s a e con-
nec ed componen s in ∆.This implies ha KSis ample o a s ic ly ex ended
Bu nia su ace S.
(3) No e ha in De ini ion 4.1 (2), he p ocedu e applied o he b anch
di iso s is ac ually ela ed o he p ocedu e o no maliza ion in he heo y
o bidouble co e s (c . [Ca 99, Sec ion 2, Rema k 3]).
Theo em 4.1. Le Sbe he minimal model o e
Sin De ini ion 4.2. Then S
is a su ace o gene al ype wi h K2
S= 3, pg(S) = q(S) = 0.
Mo eo e , π op
1(S)∼
=H8×Z/2Z,whe e H8is he qua e nion g oup o
o de 8.
Fo he i s s a emen see [BC10-b], o apply Co olla y 1.4. Fo he
second s a emen see [BC11, Theo em 3.2]. See also [In94].
Co olla y 4.2 ([BC10-b, Rema k 1.5]).I Xis he canonical model o an
ex ended Bu nia su ace o a nodal Bu nia su ace Swi h K2
S= 3, hen
he bicanonical map o X ealizes Xas a ini e bidouble co e o a 3A1- ype
cubic su ace Y.
18

5. ONE PARAMETER LIMITS
In [BC10-b], Baue and Ca anese p o ed, among o he hings, he ollow-
ing heo em abou he subse in he moduli space co esponding o ex ended
Bu nia su aces and nodal Bu nia su aces wi h K2= 3.
Theo em 4.3 ([BC10-b, P oposi ion 5.7, Theo em 0.1 and Theo em 0.2]).
(1) The subse N EB3o he moduli space o canonical su aces o gene al
ype Mcan
1,3co esponding o ex ended Bu nia su aces and nodal Bu -
nia su aces wi h K2= 3 is an i educible open se , no mal, uni a ional
o dimension 4.
(2) Le Sbe an ex ended Bu nia su ace o a nodal Bu nia su ace wi h
K2
S= 3.Then h1(S, ΘS) = 4, h2(S, ΘS) = 0 and he base o he Ku an-
ishi amily o such a minimal model Sis smoo h.
(3) I Xis he canonical model o an ex ended Bu nia su ace o a nodal
Bu nia su ace Swi h K2
S= 3, hen De (X, (Z/2Z)2) = De (X).
Rema k 4.2. (1) He e we gi e a geome ic explana ion o he dimension o
NEB3: a 3A1- ype cubic su ace has one pa ame e (c . Sec ion 3), and each
Cimo es in a pencil o cu es. This gi es he 4 dimensions.
(2) Theo em 4.3 is ob ained by a mo e ca e ul s udy o de o ma ions o he
ex ended Bu nia su aces (c . [BC10-b, P oposi ion 5.7]), using bidouble
co e heo y. We will ollow his me hod in Pa III.
(3) Deno e by SEB he subse o N EB3co esponding o he s ic ly ex ended
Bu nia su aces. Then SEB is a p ope open subse o N EB3o dimension 4.
Theo em 4.3 (1) and (2) imply ha NEB3is an i educible componen in
Mcan
1,3.He e comes a na u al ques ion: is N EB3is closed in Mcan
1,3? Baue
and Ca anese al eady showed ha he answe is No (c . [BC10-b, Sec ion 7]).
The aim o Pa II is o comple e he ollowing ask.
Task : De e mine he i educible componen N EB3in Mcan
1,3,i.e., desc ibe
all he su aces co esponding o N EB3 N EB3.
19
5. ONE PARAMETER LIMITS
5 One Pa ame e Limi s o
he Ex ended Bu nia Su aces
This sec ion is he i s s ep o s udy limi s o ex ended Bu nia su aces wi h
K2= 3 in he moduli space. We need he ollowing p oposi ion conce ning
no mal Del Pezzo su aces.
Le Ybe a no mal Q-Go ens ein su ace. Deno e he dualizing shea o
Yby ωY,and deno e he associa ed Weil di iso by KY.Then he e is a
minimal posi i e in ege msuch ha ω⊗m
Yis an in e ible shea . So i makes
sense o de ine KY o be ample o an i-ample. I KYis an i-ample, we call
YaDel Pezzo su ace. Also no e ha Yis Go ens ein i and only i m= 1.
P oposi ion 5.1 ([HW81, Theo em 4.4 (ii)]).Le Ybe a no mal Go ens ein
Del Pezzo su ace wi h K2
Y= 3.Then Yis a cubic su ace in P3.
The main esul o his sec ion is he ollowing Theo em.
Theo em 5.2. Le Tbe a smoo h a ine cu e and o∈T, and le F:X → T
be a la amily o canonical su aces. Suppose ha X is he canonical model
o an ex ended Bu nia su ace o a nodal Bu nia su ace wi h K2
X = 3
o 6=o∈T. Then (a e possibly sh inking T) he e is a g oup ac ion o
G:= (Z/2Z)2on Xand he quo ien map Π: X → Y := X/G yields a one
pa ame e amily o ini e (Z/2Z)2-co e s,
XΠ//
F
@
@
@
@
@
@
@Y
F0







T
(i.e., Π :X → Y is a ini e (Z/2Z)2-co e ), such ha o each 6=o, Y is
a3A1- ype cubic su ace, and Yois a no mal cubic su ace.
Rema k 5.1. To s udy he limi s o he ex ended Bu nia su aces wi h
K2= 3,i su ices o equi e ha X is a s ic ly ex ended Bu nia su ace
o 6=oin Theo em 5.2. In ac , Rema k 4.2 (3) implies ha SEB =N EB3.
P oo . No e ha Xis Go ens ein, since he base Tis smoo h and he ib es
ha e only a ional double poin s.
20
5. ONE PARAMETER LIMITS
Since X F−1(o)→T {o}is a amily o canonical models o ex ended
Bu nia su aces o nodal Bu nia su aces wi h K2= 3,we ha e a (Z/2Z)2-
ac ion on X F−1(o).This is he Galois g oup ac ion inducing he bicanonical
mo phism ( he key poin is ha we wo k on he canonical models, c . [BC10-
b, Theo em 0.2]).
Hence, by [Ca 83, Theo em 1.8], he (Z/2Z)2-ac ion ex ends o X.
Le Ybe he quo ien o Xby he g oup ac ion, and le Π: X → Y be
he quo ien map. Se X := F−1( ) and Y := F0−1( ) o all ∈T. Then we
ha e o all ∈T:KY =KY|Y , KX =KX|X .
Mo eo e , 2KX= Π∗(2KY+B),whe e Bis he b anch di iso o Π: X →
Y(c . Theo em 1.3). Since o 6=o, we ha e 2KX = Π∗
(−KY ) (c . Co ol-
la y 4.2), i ollows ha 2KX+ Π∗KY≡0 on X Xo.
Since Xois i educible, we ob ain (a e possibly sh inking T) ha
2KX+ Π∗KY≡0 on X.In pa icula ,
2KX = Π∗
(−KY ) o all ∈T, (5.1)
which implies ha −KY is ample and K2
Y =K2
X = 3 o all ∈T.
By cons uc ion, as he bicanonical image o X (c . Co olla y 4.2), Y
is a cubic su ace wi h h ee A1-singula i ies o 6=o, and Yois a no mal
Q-Go ens ein su ace.
We claim ha Yois Go ens ein. Then Yois a no mal cubic su ace
by P oposi ion 5.1.
We shall p o e he claim by con adic ion. Assume ha Yois non-
Go ens ein. Recall ha
2KXo≡Π∗
o(−KYo), K2
Yo= 3,(5.2)
and −KYois ample.
S ep 1: All he possibili ies o he non-Go ens ein locus o Yoa e (c . Theo-
em 2.3, Theo em 2.4 and Rema k 2.1)
(a) one B1-singula i y.
(b) one B1-singula i y and one Y1-singula i y.
21
5. ONE PARAMETER LIMITS
(c) one Y2-singula i y.
(d) one Y1-singula i y.
(e) wo Y1-singula i y.
In ac , Xohas a mos a ional double poin s. Hence by Rema k 2.2, o
a non-Go ens ein poin qon Yo,Π−1
o(q) consis s o wo poin s p1, p2,and
he s abilize s o p1and p2in Ga e isomo phic o Z/2Z.By Theo em
2.3 and Theo em 2.4, ei he qis a Bk-singula i y and bo h p1and p2
a e A2k-singula i ies o Xo,o qis a Yk+1-singula i y and bo h p1and
p2a e A2k+1-singula i ies o Xo o some k≥0.
Hence an uppe bound o he numbe o singula i ies o Xowould
bound he numbe o non-Go ens ein singula i ies o Yo.Since he min-
imal esolu ion Soo Xohas Pica d numbe 7, Sohas a mos six (−2)-
cu es (c . [BHPV, Page 272, P oposi ion 2.5]). An easy calcula ion
shows ha he lis o he non-Go ens ein singula i ies o Yos a ed
abo e is comple e.
S ep 2: Le ˜
Yobe he minimal esolu ion o Yo.Then K2
˜
Yois an in ege . The
esolu ion o a a ional double poin does no change K2,while he
esolu ion o a B1-singula i y ( espec i ely, a Y1-singula i y) con ibu es
−1
3( espec i ely, −1) o K2( o example, c . [Ba low99, Sec ion 6]).
Since K2
Yo= 3 is an in ege , case (a) and case (b) canno occu .
S ep 3: Assume ha Yohas exac ly one Y2-singula i y q. The discussion in
S ep 1 shows ha Π−1
o(q) consis s o wo A3-singula i ies p1, p2o Xoand
p1, p2a e he only singula i ies o Xo.Mo eo e , he e is an in olu ion
g∈Gpe mu ing p1and p2.
Li g o he minimal esolu ion Soo Xo,and deno e i by ˆg. Deno e
by R he di iso ial pa o he ix locus o ˆgand by he ace o
ˆg∗:H2(So,C)→H2(So,C).Deno e by N1, N2, N3( espec i ely, Z1, Z2,
Z3) he (−2)-cu es o Solying o e p1( espec i ely, p2).
No e ha c1(KSo) and c1(N1),...,c1(Z3) a e a basis o H2(So,C).Since
gpe mu es p1and p2on Xo, N1,...,Z3a e disjoin om he ix lo-
22
6. EXCLUSION OF CERTAIN CUBIC SURFACES
(1) An A5- ype cubic su ace x3x0x1−(x3
0+x3
1−x1x2
2) = 0 has h ee lines,
l1:x0=x1= 0, l2:x0= 0, x1−x2= 0, l3:x0= 0, x1+x2= 0,
which all pass h ough he A5-singula i y P= (0 : 0 : 0 : 1).
(2) A 3A2- ype cubic su ace x3x0x1−x3
2= 0 has h ee lines, l1:x0=
x2= 0, l2:x1=x2= 0, l3:x2=x3= 0,which o m a iangle wi h
he h ee A2-singula i ies P1= (0 : 0 : 0 : 1), P2= (1 : 0 : 0 : 0),
P3= (0 : 1 : 0 : 0) as he e ices.
(3) An (A1+A4)- ype cubic su ace x3(x0x2−x2
1)−x2
0x1= 0 has ou lines,
l1:x0=x1= 0, l2:x0=x3= 0, l3:x1=x2= 0, l4:x1=x3= 0.
l1, l3con ain he A1-singula i y P1= (0 : 0 : 0 : 1) and l2, l4con ain he
A4-singula i y P2= (0 : 0 : 1 : 0).
(4) A (2A1+A3)- ype cubic su ace x3(x0x2−x2
1)−x0x2
1= 0 has i e lines,
l1:x0=x1= 0, l2:x1=x2= 0, l3:x1=x3= 0, l4:x0=x3= 0,
l5:x2=x0+x3= 0,and h ee singula i ies P1= (0 : 0 : 1 : 0)(A3),
P2= (1 : 0 : 0 : 0)(A1), P3= (0 : 0 : 0 : 1)(A1). l1, l2, l3 o m a iangle
wi h e ices P1, P2, P3,and l4con ains P1.The e is only one line l5
which does no con ain any singula i y.
(5) An (A1+ 2A2)- ype cubic su ace x3(x0x2−x2
1)−x3
1= 0 has i e lines,
l1:x0=x1= 0, l2:x1=x2= 0, l3:x1=x3= 0, l4:x0=x1+x3= 0,
l5:x2=x1+x3= 0,and i has h ee singula i ies
P1= (0 : 0 : 0 : 1)(A1), P2= (0 : 0 : 1 : 0)(A2), P3= (1 : 0 : 0 : 0)(A2).
l1, l2, l3 o m a iangle wi h e ices P1, P2, P3, l4con ains P2and l5
con ains P3.
(6) An (A1+A3)- ype cubic su ace
x3(x0x2−x2
1)−(x0−x1)(−x1+x2)(x0−2x1+x2) = 0
has wo singula i ies P= (0 : 0 : 0 : 1)(A1), Q = (1 : 1 : 1 : 0)(A3).
I has se en lines, l1:x3=x0−x1= 0, l2:x3=−x1+x2= 0,
l3:x3=x0−2x1+x2= 0, l4:x0=x1= 0, l5:x0=x1=x2,
l6:x1=x2= 0, l7:x1=x0+x2−x3= 0.
No e ha l1, l2, l3, l5mee a Q, and l4, l6mee a P. The e is only one
line l7which does no con ain any singula i y.
29

7. D4-GENERALIZED BURNIAT SURFACES
(7) A (2A1+A2)- ype no mal cubic su ace x3(x0x2−x2
1)−x2
1(x0−x1) = 0
has wo A1-singula i ies P1= (0 : 0 : 0 : 1), P2= (1 : 0 : 0 : 0),and one
A2-singula i y Q= (0 : 0 : 1 : 0).I has eigh lines, l1:x0=x1= 0,
l2:x1=x2= 0, l3:x1=x3= 0, l4:x0−x1=x3= 0, l5:x0=x1=
x2, l6:x0=x1−x3= 0, l7:x1=x2=x3, l8:−x0+x1−x3=x2= 0.
No e ha l1, l3, l4, l6mee a Q, l1, l2, l5mee a P1, l2, l3, l7mee a P2.
The e is only one line l8which does no con ain any singula i y.
Combining hese h ee p oposi ions wi h he classi ica ion o cubic su -
aces (c . Theo em 3.3), Theo em 6.1 ollows.
7D4-gene alized Bu nia Su aces
By Theo em 6.1, Yocan be only one o he ollowing ypes: 3A1, D4(1),
D4(2) and 4A1.Fo each case we will ei he exclude i o ind all he possible
b anch loci such ha he associa ed bidouble co e Xocan be de o med o
ex ended Bu nia su aces wi h K2= 3.
In o de o apply he heo y o Sec ion 1 o smoo h su aces, we make
he ollowing con en ions o he emaining sec ions o Pa II.
Con en ions Le Πo:Xo→ Yobe he bidouble co e as in Theo em
5.2. Le µ:e
Y→ Yobe he minimal esolu ion o Yo.Deno e by e
S he
no maliza ion o he ibe p oduc o Xoand e
Yo e Yo,and π:e
S→e
Y he
induced bidouble co e . Mo eo e , le ∆ be he b anch locus o he bidouble
co e π:e
S→e
Y . W i e ∆ as ∆=∆1+ ∆2+ ∆3acco ding o he g oup ac ion
(c . Theo em 1.1, Sec ion 1).
In iew o Co olla y 5.4, ∆ has he ollowing p ope ies.
P oposi ion 7.1. (1) E e y i educible componen o ∆is a (−1)-cu e,
o a (−2)-cu e o a 0-cu e.
(2) −Ke
Y.∆i= 3 o i= 1,2,3.
(3) µ∗(∆) ≡ −3KYo.
30
7. D4-GENERALIZED BURNIAT SURFACES
P oo . By adjunc ion, o a smoo h a ional cu e D, −Ke
Y.D =D2+ 2.
Hence a (−1)-cu e on e
Yco esponds o a line on Yo,and a 0-cu e co e-
sponds o a smoo h conic. Thus (1) ollows om Co olla y 5.4. E ec i e
di iso s in he linea sys em |−Ke
Y|co espond o hype plane sec ions o Yo.
No e ha OYo(KYo) is in e ible, µ∗(Ke
Y) = KYoand µ∗(KYo) = Ke
Y.Since
µ∗(∆i) = Bi(c . Co olla y 5.4), (2) ollows om he p ojec ion o mula and
Co olla y 5.4 (1), and (3) ollows om Co olla y 5.4 (4).
Rema k 7.1. By P oposi ion 1.2, ∆ = ∆1+ ∆2+ ∆3is educed. By The-
o em 1.1, ∆i’s a e di iso s such ha o any i= 1,2,3,∆i+ ∆i+1 is e en
in Pic(e
Y).See also Rema k 1.1. We will use his ema k equen ly in he
ollowing sec ions.
In his sec ion we i s deal wi h he case when Yohas a D4-singula i y.
7.1 Con igu a ion o B anch Di iso s
Assume ha Yois o D4(1)- ype o o D4(2)- ype. Le yobe he D4-singula i y
and e
Ybe i s minimal esolu ion. Recall he no a ion in oduced in Sub-
sec ion 3.2. See Figu e 4 and Figu e 5.
Lemma 7.2. Π−1
o(yo)consis s o one poin xoand xois an A1-singula i y o
Xo.Mo eo e , locally, Πo: (Xo, xo)→(Yo, yo)is isomo phic o
(X0, x0) : z2+x2+y2= 0 →(Y0, y0) : w2+u (u+ ) = 0,
(x, y, z)7→ (u, , w) = (x2, y2, xyz),
wi h he G-ac ion on (X0, x0)gi en by g1: (x, y, z)7→ (x, −y, −z),
g2: (x, y, z)7→ (−x, y, −z), g3: (x, y, z)7→ (−x, −y, z).
P oo . Conside he amily o bidouble co e s Π: X → Y in Theo em 5.2.
Fo 6=o, Y has h ee nodes n1( ), n2( ), n3( ).Thei limi s in Yomus be he
singula i y yo.Thus hei in e se images unde Π mus ha e limi poin s in
Π−1
o(yo).By he cons uc ion (c . De ini ion 4.1), o each ni( ),e e y poin
o Π−1
(ni( )) is ixed by gi.No e ha o any i, giand gi+1 gene a es G. Since
Π−1
o(yo) o ms an o bi unde he g oup ac ion, he ca dinali y o Π−1
o(yo) can
only be 4,2,o 1.The a gumen abo e shows ha Π−1
o(yo) consis s o one
31
7. D4-GENERALIZED BURNIAT SURFACES
poin xo.By looking a Theo em 2.5 Table 3 whe e he quo ien (Y0, y0) is a
D4-singula i y, he conclusion ollows.
I is easy o see ha u=x2, =y2, w =xyz gene a e he ing o
in a ian s o he ac ion, and sa is y he equa ion w2+u (u+ ) = 0.
Theo em 7.3. Assume ha Yohas a D4-singula i y. Then
(1) Yomus be o D4(2)- ype.
(2) π:e
S→e
Yis isomo phic o he bidouble co e associa ed o he ollowing
b anch di iso s:
∆1= Γ1+N2+C3,∆2= Γ2+N3+C1,∆3= Γ3+N1+C2,
whe e all Ci’s a e i educible smoo h cu es.
P oo . Fi s we conside he (−2)-cu es. Lemma 7.2 and Example 2.1 show
ha one may assume ha ∆1≥N2,∆2≥N3,∆3≥N1,∆6≥ Z, ha
N1, N2, N3a e connec ed componen s o ∆,and ha any i educible com-
ponen in ∆ −N1−N2−N3does no in e sec any o he ou (−2)-cu es
N1, N2, N3, Z.
This shows ha (∆−Ni).Ni= 0, i = 1,2,3 and (∆−N1−N2−N3).Z = 0.
I ollows ha ∆ ≡ −3Ke
Y+N1+N2+N3.In ac , by P oposi ion 7.1 (3)
we may assume ha
∆≡ −3Ke
Y+x1N1+x2N2+x3N3+yZ, whe e x1, x2, x3, y a e in ege s.
The condi ions abo e show ha x1=x2=x3= 1, y = 0.
Second, we conside he (−1)-cu es. Recall ha e
Ycon ains exac ly six
(−1)-cu es: E0
1, E0
2, E0
3,Γ1,Γ2,Γ3.Since E0
i.Ni= 1, he discussion abo e
shows ha ∆ 6≥ Ei o i= 1,2,3.Bu ∆ con ains a leas h ee (−1)-cu es,
hus ∆ ≥Γ1+ Γ2+ Γ3.
Le ∆0:= ∆ −N1−N2−N3−Γ1−Γ2−Γ3≡ −2Ke
Y.Since we ha e
conside ed all he (−2)-cu es and all he (−1)-cu es, ∆0consis s o 0-cu es.
No e ha ∆0is e ec i e, educed and is disjoin om all (−2)-cu es. An
easy a gumen using he ollowing Lemma 7.4 shows ha ∆0=C1+C2+C3.
32
7. D4-GENERALIZED BURNIAT SURFACES
Lemma 7.4. Assume ha Cis a smoo h a ional cu e on e
Ywi h C2= 0.
I C.N1=C.N2=C.N3=C.Z = 0, hen Cbelongs o one o he ollowing
linea sys ems: |2L−Ei+1 −Ei+2 −E0
i+1 −E0
i+2| o i= 1,2,3.
P oo . We may assume ha C≡λL −P3
i=1(xiEi+yiE0
i) in Pic(e
Y), λ and
xi, yia e in ege s. C.N1=C.N2=C.N3=C.Z = 0 show ha xi=yi o
i= 1,2,3 and λ−x1−x2−x3= 0.Thus C≡(x1+x2+x3)L−P3
i=1 xi(Ei+E0
i).
Then C2= 0 and −Ke
Y.C = 2 imply x1+x2+x3= 2, x2
1+x2
2+x2
3= 2.
Since Cis e ec i e and i educible, he conclusion ollows.
We ha e seen
∆ = N1+N2+N3+ Γ1+ Γ2+ Γ3+C1+C2+C3
≡9L−2E1−4E0
1−2E2−4E0
2−2E3−4E0
2.
By Co olla y 5.4 (1) and P oposi ion 7.1 (2), we ha e
∆1=N2+Cj+ Γα,∆2=N3+Ck+ Γβ,∆3=N1+Cl+ Γγ,
whe e {j, k, l}={α, β, γ}={1,2,3}.By Rema k 7.1, each ∆ihas e en
coe icien s in E1, E0
1, E2, E0
2, E3, E0
3.So he e a e only wo possibili ies:
(a) ∆i= Γi+Ni+1 +Ci+2,(b) ∆i= Γi+2 +Ni+1 +Ci, o each i= 1,2,3.
I Yois o D4(1)- ype, Γ1,Γ2,Γ3mee a a poin Pon e
Y . No e ha any
o he i educible componen o ∆ does no pass h ough P. Then Example 1.1
shows ha e
Shas a 1
4(1,1)-singula i y P0,which is no a a ional double poin .
Since he Γi’s a e disjoin om any (−2)-cu es, e
S→ Xois locally isomo phic
a P0.This con adic s ha Xois a canonical su ace.
Thus Yomus be o D4(2)- ype.
No e ha he e is an in olu ion τ:P2→P2such ha τ(P1) = P1, τ(P0
1) =
P0
1, τ(P2) = P3, τ(P0
2) = P0
3, τ(P3) = P2, τ(P0
2) = P0
3( o example, in he
no a ion o Subsec ion 3.2, τis de ined by (y0:y1:y2)7→ (y0:y2:y1)). τ
induces an in olu ion on e
Y . I maps he di iso classes o ∆1,∆2,∆3in case
(a) o he ones o ∆2,∆1,∆3in case (b) espec i ely. Hence he bidouble
co e s associa ed o he wo kinds o b anch loci a e essen ially he same.
33
7. D4-GENERALIZED BURNIAT SURFACES
Rema k 7.2. I Yois o D4(1)- ype, hen we al eady see ha e
Shas a 1
4(1,1)-
singula i y. I we esol e his singula i y and blow down he (−1)-cu es
π−1Ni,we ge a amily o minimal smoo h su aces o gene al ype wi h
K2= 2, pg=q= 0.We ema k ha he undamen al g oup o such a
su ace is isomo phic o (Z/2Z)3.
7.2 D4-gene alized Bu nia Su aces
Assume ha Yois he D4(2)- ype cubic su ace, and e
Yis i s minimal esolu-
ion. Recall he no a ion in oduced in Subsec ion 3.2 and Figu e 5.
We de ine h ee e ec i e di iso s on e
Y ,
∆i= Γi+Ni+1 +Ci+2 ≡3L−2Ei−2E0
i−2E0
i+1, i = 1,2,3,(7.1)
whe e all Ci’s a e i educible smoo h cu es. And de ine h ee di iso s
Li=−Ke
Y+Ei−E0
i+2, i = 1,2,3.(7.2)
Theo em 7.5 ([BC10-b, Sec ion 7]).Le π:e
S→e
Ybe he bidouble co e as-
socia ed o he abo e da a ∆1,∆2,∆3,L1,L2,L3.Then e
Sis a smoo h su ace
wi h K2
e
S=−3, pg(e
S) = q(e
S) = 0.
Mo eo e , |2Ke
S|=π∗| − Ke
Y|+π∗(N1+N2+N3)and P2(e
S) = 4.
P oo . Fi s no e ha ∆i’s and Li’s sa is y he equa ions (1.1) and (1.2).
Since he o al b anch di iso ∆ is no mal c ossing and each ∆iis smoo h,
e
Sis smoo h by P oposi ion 1.2 (2).
No e ha L2
i= 1, Ke
Y.Li=−3.By Co olla y 1.4, K2
e
S=−3 and
χ(Oe
S) = 1.F om (7.2), one sees ha Ke
Y+Liis no e ec i e o all i= 1,2,3.
Hence by Co olla y 1.4, pg(e
S) = pg(e
Y) = 0.I ollows ha q(e
S) = 0.
F om (7.2), one sees ha 2Ke
S+Li+Li+1 is no e ec i e o all iand
L1+L2+L3≡ −3Ke
Y+N1+N2+N3.By Theo em 1.3 (2) and Co olla y 1.4,
2Ke
S≡π∗(−Ke
Y+N1+N2+N3),
P2(e
S) = h0(e
Y , −Ke
Y+N1+N2+N3) = h0(e
Y , −Ke
Y) = 4.
I ollows ha |2Ke
S|=π∗|−Ke
Y+N1+N2+N3|=π∗|−Ke
Y|+π∗(N1+N2+N3),
since N1+N2+N3is he ixed pa o | − Ke
Y+N1+N2+N3|.
34

8. 4A1-GENERALIZED BURNIAT SURFACES
De ini ion 7.1. The minimal model o e
Sin he Theo em 7.5 is called a
D4-gene alized Bu nia su ace.
Co olla y 7.6 ([BC10-b, Sec ion 7]).Le :e
S→Sbe he blow down o he
six (−1)-cu es π−1Ni o i= 1,2,3.Then Sis a smoo h minimal su ace o
gene al ype wi h K2
S= 3, pg(S) = q(S) = 0 and P2(S) = 4. S has exac ly
one (−2)-cu e Z0.Mo eo e , ∗|2KS|=π∗|−Ke
Y|and he bicanonical linea
sys em o Sis base-poin - ee.
P oo . Since each Ni, i = 1,2,3, o ms a connec ed componen o he b anch
locus, each π−1Niis a disjoin union o wo (−1)-cu es. No e ha Zis no
in he b anch locus, and Z.Ni= 1, i = 1,2,3.Then Hu wi z’s Theo em
shows ha π∗Zis a smoo h a ional cu e wi h sel -in e sec ion numbe −8.
Le :e
S→Sbe he blow down o he six (−1)-cu es. Then K2
S= 3 and
he image o π∗Zis a (−2)-cu e Z0.
Since pg, q, P2a e bi a ional in a ian s, pg(S) = 0 and P2(S) = 4.Mo e-
o e , since |2Ke
S|= ∗|2KS|+π∗(N1+N2+N3) by he Theo em 7.5, we ha e
∗|2KS|=π∗|−Ke
Y|.|−Ke
Y|is base-poin - ee, hus |2KS|is base-poin - ee.
Mo eo e , −Ke
Yis ne and big, so is KS.Thus Sis minimal and o gene al
ype.
Co olla y 7.7 ([BC10-b, Sec ion 7]).Le ϕ:S→Xbe he con ac ion o
he (-2)-cu e Z0,i.e., Xis he canonical model o S. Then Xis a bidouble
co e o he D4(2)- ype cubic su ace Yoby he bicanonical mo phism.
Mo eo e , Xhas an A1-singula i y, lying o e he D4-singula i y o Yo,
whe e he bicanonical mo phism is o ally ami ied.
P oo . I ollows om Co olla y 7.6 and Lemma 7.2.
84A1-gene alized Bu nia Su aces
8.1 Con igu a ion o B anch Di iso s
Assume ha Yois he 4A1- ype cubic su ace. Le µ:e
Y→ Yobe i s minimal
esolu ion. Recall he no a ion in oduced in Subsec ion 3.3 and
Figu e 6.
35
8. 4A1-GENERALIZED BURNIAT SURFACES
Theo em 8.1. π:e
S→e
Yis isomo phic o he bidouble co e associa ed o
he ollowing b anch di iso s:
∆1= Γ1+N2+C3,∆2= Γ2+N3+C1,∆3= Γ3+N1+C2,
whe e all Ci’s a e i educible smoo h cu es.
Be o e gi ing he p oo , we make he ollowing ema k.
Rema k 8.1. (1) By Theo em 3.3, up o an isomo phism, he e is exac ly
one 4A1- ype cubic su ace.
(2) I well known (c . [Sak10, Theo em 3]), he au omo phism g oup o a
4A1- ype cubic su ace is isomo phic o he symme y g oup o ou le e s,
which pe mu es he 4 nodes o he su ace.
P oo . Fi s we conside he (−1)-cu es. All he (−1)-cu es excep Γ1,
Γ2,Γ3in e sec a leas one (−2)-cu e, which co espond o he lines in Yo
passing h ough singula i ies. By Co olla y 5.4 (2) and Lemma 6.4, ∆ ≥
Γ1+ Γ2+ Γ3.
Nex we conside 0-cu es.
Lemma 8.2. Fix k∈ {1,2,3}.Assume ha Cis a educed cu e o e
Ysuch
ha C6≥ Γi, Ni, o i= 1,2,3.I µ(C+ Γk)is a hype plane sec ion o Yo,
hen Cis a smoo h i educible cu e in he linea sys em |2L−Ek+1 −Ek+2 −
E0
k+1 −E0
k+2|.I ollows ha Cis disjoin om all Ni’s and Z.
P oo . Wi hou loss o gene ali y, assume ha k= 1.No e ha elemen s in
| − Ke
Y|co espond o hype plane sec ions o Yoand Γ1+ (2L−E2−E3−
E0
2−E0
3)≡ −Ke
Y.I Cis a singula elemen in |2L−E2−E3−E0
2−E0
3|,
hen C=N1+Z+ 2E1,o C=N2+N3+ 2E0
1o C= Γ2+ Γ3.Thus he
i s conclusion ollows. The second conclusion ollows om he calcula ion
o in e sec ion numbe s.
By Co olla y 5.4 (3) and Lemma 8.2, one sees ha ∆ mus con ain a
smoo h cu e Ciin he linea sys em |2L−Ei+1 −Ei+2 −E0
i+1 −E0
i+2| o
each i.
36
8. 4A1-GENERALIZED BURNIAT SURFACES
We ha e shown ha ∆ ≥Γ1+ Γ2+ Γ3+C1+C2+C3.By Co olla y
5.4 (1), (2) and (3), up o a pe mu a ion o 1,2,3,one o he ollowing wo
holds:
(a) ∆1≥Γ1+C3,∆2≥Γ2+C1,∆3≥Γ3+C2,
(b) ∆1≥Γ3+C1,∆2≥Γ1+C2,∆3≥Γ2+C3.
Since he 3 nodes on Y a e in he b anch locus o π , hus a leas 3 nodes
o Yoa e in he b anch locus o πo.Equi alen ly, ∆ con ains a leas h ee
(−2)-cu es. We dis inguish wo cases.
Case I: One o he ou (−2)-cu es is no in ∆.
Wi hou loss o gene ali y (c . Rema k 8.1 (2)), assume ha ∆ 6≥ Z. Then
∆ =
3
X
i=1
(Γi+Ci+Ni)≡12L−
3
X
i=1
(4Ei+ 5E0
i).
Thus ∆ihas e en coe icien s in E1, E2, E3and odd coe icien s in E0
1, E0
2, E0
3
by Rema k 7.1. I ollows ha i (a) holds hen ∆i= Γi+Ni+1 +Ci+2,and
i (b) holds hen ∆i= Γi+2 +Ni+1 +Ci.
Take an in olu ion τo P2such ha τ(P1) = P1, τ(P0
1) = P0
1, τ(P2) = P3,
τ(P0
2) = P0
3.Then i ollows ha τ(P3) = P2, τ(P0
3) = P0
2.I induces an
in olu ion on e
Ywhich maps he di iso classes o ∆1,∆2,∆3in case (a) o
he ones o ∆2,∆1,∆3in case (b). Hence he bidouble co e s associa ed o
he wo kinds o b anch loci a e essen ially he same.
Case II: All he 4 nodes a e con ained in he b anch locus, i.e,
∆≥N1+N2+N3+Z. We in end o exclude his case.
Assume ha (a) holds. Then we may assume ha
∆1= Γ1+C3+
3
X
i=1
aiNi+a4Z,
∆2= Γ2+C1+
3
X
i=1
biNi+b4Z,
∆3= Γ3+C2+
3
X
i=1
ciNi+c4Z,
37
8. 4A1-GENERALIZED BURNIAT SURFACES
o each k= 1,2,3,4,exac ly one o he ak, bk, ckis 1 and he o he wo is
0,since ∆ is e ec i e and educed. The ollowing able gi es he coe icien s
(up o sign) o E1, E2, E3in he b anch di iso s.
∆1∆2∆3
E1a1+a4+ 2 b1+b4c1+c4+ 1
E2a2+a4+ 1 b2+b4+ 2 c2+c4
E3a3+a4b3+b4+ 1 c3+c4+ 2
By Rema k 7.1, a1+a4+ 2, b1+b4, c1+c4+ 1 mus be o he same pa i y.
Since hei sum is 5, hey mus be all odd in ege s. Thus ei he
(a1, b1, c1) = (1,0,0) and (a4, b4, c4) = (0,1,0),o (a1, b1, c1) = (0,1,0) and
(a4, b4, c4) = (1,0,0).
I he o me holds, hen he coe icien s a3, b3+ 2, c3+ 2 o E3canno
ha e he same pa i y. I he la e holds, hen he coe icien s a2+2, b2+2, c2
o E2canno ha e he same pa i y. So his case is excluded.
I (b) holds, a simila a gumen shows ha Case II can be excluded.
Rema k 8.2. In he cou se o excluding Case II, we ind ano he amily o
su aces o gene al ype which a e also bidouble co e s o he 4A1- ype cubic
su ace, bu b anched on all he nodes. Fi s cons uc he bidouble co e
π:e
S→e
Yassocia ed o he ollowing da a,
∆1=C1+ Γ2+N1+N2,∆2=C2+ Γ1+N3+Z, ∆3=C3+ Γ3.
Then blow down he eigh (−1)-cu es π−1Niand π−1Z, :e
S→S. S is
o gene al ype wi h K2
S= 3 and pg(S) = 0. S has 4 nodes coming om
he nodes o he cu e ∆3.Howe e , no e ha ∆3≡ −Ke
Y,we can de o m
S o smoo h su aces by de o ming ∆3 o smoo h cu es. Fo de ails, see
Sec ion 14 in Pa IV.
8.2 4A1-gene alized Bu nia Su aces
Assume ha Yois he 4A1- ype cubic su ace, and e
Yis i s minimal esolu ion.
Recall he no a ion in oduced in Subsec ion 3.3 and Figu e 6.
38
10. KEY TOOLS
Mo eo e , le Mbe a di iso on e
Ysuch ha (Ke
Y+ 2C+M).C < 0.Then
H0(Ω1
e
Y(log(∆ −C))(C+M)) ∼
=H0(Ω1
e
Y(log ∆)(M)).
P oo . Since Cis a connec ed componen o a smoo h di iso ∆,we ha e he
ollowing exac sequence,
0→Ω1
e
Y(log ∆) →Ω1
e
Y(log(∆ −C))(C)→Ω1
C(C)→0
Tenso i wi h he in e ible shea Oe
Y(M) and use he adjunc ion o mula
Ω1
C=OC(Ke
Y+C), o ge he exac sequence,
0→Ω1
e
Y(log ∆)(M)→Ω1
e
Y(log(∆ −C))(C+M)→ OC(Ke
Y+ 2C+M)→0
Since (Ke
Y+ 2C+M).C < 0, H0(C, OC(Ke
Y+ 2C+M)) = 0, he associ-
a ed exac sequence o cohomology g oups shows ha H0(Ω1
e
Y(log ∆)(M)) ∼
=
H0(Ω1
e
Y(log(∆ −C))(C+M)).
Lemma 10.3 ([Ca 84, Lemma 3.7], [CHKS06, Lemma 3, page 675]).Le
∆ = ∪i∆ibe a union o smoo h di iso s ∆1,...,∆kon a smoo h su ace e
Y ,
such ha ∆has only no mal c ossing singula i ies. Then
(1) he e is an exac sequence
0→Ω1
e
Y→Ω1
e
Y(log ∆1,...,log ∆k)→ ⊕k
i=1O∆i→0.
(2) In he cohomology exac sequence associa ed o he abo e exac sequence
∂:⊕k
i=1 H0(O∆i)→H1(Ω1
e
Y),i 1∆iis he unc ion which is ≡1on ∆i
and 0elsewhe e, hen ∂(1∆i) = c1(∆i).
The nex heo em s udies how he dimensions o he cohomology g oups
o angen shea change when blowing down a (−1)-cu e.
Theo em 10.4 (c . [Ca 88, Lemma 9.22]).Le Sbe a smoo h su ace, and
:e
S→Sbe he blowup o Sa a poin p. Then R1 ∗Θe
S= 0.
Mo eo e , i Sis o gene al ype, hen h0(e
S, Θe
S) = h0(S, ΘS) = 0,
h1(e
S, Θe
S) = h1(S, ΘS) + 2 and h2(e
S, Θe
S) = h2(S, ΘS).
45

10. KEY TOOLS
P oo . Le Ebe he excep ional cu e o . The shea R1 ∗Θe
Sis suppo ed
on he poin p, by o mal unc ion heo em (c . [Ha 77, Theo em 11.1]), i
su ices o show H1(En,Θe
S⊗ OEn) = 0,whe e Enis he closed subscheme o
e
Sde ined by In,whe e Iis he ideal shea o E.
The e is an exac sequence
0→In
In+1 → OEn+1 → OEn→0.
o all n≥0.Tenso he exac sequence by Θe
S,i emains exac since Θe
Sis
a locally ee shea . No e ha E1=Eand In
In+1 ∼
=OE(n),i su ices o show
H1(E, Θe
S⊗ OE(n)) = 0 o all n≥0.
We ha e a no mal exac sequence
0→ΘE→Θe
S⊗ OE→ OE(E)→0.
Tenso i wi h OE(n),we ge
0→ OE(n+ 2) →Θe
S⊗ OE(n)→ OE(n−1) →0.
Since H1(E, OE(n−1)) = 0 o n≥0,one sees ha H1(E, Θe
S⊗OE(n)) = 0.
Hence we ha e shown ha R1 ∗(Θe
S) = 0.(See [Ca 88, Lemma 9.22] o
ano he p oo ).
The e is an exac sequence (c . [Se 06, page 73]),
0→Θe
S→ ∗ΘS→ OE(−E)→0.
By [Ha 77, P oposi ion 3.4, Chap e V] Rk ∗Oe
S= 0 o k≥1, hen he
p ojec ion o mula shows ha Rk ∗( ∗ΘS) = Rk ∗Oe
S⊗ΘS= 0.Thus we
ha e an exac sequence
0→ ∗Θe
S→ΘS→ ∗OE(−E)→0.
I Sis o gene al ype, h0(e
S, Θe
S) = h0(S, ΘS) = 0 (c . [Ma s63]). No e
ha ∗OE(−E) is suppo ed on p, hus Hk(S, ∗OE(−E)) = 0 o k≥1.By
he long exac sequence o cohomology associa ed o he las exac sequence
abo e, and OE(−E)∼
=OE(1),we ha e h1(S, ∗Θe
S) = h1(S, ΘS) + 2 and
h2(S, ∗Θe
S) = h2(S, ΘS).
Finally since Rk ∗Θe
S= 0 o k≥1,Le ay spec al sequence shows ha
h1(e
S, Θe
S) = h1(S, ∗Θe
S) and h2(e
S, Θe
S) = h2(S, ∗Θe
S).Hence he conclusion
ollows.
46
11. DEFORMATIONS OF THE D4-GENERALIZED BURNIAT SURFACES
The las heo em desc ibes how he dimensions o he cohomology g oups
o he angen shea change when con ac ing a (−2)-cu e o a node.
Theo em 10.5 ([BW74, P oposi ion 1.10, Theo em 2.14]).Le Sbe a min-
imal su ace o gene al ype, and le ϕ:S→Xbe a mo phism con ac ing a
(−2)-cu e No S o an A1-singula i y on X. Then
ϕ∗ΘS= ΘX, H1(S, ΘS)∼
=H1(X, ΘX)⊕H1
N(ΘS), H2(S, ΘS)∼
=H2(X, ΘX).
Mo eo e , dim H1
N(ΘS) = 1.
11 De o ma ions o he D4-gene alized
Bu nia Su aces
Th oughou his sec ion, we use he no a ion in oduced in Sub-
sec ion 3.2 and Subsec ion 7.2. See Figu e 5.
We s a o s udy he local de o ma ions o he D4-gene alized Bu nia
su aces. Le Xbe he canonical model o a D4-gene alized Bu nia su ace S.
We in end o calcula e he dimension o he angen space o he base o he
Ku anishi amily o X, i.e., dim Ex 1
OX(Ω1
X,OX).Fo his we i s calcula e
hi(e
S, Θe
S) (c . Theo em 7.5), using he bidouble co e s uc u e as desc ibed
in Theo em 10.1. Then we pass om e
S o he minimal model S, calcula e
hi(S, ΘS) by Theo em 10.4. Finally, we pass om S o he canonical model
Xby Theo em 10.5, and use he spec al sequence
Epq
2=Hp(X, Ex q
OX(Ω1
X,OX)) ⇒Ex p+q
OX(Ω1
X,OX).
By Se e Duali y and Theo em 10.1,
Hk(e
S, Θe
S)in =H2−k(e
Y , Ωe
Y(log ∆1,log ∆2,log ∆3)⊗Ω2
e
Y),(11.1)
Hk(e
S, Θe
S)χi=H2−k(e
Y , Ωe
Y(log ∆i)(Ke
Y+Li)),(11.2)
o k= 0,1,2 and i= 1,2,3.
Since e
Sis a su ace o gene al ype, H0(e
S, Θe
S) = 0.The e o e he igh -
hand sides o he equa ions equal 0 when k= 0.
P oposi ion 11.1. h0(e
Y , Ω1
e
Y(log ∆1,log ∆2,log ∆3)⊗Ω2
e
Y) = 0 and
h1(e
Y , Ω1
e
Y(log ∆1,log ∆2,log ∆3)⊗Ω2
e
Y) = 4.
47
11. DEFORMATIONS OF THE D4-GENERALIZED BURNIAT SURFACES
P oo . By Lemma 10.3 (1), we ha e an exac sequence
0→Ω1
e
Y(Ke
Y)→Ω1
e
Y(log ∆1,log ∆2,log ∆3)(Ke
Y)→ ⊕3
i=1O∆i(Ke
Y)→0
(11.3)
No e ha H0(e
Y , Ω1
e
Y) = 0 and −Ke
Yis e ec i e, hus H0(e
Y , Ω1
e
Y(Ke
Y)) = 0.
To p o e he i s equali y, i su ices o show he bounda y map
δ:H0(e
Y , ⊕3
i=1O∆i(Ke
Y)) →H1(e
Y , Ω1
e
Y(Ke
Y))
is injec i e.
Since ∆iis a disjoin union o h ee smoo h a ional cu es Γi, Ni+1, Ci+2,
H0(e
Y , O∆i(Ke
Y)) ∼
=H0(e
Y , ONi+1 )∼
=C.
| − Ke
Y|is base-poin - ee, he e o e he e is a mo phism Oe
Y(Ke
Y)→ Oe
Y,
which is no iden ically ze o on any componen o ∆i’s, in pa icula on Ni’s.
Now conside he commu a i e diag am coming om he abo e mo phism
Oe
Y(Ke
Y)→ Oe
Y,
0//Ω1
e
Y(Ke
Y)

//Ω1
e
Y(log ∆1,log ∆2,log ∆3)(Ke
Y)

//⊕3
i=1O∆i(Ke
Y)

//0
0//Ω1
e
Y
//Ω1
e
Y(log ∆1,log ∆2,log ∆3)//⊕3
i=1O∆i
//0.
I gi es a commu a i e diag am o cohomology g oups,
C3∼
=H0(e
Y,⊕3
i=1O∆i(Ke
Y))
ψ2

δ//
ψ
**
V
V
V
V
V
V
V
V
V
V
V
V
V
V
V
V
H1(e
Y,Ω1
e
Y(Ke
Y))

H0(e
Y , ⊕3
i=1O∆i)ψ1//H1(e
Y , Ω1
e
Y).
By Lemma 10.3 (2), he image o he unc ion iden ically equal o 1 on Ni
maps unde ψ1 o he i s Che n class o Ni.Because Ni’s a e disjoin (−2)-
cu es, hei Che n classes a e linea ly independen in H1(e
Y , Ω1
e
Y).Thus he
composi e map ψis injec i e.
Hence δis also injec i e and H0(e
Y , Ω1
e
Y(log ∆1,log ∆2,log ∆3)⊗Ω2
e
Y) = 0.
Since H2(e
Y , Ω1
e
Y(log ∆1,log ∆2,log ∆3)⊗Ω2
e
Y) = 0, o calcula e he dimen-
sion o H1(e
Y , Ω1
e
Y(log ∆1,log ∆2,log ∆3)⊗Ω2
e
Y) is he same as o calcula e
χ(Ω1
e
Y(log ∆1,log ∆2,log ∆3)⊗Ω2
e
Y).By he exac sequence (11.3),
χ(Ω1
e
Y(log ∆1,log ∆2,log ∆3)⊗Ω2
e
Y) = χ(Ω1
e
Y(Ke
Y)) +
3
X
i=1
χ(O∆i(Ke
Y)).
48
11. DEFORMATIONS OF THE D4-GENERALIZED BURNIAT SURFACES
Se e’s Duali y and Riemann-Roch heo em show ha ,
χ(Ω1
e
Y(Ke
Y)) = χ(Θe
Y) = 1
2c1(e
Y)(c1(e
Y)−Ke
Y)−c2(e
Y) + 2χ(Oe
Y) = −4.
No e ha ∆iis a disjoin union o h ee smoo h a ional cu es Γi, Ni+1, Ci+2.
I ollows ha χ(O∆i(Ke
Y)) = 0 o i= 1,2,3.
Hence χ(Ω1
e
Y(log ∆1,log ∆2,log ∆3)⊗Ω2
e
Y) = −4 and i ollows ha
h1(e
Y , Ω1
e
Y(log ∆1,log ∆2,log ∆3)⊗Ω2
e
Y) = 4.
In o de o calcula e h0(e
Y , Ω1
e
Y(log ∆i)(Ke
Y+Li)) o i= 1,2,3, we need
he ollowing lemmas.
Lemma 11.2. Le p1:W→C2be he blowup o C2a (0,0),and le p2: Σ →
Wbe he blowup o Wa he in e sec ion poin O0o he s ic ans o m o
he line l:y= 0 wi h he excep ional cu e Eo p1.
Deno ed by E0 he excep ional cu e o p2and by Γ he s ic ans o m
o he line lunde he mo phism p=p2◦p1: Σ →W→C2.Then
(1) p∗Ω1
Σ(−E0)⊆Ω1
C2is he subshea o o ms
{ω∈Ω1
C2|ω=α(x, y)dx +β(x, y)dy, α(0,0) = 0}.
(2) p∗Ω1
Σ(−2E0)⊆Ω1
C2is he subshea o o ms
{ω∈Ω1
C2|ω=α(x, y)dx +β(x, y)dy, α(0,0) = 0,
∂α
∂x(0,0) = 0, β(0,0) = 0}.
(3) p∗Ω1
Σ(log Γ)(−E0)⊆Ω1
C2(log l)is he subshea o o ms
{ω∈Ω1
C2(log l)|ω=α(x, y)dx +β(x, y)dy
y,
β(0,0) = 0, α(0,0) + 2∂β
∂x(0,0) = 0}.
P oo . Wcan be co e ed by wo a ine coo dina e cha s V1∼
=C2(x, ) and
V2∼
=C2(s, y),such ha p1is gi en by
V1→C2,(x, )7→ (x, x),
V2→C2,(s, y)7→ (sy, y).
49
11. DEFORMATIONS OF THE D4-GENERALIZED BURNIAT SURFACES
p−1
2(V1) can be co e ed by wo a ine coo dina e cha s U11 ∼
=C2(x, u) and
U12 ∼
=C2( , ) such ha he mo phism p: Σ →C2is gi en by
U11 →V1→C2,(x, u)7→ (x, ux)7→ (x, x2u),
U12 →V1→C2,( , )7→ ( , )7→ ( , 2).
And simila ly o p−1
2(V2) = U21 ∪U22.No e ha bo h E0and Γ a e con ained
in U11 ∪U12.
Fi s use he coo dina e cha U11.Locally E0is de ined by x= 0 and Γ
is de ined by u= 0.
(1) By Riemann’s ex ension heo em, p∗Ω1
Σ(−mE0)⊆Ω1
C2 o all m≥0.
Assume ha ω=α(x, y)dx +β(x, y)dy o some holomo phic unc ion
α(x, y) and β(x, y).Then
p∗ω=α(x, x2u)dx +β(x, x2u)(x2du + 2xudx)
= (α(x, x2u) + 2xuβ(x, x2u))dx +β(x, x2u)x2du,
Hence locally p∗ωbelongs o he OΣ-module gene a ed by xdx, xdu i
and only i α(0,0) = 0.
(2) By he calcula ion abo e, locally p∗ωbelongs o he OΣ-module gene -
a ed by x2dx, x2du i and only i α(x, x2u) + 2xuβ(x, x2u) is di isible
by x2.Assume ha
α(x, y) = a+bx +cy +highe deg ee e ms, (11.4)
β(x, y) = A+Bx +Cy +highe deg ee e ms, (11.5)
a=α(0,0), b =∂α
∂x(0,0), c =∂α
∂y (0,0),
A=β(0,0), B =∂β
∂x(0,0), C =∂β
∂y (0,0),
hen
α(x, x2u) + 2xuβ(x, x2u) = a+bx + 2Axu +x2h(x, u),
o some holomo phic unc ion h(x, u).Thus p∗ωbelongs o he OΣ-
module gene a ed by x2dx, x2du, i and only i a=b=A= 0.
50

11. DEFORMATIONS OF THE D4-GENERALIZED BURNIAT SURFACES
(3) Obse e ha p∗Ω1
Σ(log Γ)(−E0) consis s o a ional di e en ial 1- o ms
ωwhich, when es ic ed o C2 {(0,0)},yield sec ions o Ω1
C2(log l).In
pa icula , yω is a egula 1- o m on C2 {(0,0)},which can be ex ended
o a egula 1- o m on C2.Assume ha ω=α1(x, y)dx
y+β(x, y)dy
y o
some holomo phic unc ion α1(x, y) and β(x, y), hen
p∗ω=α1(x, x2u)
x2udx + 2β(x, x2u)dx
x+β(x, x2u)du
u
= (α1(x, x2u)
x3u+2β(x, x2u)
x2)xdx +β(x, x2u)
xxdu
u.
Thus p∗ωbelongs o he OΣ-module gene a ed by xdx, xdu
ui and only
i α1(x, x2u)+2xuβ(x, x2u) is di isible by x3uand β(x, x2u) is di isible
by x.
I α1(x, x2u) + 2xuβ(x, x2u) is di isible by x3u, hen α1(x, x2u) is di-
isible by u. This implies α1(x, y) = yα(x, y) o some holomo phic
unc ion α(x, y).Then
ω=α(x, y)dx +β(x, y)dy
y,
p∗ω=α(x, x2u)dx + 2β(x, x2u)dx
x+β(x, x2u)du
u.
I we w i e α(x, y), β(x, y) as (11.4) and (11.5), hen one sees ha
p∗ωbelongs o he OΣ-module gene a ed by xdx, xdu
ui and only i
A= 0, a + 2B= 0.
Hence we see ha (1),(2),(3) hold locally. Simila calcula ion wi h o he
coo dina e cha s show he same esul s.
Lemma 11.3. Le ldeno e he line on he p ojec i e plane P2de ined by
x1= 0.Then any ω∈H0(Ω1
P2(log l)(2)) is o he o m
ω= (−Ax1x2−Cx1x3+Dx2
2+Ex2
3+Fx2x3)dx1
x1
+ (Ax1−Dx2−Bx3)dx2+ (Cx1+Bx2−Fx2−Ex3)dx3,(11.6)
whe e A, B, C, D, E, F ∈C.
51
11. DEFORMATIONS OF THE D4-GENERALIZED BURNIAT SURFACES
P oo . By [BC10-b, Lemma 5.2 (1)], he ec o space H0(Ω1
P2(2)) is
3-dimensional wi h a basis: −x2dx1+x1dx2,−x3dx2+x2dx3,−x3dx1+x1dx3.
By he exac sequence 0 →Ω1
P2(2) →Ω1
P2(log l)(2) → Ol(2) →0 and
since h1(Ω1
P2(2)) = 0 and h0(Ol(2)) = 3,we see ha h0(Ω1
P2(log l)(2)) = 6.
Mo eo e , i is easy o show ha he ollowing o ms
x2
2
dx1
x1
−x2dx2, x2
3
dx1
x1
−x3dx3, x2x3
dx1
x1
−x2dx3
in he ec o space H0(Ω1
P2(log l)(2)),a e mapped o a basis o H0(Ol(2)).
Hence hese o ms and he abo e basis o H0(Ω1
P2(2)) a e linea ly independen
in H0(Ω1
P2(log l)(2)).Then hei linea combina ion
A(−x2dx1+x1dx2) + B(−x3dx2+x2dx3) + C(−x3dx1+x1dx3)
+D(x2
2
dx1
x1
−x2dx2) + E(x2
3
dx1
x1
−x3dx3) + F(x2x3
dx1
x1
−x2dx3)
is o he o m (11.6).
P oposi ion 11.4. h0(e
Y , Ω1
e
Y(log ∆i)(Ke
Y+Li)) = 0 and
h1(e
Y , Ω1
e
Y(log ∆i)(Ke
Y+Li)) = 4, o i= 1,2,3.
P oo . To p o e he i s equali y o i= 3,no e ha by (7.2),
H0(e
Y , Ω1
e
Y(log ∆3)(Ke
Y+L3)) = H0(e
Y , Ω1
e
Y(log N1,log C2,log Γ3)(E3−E0
2)).
Apply Lemma 10.2 o he cu e C2and hen o N1,
H0(e
Y , Ω1
e
Y(log ∆3)(Ke
Y+L3)) = H0(e
Y , Ω1
e
Y(log Γ3)(2L−2E0
1−E0
2−E0
3)).
Wi hou loss o gene ali y, we may assume ha
P1= (1:0:0),P2= (1:1:0),Q1= (0:1:1),Q2= (0:0:1).
I ollows ha P3= (0:1:0)and Q3= (1:0:−1).See Figu e 7.
No e ha σ∗(Ω1
e
Y(log Γ3)(2L−2E0
1−E0
2−E0
3)) is a subshea o Ω1
P2(log l)(2),
hus we can apply Lemma 11.3.
Any ω∈H0(e
Y , Ω1
e
Y(log ∆3)(Ke
Y+L3)),conside ed as an elemen o
H0(P2,Ω1
P2(log l)(2)),is o he o m (11.6).
52
11. DEFORMATIONS OF THE D4-GENERALIZED BURNIAT SURFACES
Locally a ound he poin P1= (1:0:0), x1= 1 and he line P1P0
1is
de ined by x2= 0.So locally we may w i e
ω=α(x2, x3)dx3+β(x2, x3)dx2,
α(x2, x3) = C+Bx2−Fx2−Ex3, β(x2, x3) = A−Dx2−Bx3.
Thus by Lemma 11.2 (2),
α(0,0) = C= 0,∂α
∂x3
(0,0) = −E= 0, β(0,0) = A= 0,
and hen
ω= (Dx2
2+Fx2x3)dx1
x1
+ (−Dx2−Bx3)dx2+ (B−F)x2dx3.
Locally a ound he poin P3= (0:1:0), x2= 1 and he line P3P0
3is
de ined by x1= 0.So locally we may w i e
ω= (D+Fx3)dx1
x1
+ (B−F)dx3.
Then by Lemma 11.2 (3), D= 0, B +F= 0,and hen
ω=F(x2x3
dx1
x1
+x3dx2−2x2dx3).
Locally a ound he poin P2= (1:1:0), x1= 1. P2is he in e sec ion
poin o he line x3= 0,and he line P2P0
2: 1−x2+x3= 0.Le x:= x3, y :=
1−x2+x3.Then locally
ω=F(−2−x+ 2y)dx +F(−x)dy.
Thus by Lemma 11.2 (1), F= 0, ω = 0.
Hence H0(e
Y , Ω1
e
Y(log ∆3)(Ke
Y+L3)) = 0.
No e ha H2(e
Y , Ω1
e
Y(log ∆3)(Ke
Y+L3)) = 0,so o calcula e he dimension
o H1(e
Y , Ω1
e
Y(log ∆3)(Ke
Y+L3)) is equi alen o calcula e
χ(Ω1
e
Y(log ∆3)(Ke
Y+L3)).Twis he ollowing exac sequence wi h he in e -
ible shea associa ed o he di iso F:= Ke
Y+L3,
0→Ω1
e
Y→Ω1
e
Y(log ∆3)→ O∆3→0,
53
11. DEFORMATIONS OF THE D4-GENERALIZED BURNIAT SURFACES
we ge
χ(Ω1
e
Y(log ∆3)(Ke
Y+L3)) = χ(Ω1
e
Y(F)) + χ(O∆3(F)).
Fo he second summand, since ∆3is he disjoin union o a ional cu es
N1, C2,Γ3, and F.N1= 0,F.C2= 1,F.Γ3= 1,we ha e
χ(O∆3(F)) = χ(ON1) + χ(OC2(1)) + χ(OΓ3(1)) = 5.
Fo he i s summand, using he spli ing p inciple, o mally w i e
Ω1
e
Y=Oe
Y(A1)⊕ Oe
Y(A2),and A1+A2=Ke
Y, A1.A2=c2(Y) = 9.
No e ha F2=−2 and F.Ke
Y= 0,Riemann-Roch Theo em gi es
χ(Ω1
e
Y(F)) = χ(Oe
Y(A1+F)) + χ(Oe
Y(A2+F))
=
2
X
i=1
1
2(Ai+F)(Ai+F − Ke
Y) + 2χ(Oe
Y)
=−9.
Hence χ(Ω1
e
Y(log ∆3)(Ke
Y+L3)) = −4 and h1(e
Y , Ω1
e
Y(log ∆3)(Ke
Y+L3)) = 4.
Simila ly, he s a emen also holds o i= 1,2.
Theo em 11.5. Le π:e
S→e
Ybe he bidouble co e as in Subsec ion 7.2.
Le Sbe he minimal model o e
Sand X he canonical model o e
S(c . Subsec-
ion 8.1). The espec i e dimensions o he cohomology g oups o he angen
shea es Θe
S,ΘS,ΘXa e as ollows.
h1(e
S, Θe
S) = 16, h1(S, ΘS) = 4, h1(X, ΘX) = 3,
h2(e
S, Θe
S) = 0, h2(S, ΘS) = 0, h2(X, ΘX) = 0.
P oo . By (11.1), (11.2), P oposi ion 11.1 and P oposi ion 11.4, h1(e
S, Θe
S) =
16 and h2(e
S, Θe
S) = 0.
Since Sis ob ained by blowing down six (−1)-cu es (c . Co olla y 7.6)
on e
S, hen by Theo em 10.4, h1(S, ΘS) = 4 and h2(S, ΘS) = 0.
Xis ob ained by con ac ing he (−2)-cu e Z0on S(c . Co olla y 7.7),
hen by Theo em 10.5, h1(X, ΘX) = 3 and h2(X, ΘX) = 0.
Co olla y 11.6. The base o he Ku anishi amily o Sis smoo h.
54
12. DEFORMATIONS OF THE 4A1-GENERALIZED BURNIAT SURFACES
Xis ob ained by con ac ing ou (−2)-cu es on S(c . Co olla y 8.5).
The g oup Gac s on he se o he ou (−2)-cu es ansi i ely. Thus he
conclusion abou X ollows by Theo em 10.5.
Lemma 12.5. The shea Ex 1
OX(Ω1
X,OX)has suppo on he 4nodes o X,
such ha e e y s alk o e a node has leng h 1.Mo eo e , we ha e a decompo-
si ion o he global sec ion g oup o Ex 1
OX(Ω1
X,OX),acco ding o he g oup
ac ion,
H0(X, Ex 1
OX(Ω1
X,OX)) = H0(X, Ex 1
OX(Ω1
X,OX))in ⊕
⊕3
i=1 H0(X, Ex 1
OX(Ω1
X,OX))χi,
and each di ec summand has dimension 1.
P oo . Since he g oup ac s ansi i ely on ou A1-singula i ies, i induces
he egula ep esen a ion on H0(X, Ex 1
OX(Ω1
X,OX)).Hence he conclusion
ollows.
Co olla y 12.6. dim Ex 1
OX(Ω1
X,OX)in = 4 and
dim Ex 2
OX(Ω1
X,OX)in = 0.
P oo . We ha e an exac sequence
0→H1(X, ΘX)→Ex 1
OX(Ω1
X,OX)→H0(X, Ex 2
OX(Ω1
X,OX))
→H2(X, ΘX)→Ex 2
OX(Ω1
X,OX)→0,
associa ed o he spec al sequence,
Epq
2=Hp(X, Ex q
OX(Ω1
X,OX)) ⇒Ex p+q
OX(Ω1
X,OX).
The exac sequence is a G-equi a ian sequence o C- ec o spaces, since
all shea es ha e a na u al G-linea iza ion. Then he conclusion ollows by
Theo em 12.4 and Lemma 12.5.
Unlike he case o he D4-gene alized su aces, we canno de e mine he
de o ma ions o he 4A1-gene alized su aces comple ely by using he bidou-
ble co e s uc u e o he 4A1- ype cubic su ace.
61

13. KEUM-NAIE-MENDES LOPES-PARDINI SURFACES
Pa IV
The I educible Componen
con aining he Keum-Naie-
Mendes Lopes-Pa dini Su aces
13 Keum-Naie-Mendes Lopes-Pa dini
Su aces
J. H. Keum and la e D. Naie ([Ke88], [Na94]) cons uc ed a amily o su aces
o gene al ype wi h K2= 3 and pg(S) = 0.These su aces a e double co e s
o nodal En iques su aces wi h 8 nodes (c . [Na94, Th´eo `eme 2.10]). Also
hese su aces a e di e en om he (ex ended) Bu nia su aces wi h K2= 3,
since hey ha e di e en undamen al g oups.
Theo em 13.1 ([Na94, Th´eo `eme 3.1]).I Sis a Keum-Naie su ace wi h
K2= 3, hen π op
1(S)∼
=(Z/2Z)2×Z/4Z.
Ano he p ope y o Keum-Naie su aces is ha hei bicanonical map
ac o s h ough he co e ing map o he nodal En iques su ace and is o
deg ee 4.La e , in he a icle [MP04], Mendes Lopes and Pa dini ga e an
explici cons uc ion o su aces o gene al ype whose bicanonical map is a
mo phism o deg ee 2.They p o ed he ollowing heo em abou he co e-
sponding subse in he moduli space.
Theo em 13.2 ([MP04, Theo em 2.1, Theo em 7.1]).Le Mcan
1,3be he mod-
uli space o canonical models o su aces o gene al ype wi h χ= 1 and
K2= 3.Le Ebe he subse o Mcan
1,3consis ing o he canonical su aces wi h
pg= 0 whose bicanonical map is composed wi h an in olu ion such ha he
quo ien su ace is bi a ional o an En iques su ace.
(1) I Xbelongs o Eand τis he in olu ion sa is ying he p ope y abo e,
hen X/τ is a nodal En iques su ace wi h 7nodes.
62
14. A SUBFAMILY OF KNMP SURFACES
(2) The se Eis cons uc ible.
(3) The closu e Ein Mcan
1,3is i educible and uni uled o dimension 6.
(4) Econ ains he Keum-Naie su aces wi h K2= 3.
As poin ed ou in [MP04, Rema k 7.2], he e is a ques ion le open:
whe he Eis an i educible componen o Mcan
1,3o no .
We will econs uc a subse E0in E h ough bidouble co e s o a 4A1- ype
cubic su ace. Then by s udying he de o ma ions o he su aces in E0,we
gi e an a i ma i e answe o his ques ion.
14 A Sub amily o KNMP Su aces
In his sec ion we will cons uc he amily o su aces o gene al ype al eady
men ioned in Rema k 8.2. The cons uc ion he e is simila o (bu di e en
om) he one in [MP04, Example 3.6].
Assume ha Yis a 4A1- ype cubic su ace, and e
Yis i s minimal esolu-
ion. Recall he no a ion in oduced in Subsec ion 3.3 and Figu e 6.
Especially ecall ha e
Yhas a pencil o a ional cu es Ciin he linea sys em
|2L−Ei+1 −Ei+2 −E0
i+1 −E0
i+2| o i= 1,2,3.
We de ine h ee e ec i e di iso s on e
Y ,
∆1=C1+ Γ2+N1+N2≡ −Ke
Y+ 2L−2E2−2E0
2−2E0
3,
∆2=C2+ Γ1+N3+Z≡ −Ke
Y+ 2L−2E1−2E3−2E0
1,
∆3=H≡ −Ke
Y,
(14.1)
whe e C1, C2, H a e i educible smoo h cu es. And de ine h ee di iso s
L1=−Ke
Y+L−E1−E3−E0
1≡ −Ke
Y+ Γ1−E3,
L2=−Ke
Y+L−E2−E0
2−E0
3≡ −Ke
Y+ Γ2−E0
3,
L3=−Ke
Y+ 2L−E1−E2−E3−E0
1−E0
2−E0
3≡ −2Ke
Y−L.
(14.2)
Thoughou he ollowing sec ions, we will assume ha he di i-
so ∆ := ∆1+ ∆2+ ∆3has only no mal c ossing singula i ies.
63
14. A SUBFAMILY OF KNMP SURFACES
Theo em 14.1. Le π:e
S→e
Ybe he bidouble co e associa ed o he abo e
da a ∆1,∆2,∆3,L1,L2,L3.Then e
Sis a smoo h su ace wi h K2
e
S=−5and
pg(e
S) = q(e
S) = 0.
Mo eo e , |2Ke
Y| ≡ π∗| − Ke
Y|+π∗(N1+N2+N3+Z)and P2(e
S) = 4.
P oo . No e ha ∆i’s and Li’s sa is y he equa ions (1.1) and (1.2). Since
he o al b anch di iso ∆ has no mal c ossings and each ∆iis smoo h, e
Sis
smoo h by P oposi ion 1.2 (2).
No e ha L1+L2+L3≡ −3Ke
Y+N1+N2+N3+Z, L2
i= 1, Ke
Y.Li=−3.
By Co olla y 1.4, K2
e
S=−5 and χ(Oe
S) = 1.F om (14.2) one sees ha Ke
Y+Li
is no e ec i e, hence by Co olla y 1.4, pg(e
S) = pg(e
Y) = 0.I ollows ha
q(e
S) = 0.
By Theo em 1.3, 2Ke
S≡π∗(−Ke
Y+N1+N2+N3+Z).Mo eo e , om
(14.2) 2Ke
Y+Li+Li+1 is no e ec i e o all i. Take i= 2 o example, assume
ha |2Ke
Y+L2+L3|con ains an e ec i e di iso D. Then D.N1=−2,
(D−N1).N2=−2 and (D−N1−N2).Z =−1 show ha D≥N1+N2+Z.
Bu D−N1−N2−Z≡E1−E0
2,which is no e ec i e. This gi es a
con adic ion. Hence 2Ke
Y+L2+L3is no e ec i e.
I ollows ha
P2(e
S) = h0(e
Y , −Ke
Y+N1+N2+N3+Z) = h0(e
Y , −Ke
Y) = 4,and
|2Ke
S|=π∗| − Ke
Y+N1+N2+N3+Z|
=π∗| − Ke
Y|+π∗(N1+N2+N3+Z),
since N1+N2+N3+Zis he ixed pa o | − Ke
Y+N1+N2+N3+Z|.
Co olla y 14.2. Le :e
S→Sbe he blow down o he eigh (−1)-cu es
π−1Nk(k= 1,2,3) and π−1Z. Then Sis a smoo h minimal su ace o gene al
ype wi h K2
S= 3, pg(S) = 0 and P2(S) = 4.
Mo eo e , KSis ample and |2KS|is base-poin - ee. Sis a bidouble co e
o he 4A1- ype cubic su ace Y h ough he bicanonical mo phism.
P oo . Since each Nk, k = 1,2,3,o Z o ms a connec ed componen o he
b anch locus, each π−1Nko π−1Zis a disjoin union o wo (−1)-cu es. Le
:e
S→Sbe he blow down o hese eigh (−1)-cu es. Then K2
S= 3.
64
14. A SUBFAMILY OF KNMP SURFACES
Since pg, q, P2a e bi a ional in a ian s, pg(S) = 0 and P2(S) = 4.Mo e-
o e , since |2Ke
S|= ∗|2KS|+π∗(N1+N2+N3+Z),by Theo em 14.1, we ha e
∗|2KS|=π∗|−Ke
Y|.Since |−Ke
Y|is base-poin - ee, |2KS|is base-poin - ee.
The minimal esolu ion µ:e
Y→Ycon ac s exac ly he (−2)-cu es
N1, N2, N3, Z. F om he cons uc ion, we ha e a ini e bidouble co e p:S→
Ysuch ha he ollowing diag am commu es:
e
S //
π

S
p

e
Yµ//Y
and 2KS≡p∗(−KY).Since −KYis ample and pis ini e, KSis ample and
hus Sis minimal. I also shows ha he bicanonical mo phism o Sis he
composi ion o pand he an icanonical embedding o Yin o P3.
We deno e by E0 he co esponding subse o smoo h su aces cons uc ed
abo e in he moduli space Mcan
1,3.
P oposi ion 14.3. E0is con ained in E.
P oo . Gi en a su ace Sin E0,conside he in e media e double co e
ˆπ:ˆ
S→e
Y
associa ed o he da a 2L3≡∆1+ ∆2.
S anda d o mulae o double co e s ( o example, see [BHPV, Page 236-
237]) show ha
Kˆ
S≡ˆπ∗(KY+L3)≡ˆπ∗(2L−E1−E2−E3−E0
1−E0
2−E0
3),
2Kˆ
S≡ˆπ∗(4L−2E1−2E2−2E3−2E0
1−2E0
2−2E0
3)≡2ˆ
E1+2 ˆ
E2+2 ˆ
E3+2 ˆ
E4,
K2
ˆ
S=−4, pg(ˆ
S) = 0,
whe e ˆ
Ek:= ˆπ−1Nkand ˆ
E4:= ˆπ−1Za e (−1)-cu es. Mo eo e , ˆ
Shas 7
nodes lying o e he nodes o he cu e C1+C2+ Γ1+ Γ2.
65
15. LOCAL DEFORMATIONS AND IRREDUCIBLE COMPONENT
Le ˆ
:ˆ
S→S0be he blow down o he ou (−1)-cu es. We ob ain a
nodal En iques su ace S0wi h 7 nodes. The ollowing diag am commu es:
e
S //
ˆπ

=
=
=
=
=
=
=
=
π

S

>
>
>
>
>
>
>
>
p
ˆ
S//









S0








e
Yµ//Y
Thus he bicanonical mo phism S→Y ,→P3o S ac o s h ough S0.
By he de ini ion o E(c . Theo em 13.2), Sbelongs o E.
15 Local De o ma ions and
I educible Componen
In his sec ion we will p o e he ollowing heo em.
Theo em 15.1. (1) Fo a gene al su ace Sin E0, h1(S, ΘS) = 6,
h2(S, ΘS) = 2 and he base o he Ku anishi amily o Sis smoo h.
(2) Eis an i educible componen o he moduli space Mcan
1,3.
The key poin is o p o e he ollowing p oposi ion.
P oposi ion 15.2. Fo a gene al su ace Sin E0, h2(S, ΘS)≤2.
P oo o Theo em 15.1 assuming P oposi ion 15.2.
Since −h1(S, ΘS) + h2(S, ΘS) = 2K2
S−10χ(S) = −4,by P oposi ion 15.2
h1(S, ΘS)≤6.Since Sis smoo h and KSis ample (c . Co olla y 14.2), he
minimal model and he canonical model o Scoincide. We ha e he ollowing
inequali ies,
6≥h1(S, ΘS)≥ he dimension o he base o he Ku anishi amily o S
= he dimension o Mcan
1,3a he poin [S]
≥ he dimension o E.
66

15. LOCAL DEFORMATIONS AND IRREDUCIBLE COMPONENT
Since he dimension o Eis 6 by Theo em 13.2, we see ha all he equali ies
hold. The second equali y shows ha he base o he Ku anishi amily o
Sis smoo h. Since locally he ge m o he complex space (Mcan
1,3,[S]) is
analy ically isomo phic o he quo ien o he base o he Ku anishi amily
by he ini e g oup Au (S),i ollows ha (Mcan
1,3,[S]) is i educible. Since E
is i educible by Mendes Lopes and Pa dini’s Theo em 13.2, he las equali y
shows ha Ecoincides wi h Mcan
1,3locally a [S].I ollows ha Eis an
i educible componen o Mcan
1,3.
By Theo em 10.4, o p o e P oposi ion 15.2, i su ices o show
h2(e
S, Θe
S)≤2.By Se e Duali y and Theo em 10.1,
H2(e
S, Θe
S) = H0(e
Y , Ω1
e
Y(log ∆1,log ∆2,log ∆3)⊗Ω2
e
Y)
⊕⊕3
i=1H0(e
Y , Ω1
e
Y(log ∆i)(Ke
Y+Li)).
Thus i su ices o calcula e he dimension o each summand.
Lemma 15.3. H0(e
Y , Ω1
e
Y(log ∆1,log ∆2,log ∆3)⊗Ω2
e
Y) = 0.
P oo . By Lemma 10.3, we ha e an exac sequence
0→Ω1
e
Y(Ke
Y)→Ω1
e
Y(log ∆1,log ∆2,log ∆3)(Ke
Y)→ ⊕3
i=1O∆i(Ke
Y)→0
(15.1)
No e ha H0(e
Y , Ω1
e
Y) = 0 and −Ke
Yis e ec i e, hus H0(e
Y , Ω1
e
Y(Ke
Y)) = 0.
To p o e he claimed equali y, i su ices o show he bounda y map
δ:H0(e
Y , ⊕3
i=1O∆i(Ke
Y)) →H1(e
Y , Ω1
e
Y(Ke
Y))
is injec i e.
By (14.1),
H0(e
Y , O∆1(Ke
Y)) ∼
=H0(e
Y , ON1⊕ ON2)∼
=C2,
H0(e
Y , O∆2(Ke
Y)) ∼
=H0(e
Y , ON3⊕ OZ)∼
=C2,
H0(e
Y , O∆3(Ke
Y)) = 0.
Since |−Ke
Y|is base-poin - ee, he e is a mo phism Oe
Y(Ke
Y)→ Oe
Y,which is
no iden ically ze o on any componen o ∆i’s. Now conside he commu a i e
67
15. LOCAL DEFORMATIONS AND IRREDUCIBLE COMPONENT
diag am coming om he mo phism Oe
Y(Ke
Y)→ Oe
Y,
0//Ω1
e
Y(Ke
Y)

//Ω1
e
Y(log ∆1,log ∆2,log ∆3)(Ke
Y)

//⊕3
i=1O∆i(Ke
Y)

//0
0//Ω1
e
Y
//Ω1
e
Y(log ∆1,log ∆2,log ∆3)//⊕3
i=1O∆i
//0.
I gi es a commu a i e diag am o cohomology g oups,
C4∼
=H0(e
Y,⊕3
i=1O∆i(Ke
Y))
ψ2

δ//
ψ
**
V
V
V
V
V
V
V
V
V
V
V
V
V
V
V
V
H1(e
Y,Ω1
e
Y(Ke
Y))

H0(e
Y , ⊕3
i=1O∆i)ψ1//H1(e
Y , Ω1
e
Y).
By Lemma 10.3, he image o he unc ion iden ically equal o 1 on Nk
(k= 1,2,3), espec i ely on Zmaps unde ψ1 o he i s Che n class o Nk,
espec i ely o Z. Because he Nk’s and Za e 4 disjoin (−2)-cu es, hei
Che n classes a e independen in H1(e
Y , Ω1
e
Y).
Thus he composi e map ψis injec i e. I ollows ha δis also injec i e
and H0(e
Y , Ω1
e
Y(log ∆1,log ∆2,log ∆3)⊗Ω2
e
Y) = 0.
To calcula e o he summands, we ix he coo dina es o Piand P0
i.Wi h-
ou loss o gene ali y, assume ha
P1= (1:−1:0),P2= (0:1:0),P3= (1:0:0),
P0
1= (0:0:1),P0
2= (1:0:1),P0
3= (0:1:1).(15.2)
See Figu e 9.
Lemma 15.4. H0(e
Y , Ω1
e
Y(log ∆3)(Ke
Y+L3)) = 0 o a gene al H∈ | − Ke
Y|.
P oo . Le M:= Ke
Y+L3= 2L−E1−E2−E3−E0
1−E0
2−E0
3(c . (14.2)).
Recall ha ∆3=H∈ | − Ke
Y|.Then Ke
Y.M = ∆3.M = 0.Fo a gene al
H, H is a smoo h ellip ic cu e and O∆3(M) is a 2- o sion elemen , hus
H0(O∆3(M)) = 0.
In ac , no e ha 2M≡N1+N2+N3+Z. Take he double co e ˜q:˜
Σ→e
Y
associa ed o he da a 2M≡N1+N2+N3+Z, and blow down he (−1)-
cu es ˜q−1Ni, i = 1,2,3 and ˜q−1Z, η :˜
Σ→Σ.We ha e a mo phism q: Σ →Y
68
15. LOCAL DEFORMATIONS AND IRREDUCIBLE COMPONENT
and he ollowing commu a i e diag am
˜
Ση//
˜q

Σ
q

e
Yµ//Y.
qonly ami ies o e he 4 nodes o Yand q∗(−KY)≡ −KΣ.Σ is a smoo h Del
Pezzo su ace o deg ee 6,i.e., K2
Σ= 6 and −KΣis e y ample. By Be ini’s
heo em, a gene al cu e Co |−KY|is smoo h and i educible and q−1Cis an
i educible smoo h cu e in | − KΣ|.Since | −Ke
Y|=µ∗| − KY|and a gene al
elemen H∈ | − Ke
Y|is disjoin om he (−2)-cu es, he commu a i e
diag am shows ha ˜q−1His an i educible smoo h cu e. Hence OH(M) is
a 2- o sion elemen .
Tenso he ollowing exac sequence wi h Oe
Y(M),
0→Ω1
e
Y→Ω1
e
Y(log ∆3)→ O∆3→0,
we see ha h0(e
Y , Ω1
e
Y(log ∆3)(M)) = h0(e
Y , Ω1
e
Y(M)).
Since σ∗Ω1
e
Y(M) is a subshea o Ω1
P2(2),one can iew H0(e
Y , Ω1
e
Y(M)) as
a subspace o H0(P2,Ω1
P2(2)).By [BC10-b, Lemma 5.2], any o m o
H0(P2,Ω1
P2(2)) can be w i en as
ω=A(x1dx2−x2dx1) + B(x2dx3−x3dx2) + C(x1dx3−x3dx1).
E alua ing a P2= (0 : 1 : 0),by Lemma 12.2 (1), we ge A=B= 0.Then
e alua e a P3= (1 : 0 : 0) and ge C= 0.
Thus we see ha H0(e
Y , Ω1
e
Y(log ∆3)(Ke
Y+L3)) = 0.
P oposi ion 15.5. h0(e
Y , Ω1
e
Y(log ∆1)(Ke
Y+L1)) ≤1and
h0(e
Y , Ω1
e
Y(log ∆2)(Ke
Y+L2)) ≤1.
Fi s we p o e he ollowing lemma.
Lemma 15.6.
h0(e
Y , Ω1
e
Y(log ∆1)(Ke
Y+L1)) =
h0(e
Y , Ω1
e
Y(log N1,log N2,log N3,log Z)(2L−E2−E0
2−E0
3)),
h0(e
Y , Ω1
e
Y(log ∆2)(Ke
Y+L2)) =
h0(e
Y , Ω1
e
Y(log N1,log N2,log N3,log Z)(2L−E1−E3−E0
1)).
69
15. LOCAL DEFORMATIONS AND IRREDUCIBLE COMPONENT
P oo . H0(e
Y , Ω1
e
Y(log ∆1)(Ke
Y+L1)) = H0(e
Y , Ω1
e
Y(log ∆1)(Γ1−E3)) by (14.2).
No e ha ∆1is he disjoin union o C1,Γ2, N1and N2.Since
(Ke
Y+ 2C1+ Γ1−E3).C1=−1<0,
(Ke
Y+ 2Γ2+ Γ1−E3+C1).Γ2=−2<0,
apply Lemma 10.2 o C1and hen o Γ2,
H0(e
Y , Ω1
e
Y(log ∆1)(Ke
Y+L1)) ∼
=
H0(e
Y , Ω1
e
Y(log N1,log N2)(Γ1−E3+C1+ Γ2)).
Since
Γ1−E3+C1+ Γ2≡4L−E1−2E2−2E3−E0
1−2E0
2−E0
3
≡N3+Z+ (2L−E2−E0
2−E0
3),
(Ke
Y+ 2N3+ (2L−E2−E0
2−E0
3) + Z).N3=−3<0,
(Ke
Y+ 2Z+ 2L−E2−E0
2−E0
3).Z =−3<0,
apply Lemma 10.2 o N3and hen o Z,
H0(e
Y , Ω1
e
Y(log N1,log N2)(4L−E1−2E2−2E3−E0
1−2E0
2−E0
3)) ∼
=
H0(e
Y , Ω1
e
Y(log N1,log N2,log N3,log Z)(2L−E2−E0
2−E0
3)).
Thus he i s equali y holds. A simila a gumen shows ha he second
equali y also holds.
P oo o P oposi ion 15.5. The e is an au omo phism
τ:P2→P2such ha τ(P1) = P0
2, τ(P2) = P1, τ(P0
1) = P2, τ(P0
2) = P0
1.
I ollows ha τ(P3) = P0
3, τ(P0
3) = P3.This au omo phism induces an au o-
mo phism o e
Yand shows ha
Ω1
e
Y(log N1,log N2,log N3,log Z)(2L−E2−E0
2−E0
3)∼
=
Ω1
e
Y(log N1,log N2,log N3,log Z)(2L−E1−E3−E0
1).
Toge he wi h Lemma 15.6, i su ices o show
H0(e
Y , Ω1
e
Y(log N1,log N2,log N3,log Z)(2L−E2−E0
2−E0
3)) ≤1.
70
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