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

Chen, Yifan

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

Chen, Yifan

Published: 2012-01-01
Repository: epub.uni-bayreuth.de
Language: de

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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 [MP01] M. Mendes Lopes and R. Pa dini, A connec ed componen o he moduli space o su aces wi h pg= 0, Topology 40 (5) (2001), 977-991. [MP04] ,A new amily o su aces wi h pg= 0 and K2= 3, Ann. Scien . Ec. No m. Sup. 4es´e ie, 37 (2004), 507-531. [Miy77] Y. Miyaoka, On nume ical Campedelli su aces, Complex analysis and alge- b aic geome y, Iwanami Sho en, Tokyo, 1977, 113-118. [Na94] D. Naie, Su aces d’En iques e une cons uc ion de su aces de ype g´en´e al a ec pg= 0,, Ma h. Z. 215 (2) (1994), 269-280. [NP11] J. Ne es and R. Pigna elli, Unp ojec ion and de o ma ions o e ia y Bu nia su aces. [PR04] S. Papadakis and M. Reid, Kus in-Mille unp ojec ion wi hou complexes, J. Algeb aic Geome y 13 (2004), 563-577. [Pa 91] R. Pa dini, Abelian co e s o algeb aic a ie ies, J. Reine Angew. Ma h. 417 (1991), 191-213. [Pe 77] C. A. M. Pe e s, On ce ain examples o su aces wi h pg= 0 due o Bu nia , Nagoya Ma h. J. 66 (1977), 109-119. [Reid79] M. Reid, Su aces wi h pg= 0, K2= 2, p ep in (1979). [Reid85] ,Young pe son’s guide o canonical singula i ies, Algeb aic geome y, Bowdoin, 1985 (B unswick, Maine, 1985), P oc. Sympos. Pu e Ma h., 46, Pa 1, Ame . Ma h. Soc., P o idence, RI, (1987), 345-414. [Sc64] L. Schl¨a li, On he dis ibu ion o su aces o he hi d o de in o species, 193- 247. [Se 06] E. Se nesi, De o ma ions o Algeb aic Schemes, Sp inge , G undleh en 334, 2006. [Weng95] L. Weng, A esul on bicanonical maps o su aces o gene al ype, Osaka J. Ma h., 32 (1995), no. 2, 467-473. [Sak10] Yoshiyuki Sakamaki, Au omo phism g oups on no mal singula cubic su aces wi h no pa ame e s, T ansac ions o he Ame ican Ma hema ical Socie y Vol- ume 362, Numbe 5, May 2010, 2641-2666. [Za 58] O. Za iski, On he pu i y o he b anch locus o algeb aic unc ions, P oc. Na . Acad. Sci. U. S. A. 44 (1958), 791-796. 77