Two Irreducible Components of the Moduli Space M can 1,3
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Two Irreducible Components of the Moduli Space Mcan 1,3 DISSERTATION zur Erlangung des DOKTORGRADES(DR. RER. NAT.) der FAKULT¨ AT F¨ UR MATHEMATIK, PHYSIK UND INFORMATIK der UNIVERSIT¨ AT BAYREUTH vorgelegt von YIFAN CHEN aus P. R. China BAYREUTH Tag der Einreichung: 13. Januar 2012 Tag des Kolloquiums: 23. Februar 2012
Angefertigt mit der Genehmigung der Fakult¨at f¨ur Mathematik, Physik und Informatik der Universit¨at Bayreuth. 1. Gutachter: Prof. Dott. Fabrizio Catanese 2. Gutachter: Prof. Dr. Miles Reid, University of Warwick 3. Gutachter: Prof. Dr. Jin-Xing Cai, Peking University
Erkl¨arung Ich versichere eidesstattlich, dass ich die Arbeit selbst¨andig verfasst und keine anderen als die von mir angegebenen Quellen und Hilfsmittel benutzt habe. Ich best¨atige, dass ich keine fr¨uhere Promotionsversuche gemacht habe. Unterschrift des Autors
Contents Zusammenfassung i Abstract iii Acknowledgements v Introduction vi Notation and conventions xi Figures xi I Preliminaries 1 1 Bidouble Covers of Surfaces 1 2 Involutions on Rational Double Points 4 3 Normal Cubic Surfaces 9 3.1 3A1-type Cubic Surfaces . . . . . . . . . . . . . . . . . . . . . 12 3.2 D4(1)-type and D4(2)-type Cubic Surfaces . . . . . . . . . . . 13 3.3 4A1-type Cubic Surface . . . . . . . . . . . . . . . . . . . . . . 15 II The Irreducible Component Containing the Extended Burniat Surfaces 16 4 Burniat Surfaces and Extended Burniat Surfaces 17 5 One Parameter Limits of the Extended Burniat Surfaces 20 6 Exclusion of Certain Cubic Surfaces 27 7D4-generalized Burniat Surfaces 30 7.1 Configuration of Branch Divisors . . . . . . . . . . . . . . . . 31 7.2 D4-generalized Burniat Surfaces . . . . . . . . . . . . . . . . . 34
84A1-generalized Burniat Surfaces 35 8.1 Configuration of Branch Divisors . . . . . . . . . . . . . . . . 35 8.2 4A1-generalized Burniat Surfaces . . . . . . . . . . . . . . . . 38 9 Irreducible Component 40 9.1 Configuration of Branch Divisors on 3A1-type Cubic Surfaces . 40 9.2 Closure of the Open Subset N EB3............... 42 III Deformations of Generalized Burniat Surfaces 43 10 Key Tools to Calculate the Cohomology Groups of the Tangent Sheaves 44 11 Deformations of the D4-generalized Burniat Surfaces 47 12 Deformations of the 4A1-generalized Burniat Surfaces 56 IV The Irreducible Component containing the Keum-Naie-Mendes Lopes-Pardini Surfaces 61 13 Keum-Naie-Mendes Lopes-Pardini Surfaces 62 14 A Subfamily of KNMP Surfaces 63 15 Local Deformations and Irreducible Component 66 References 75
ZUSAMMENFASSUNG Zusammenfassung Das Ziel dieser Dissertation ist es zwei Familien von Fl¨achen von allgemeinem Typ mit pg= 0 und K2= 3 zu studieren. Genauer gesagt handelt es sich um die erweiterten Burniat Fl¨achen mit K2= 3 und die Keum-Naie-Mendes Lopes-Pardini Fl¨achen. Wir konzentrieren uns auf die lokalen Deformationen dieser Fl¨achen und auf die Modulr¨aume, die diesen Fl¨achen entsprechen. Die erweiterten Burniat Fl¨achen mit K2= 3 wurden zuerst von Bauer und Catanese in [BC10-b] konstruiert, wo sie auch Burniat Fl¨achen mit K2= 3 studierten (vgl. auch [Bu66] und [Pet77]). Sie haben gezeigt, dass der entsprechende Modulraum in dem Modulraum von Fl¨achen von allgemeinem Typ irreduzibel, offen und von der Dimension 4 ist, und, dass der Abschluss dieses Modulraums eine irreduzible Komponente des Modulrams von Fl¨achen von allgemeinem Typ ist. Das erste Ziel dieser Arbeit ist, alle Degenerationen der erweiterten Burniat Fl¨achen mit K2= 3 zu beschreiben. Dazu zeigen wir zuerst, dass die einparametrige Degeneration der kanonischen Modelle dieser Fl¨achen eine endliche, flache (Z/2Z)2-¨ Uberlagerung von normalen singul¨aren kubischen Fl¨achen ist. Danach zeigen wir mittels der Klassifikationstheorie der kubischen Fl¨achen und durch die Untersuchung des Verzweigungsorts dieser ¨ Uberlagerungen, dass genau zwei Familien von Degenerationen existieren, die in [BC10-b] beschrieben wurden. Somit beweisen wir, dass die Vereinigung der R¨aume, beschrieben in [BC10-b], tats¨achlich die ganze irreduzible Komponente des Modulrams ist. Dar¨uber hinaus studieren wir die lokalen Deformationen der Degenerationen der erweiterten Burniat Fl¨achen mit K2= 3. Unter Zuhilfenahme des Struktursatzes der (Z/2Z)2-¨ Uberlagerungen sind wir in der Lage, die Dimensionen der Eigenr¨aume der Kohomologiegruppen der Tangentialgarbe zu bestimmen. Wir zeigen, dass der Basisraum der Kuranishi Familie einer Fl¨ache in einer der zwei Familien der Degenerationen glatt ist. Im zweiten Teil der Dissertation untersuchen wir Keum-Naie Fl¨achen mit K2= 3 ([Ke88] und [Na94]) und deren Deformationen, die von Mendes Lopes und Pardini konstruiert wurden. Wir nennen wir diese Fl¨achen Keum-Naiei
ZUSAMMENFASSUNG Mendes Lopes-Pardini Fl¨achen. In [MP04] wurde gezeigt, dass der Abschluss der entsprechenden Teilmenge dieser Fl¨achen im Modulraum irreduzibel, uniruled und der Dimension 6 ist. Wir konstruieren eine Unterfamilie dieser Fl¨achen. Die Fl¨achen in unserer Familie sind endliche flache (Z/2Z)2-¨ Uberlagerungen einer kubischen Fl¨ache mit vier Knoten. Sie haben einen amplen kanonischen Divisor. Die bikanonische Abbildung dieser Fl¨ache ist die Komposition der ¨ Uberlagerung mit der antikanonischen Einbettung der kubischen Fl¨ache. Daraus folgt, dass die bikanonische Abbildung dieser Fl¨ache eine Komposition mit einer Involution aus der Galoisgruppe der ¨ Uberlagerung ist, so dass die Quotientenfl¨ache dieser Involution eine Enriques Fl¨ache mit A1-Singularit¨aten ist. Diese Eigenschaft charakterisiert alle Mendes Lopes-Pardini Fl¨achen [MP04]. Unter Zuhilfenahme des Struktursatzes der (Z/2Z)2-¨ Uberlagerungen sind wir in der Lage eine obere Schranke f¨ur die Dimension der Kohomologiegruppen der Tangentialgarbe dieser Fl¨achen zu geben. Durch Kombination unserer Ergebnisse und den Ergebnissen aus [MP04] zeigen wir, dass f¨ur eine generische Fl¨ache Sin unserer Unterfamilie h1(S, ΘS) = 6, h2(S, ΘS) = 2 gilt, und der Basisraum der Kuranishi Familie glatt ist. Somit zeigen wir, dass der Abschluss der Teilmenge des Modulraums, die den Keum-Naie-Mendes Lopes-Pardini Fl¨achen entspricht, eine irreduzible Komponente ist. ii
ABSTRACT Abstract This thesis is devoted to the study of two families of surfaces of general type with pg= 0 and K2= 3: extended Burniat surfaces with K2= 3 and KeumNaie-Mendes Lopes-Pardini surfaces. We focus on the local deformations of these surfaces and the corresponding subsets in the Gieseker moduli space. Extended Burniat surfaces with K2= 3 were constructed by Bauer and Catanese [BC10-b] in the course of studying Burniat surfaces with K2= 3 (cf. [Bu66] and [Pet77]). They showed that the corresponding subset in the moduli space is an irreducible open subset of dimension 4,and its closure is an irreducible component of the moduli space. The first goal of this thesis is to describe all the degenerations of the extended Burniat surfaces with K2= 3.For this, we first show that the one parameter limits of the canonical models of these surfaces are finite flat (Z/2Z)2-covers of normal singular cubic surfaces. Then by applying the classification theory of cubic surfaces and by investigating the branch loci of such covers, we show that there are exactly two families of degenerations, which had been described in [BC10-b]. Thus we prove that the union of the loci described in [BC10-b] is indeed the full irreducible component in the moduli space. We also study the local deformations of the degenerations of extended Burniat surfaces with K2= 3.Using the structure theorem for (Z/2Z)2covers, we are able to calculate the dimensions of the eigenspaces of the cohomology groups of the tangent sheaves. We show that the base of the Kuranishi family of a surface in one of the two families of degenerations is smooth. Another topic of this thesis is to study the Keum-Naie surfaces with K2= 3 (cf. [Ke88] and [Na94]) and their deformations constructed by Mendes Lopes and Pardini [MP04]. We call all these surfaces Keum-Naie-Mendes Lopes-Pardini surfaces. It is showed in [MP04] that the closure of the corresponding subset of such surfaces in the moduli space is irreducible, uniruled and of dimension 6. We construct a subfamily of such surfaces. The surfaces in our family iii
ABSTRACT are finite flat (Z/2Z)2-covers of a 4-nodal cubic surface. They have ample canonical divisors. Moreover, the bicanonical maps of these surfaces are the composition of the covering morphisms and the anticanonical embedding of the 4-nodal cubic surface. It follows that the bicanonical map of such a surface is composed with an involution in the Galois group (∼ =(Z/2Z)2) of the cover, such that the quotient of the surface by the involution is a nodal Enriques surface. This is a property characterizing all the Mendes LopesPardini surfaces [MP04]. Again using the structure theorem for (Z/2Z)2-covers, we give upper bounds for the dimensions of the cohomology groups of the tangent sheaves of these surfaces. Combining the results in [MP04], we show that for a general surface Sin our subfamily, h1(S, ΘS) = 6, h2(S, ΘS) = 2 and the base of the Kuranishi family of Sis smooth. We thus show that the closure of the corresponding subset of the Keum-Naie-Mendes Lopes-Pardini surfaces is an irreducible component of the moduli space. iv
FIGURES Notation and conventions •A surface will mean a projective, irreducible and reduced surface defined over the complex number field Cunless otherwise specified. •A canonical surface will mean the canonical model of a minimal smooth surface of general type. •We will only treat (extended) Burniat surfaces with K2= 3,so sometimes we call them briefly (extended) Burniat surfaces. The same convention will be used for Keum-Naie surfaces. •For a smooth surface Sand a sheaf Fon S, we will denote by hk(S, F) the dimension of the cohomology group Hk(S, F). •For a surface S, we will denote by ΘSthe sheaf associated to the tangent bundle, Ωp Sthe sheaf of holomorphic p-forms on S, pg(S) := h0(S, Ω2 S) the geometric genus, q(S) := h0(S, Ω1 S) the irregularity of S, χ(S) := 1 + pg(S)−q(S) the holomorphic Euler-Poincar´e characteristic and by K2 Sthe self-intersection number of the canonical divisor. •Denote by ≡the linear equivalence for divisors and by num ≡the numerical equivalence for divisors. •An An-singularity of a surface is a singularity analytically isomorphic to x2+y2+zn+1 = 0.An A1-singularity is also called a node. •A−m-curve on a smooth surface is an irreducible smooth rational curve with self-intersection number −m, where mis a non-negative integer. •The indices i∈ {1,2,3}should be understood as residue classes modulo 3 through the whole thesis. •Denote by G={0,g1, g2, g3}a group, which is isomorphic to (Z/2Z)2. And let G∗={1,χ1, χ2, χ3}be the group of characters of G, where χi(gi) = 1 and χi(gi+1) = χi(gi+2) = −1. Figures xi
FIGURES P1P2 P3 P0 1 P0 2 P0 3 Figure 1: A plane model for a general 3A1-type cubic surface. P1 P2 P3 P0 1 P0 2 P0 3 Figure 2: A plane model for a special 3A1-type cubic surface. P1P2 P3 Q1 Q2 Q3 P0 1P0 2 P0 3 Figure 3: Another plane model for a general 3A1-type cubic surface. xii
FIGURES P1P2P3 P0 1P0 2P0 3 Figure 4: A plane model for the D4(1)-type cubic surface. P1P2P3 Q1 Q2 Q3 P0 1P0 2P0 3 Figure 5: A plane model for the D4(2)-type cubic surface. P1 P2 P3 P0 2P0 3 P0 1 Figure 6: A plane model for the 4A1-type cubic surface. 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 Figure 7: Coordinates for the proof of Proposition 11.4 and the calculation of 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) Figure 8: Coordinates for the proof of Proposition 12.3 and the calculation of 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) Figure 9: Coordinates for the proof of Proposition 15.2 and the calculation of h0(e Y , Ω1 e Y(log N2,log N3,log Z)(2L−E2−E0 2−E0 3)). xiv
1. BIDOUBLE COVERS OF SURFACES Part I Preliminaries 1 Bidouble Covers of Surfaces This section gives a brief introduction to the theory of bidouble covers. For simplicity, we restrict ourselves to the case of algebraic surfaces. We quote the results in [Cat84], [Par91] and [Cat99] without proof. Definition 1.1 ([Cat84], [Par91, Definition 1.1]).Let e Ybe a normal surface. A bidouble cover of e Yis a finite morphism π:e S→e Y , together with a faithful G-action on e Ssuch that πexhibits e Yas the quotient of e Sby G. Definition 1.2. (1) Assume that π:e S→e Yis a bidouble cover between normal surfaces. We define the ramification locus Rof π, to be the locus of points of e Swhich have nontrivial stabilizers. The branch locus Bof πis the image of Ron e Y . (2) For i= 1,2,3,define a branch divisor Bicorresponding to gi,to be the image of all the 1-dimensional irreducible components of R, whose inertia groups are the subgroup {0, gi}. Here for a 1-dimensional irreducible component Dof R, the inertia group Hof Dis defined as follows: H={g∈G|gx =xfor any x∈D} (cf. [Par91, Definition 1.2]). Assume that e Yis smooth and e Sis normal. Then by [Ber, Section 3], πis flat, and the ramification locus of πis of pure codimension 1 (cf. [Zar58]). It follows that the branch locus is also of pure codimension 1.The next theorem describes the structure of a bidouble cover under this assumption. Theorem 1.1 ([Cat84, Section 1], [Par91, Theorem 2.1], [Cat99, Theorem 2]).Let π:e S→e Ybe a bidouble cover of surfaces. Assume that e Y is smooth. (1) Assume that e Sis normal. Then π∗(Oe S)∼ =Oe Y⊕ Oe Y(−L1)⊕ Oe Y(−L2)⊕ Oe Y(−L3), 1
1. BIDOUBLE COVERS OF SURFACES where Li’s are divisors on e Y , and Gacts on Oe Y(−Li)via the character χi.Moreover, there are three effective divisors ∆1,∆2,∆3on e Ysuch that 2Li≡∆i+1 + ∆i+2,(1.1) Li+ ∆i≡ Li+1 +Li+2,(1.2) for i= 1,2,3,and ∆iis the branch divisor corresponding to gi. (2) Conversely, given three divisors L1,L2,L3and three effective divisors ∆1,∆2,∆3on e Y , satisfying (1.1) and (1.2), we can associate a bidouble cover π:e S→e Yas follows (cf. [BC11, Section 2]): for each i= 1,2,3,locally let ∆i= div(δi)and let uibe a fibre coordinate of the geometric line bundle Li,whose sheaf of holomorphic sections is Oe Y(Li).Then e S⊂L1⊕L2⊕L3is given by the equations: u1u2=δ3u3, u2 3=δ1δ2, u2u3=δ1u1, u2 1=δ2δ3, u3u1=δ2u2, u2 2=δ3δ1. (1.3) According to this theorem, to construct a bidouble cover over a smooth surface e Y , it suffices to find divisors L1,L2,L3and effective divisors ∆1,∆2, ∆3satisfying equations (1.1) and (1.2). Remark 1.1. (1) If we sum up the left hand side and the right hand side of (1.2) for all i= 1,2,3,we obtain L1+L2+L3≡∆1+ ∆2+ ∆3. (2) In the following sections, e Ywill be a rational surface, and thus P ic(e Y) has no torsion. Hence the equations (1.1) and (1.2) are equivalent. We usually just refer to equations (1.1), or just refer to ∆1,∆2,∆3,such that the sum of any two is even in P ic(e Y),without mentioning the Li’s. Concerning the construction of a bidouble cover in Theorem 1.1 (2), the following proposition gives a criterion for the normality (respectively, smoothness) for e S. Proposition 1.2 ([Par91, Proposition 3.1], [Cat99, Theorem 2]).Let e Ybe a smooth surface, and let π:e S→e Ybe the bidouble cover corresponding to the data L1,L2,L3and ∆1,∆2,∆3,satisfying (1.1) and (1.2). Then 2
2. INVOLUTIONS ON RATIONAL DOUBLE POINTS (1) e Sis normal if and only if the total branch divisor ∆ = ∆1+ ∆2+ ∆3 is reduced. (2) e Sis smooth if and only if each ∆iis smooth for i= 1,2,3,and the total branch divisor ∆has only normal crossing singularities. In Proposition 1.2 (2), if we do not require the condition “∆ has only normal crossing singularities”, then e Smight have singularities. Example 1.1. Assume that ∆iintersects ∆i+1 transversely at a common point Pfor i= 1,2,3.Then the local equations (1.3) of e Sshows that π−1(P) consists of one point Q, which is a 1 4(1,1)-singularity on e S. See [BC11, Section 2] for details. The following theorem shows how to calculate the invariants of e Sfrom the covering data ∆i’s and Li’s. Theorem 1.3 ([Cat84, Lemma 2.15], [Cat99, Section 2]).Let e Ybe a smooth surface, and let π:e S→e Ybe the bidouble cover associated to the data L1,L2,L3and ∆1,∆2,∆3,satisfying (1.1) and (1.2). Assume that ∆ = ∆1+ ∆2+ ∆3is reduced and has only normal crossing singularities. 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). Corollary 1.4. In the situation of Theorem 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 Involutions on Rational Double Points The previous section considered a bidouble cover π:e S→e Ywhen e Yis a smooth surface. In our applications, both e Sand e Ymight have singularities. We would like to know when the quotient of a rational double point by aZ/2Z-action or a (Z/2Z)2-action remains a rational double point. This problem has been studied and solved in [Cat87]. We quote the main results and follow the notation in [Cat87] for convenience. Let us first give a list of rational double points. Table 1: Singularities (X0, x0) Equation 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,or uv +yn+1 = 0 Definition 2.1 ([Cat87, Definition 1.3]).The involution τof a rational double point (X0, x0) such that τ∗(z) = −z, τ∗(x) = x, τ∗(y) = yis called the trivial involution. Any involution σconjugate to τis also said to be trivial, and has the property that X0/σ ∼ =(C2,0). The next theorem classifies all the involutions on rational double points. Theorem 2.1 ([Cat87, Theorem 2.1]).The only involution acting on E7, E8is the trivial one. The other rational double points admit the following nontrivial conjugacy classes of involutions: (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, v, y)7→ (−u, v, −y) (A2n), (d) (x, y, z)7→ (−x, y, −z) (An), (e) (u, v, y)7→ (−u, −v, −y) (A2k+1). The following theorems classify the quotients of rational double points by involutions. We also calculate the ramification loci of the quotient maps. 4
2. INVOLUTIONS ON RATIONAL DOUBLE POINTS Theorem 2.2 ([Cat87, Theorem 2.2]).The quotient of a rational double point by a nontrivial involution not of type (c),(e), is again a rational double point according to Table 2. Table 2: Singularities (X0, x0)Involutions Quotients (Y0, y0)Ramification 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) Theorem 2.3 ([Cat87, Theorem 2.4]).The quotient Bkof the singularity A2kby an involution of type (c) is defined in C4,with coordinates (u, w, t, η) by the ideal Ik= (ηw −t2, uw +tηk, ut +ηk+1). The (reduced) exceptional divisor Dof its minimal resolution Thas normal crossings, consists of ksmooth rational curves, and its Dynkin diagram is ◦◦ · · · ◦ ◦ ◦ −3 Theorem 2.4 ([Cat87, Theorem 2.5]).Let Zbe the affine cone over the Veronese surface, i.e., the set of symmetric matrices x1x2x6 x2x3x4 x6x4x5 of rank ≤1. Then the quotient Yk+1 of the singularity A2k+1 by the involution (e) is the intersection of Zwith the hypersurface φ=x6−xk+1 3= 0.In particular, 5
2. INVOLUTIONS ON RATIONAL DOUBLE POINTS Yk+1 can also be defined as the singularity in C5defined by the 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 exceptional divisor Din the minimal resolution Tof Yk+1 has normal crossings, consists of (k+ 1) smooth rational curves, and the associated Dynkin diagram is ◦for k= 0 −4 ◦◦ · · · ◦ ◦ for k≥1 −3−3 Remark 2.1. The Y1-singularity (respectively, B1-singularity) is the 1 4(1,1)- singularity (respectively, the 1 3(1,1)-singularity), i.e., the cone over the rational normal curve of degree 4 in P4(respectively, of degree 3 in P3). Consider the involution of type (e) on an A1-singularity: σ: (X0, x0) : uv +y2= 0 →(X0, x0) : uv +y2= 0, (u, v, y)7→ (−u, −v, −y). Then by Theorem 2.4, the quotient Y0:= X0/σ has a Y1-singularity y0.Let ρ:X0→X0be the minimal resolution of x0and denote by Nthe (−2)-curve. Since Ncan be viewed as the projectivization of the tangent cone of X0to x0, σcan be lifted to X0and it has Nas fixed locus. We see that the image of N on the quotient X0/σ is a (−4)-curve. Hence X0/σ is the minimal resolution of (Y0, y0). Theorem 2.5 ([Cat87, Theorem 2.7]).Let (X0, x0)be a rational double point and let Hbe a subgroup of Aut(X0, x0),which is isomorphic to (Z/2Z)2.Then His conjugate to a subgroup listed in Table 3. Remark 2.2 ([Cat87, Remark 2.8]).From Theorem 2.5 Table 3, we conclude that the quotient of a rational double point (X0, x0) by a faithful (Z/2Z)2action is again a rational double point or a smooth point. This statement also holds for the case (X0, x0)∼ =(C2,0).This remark will be very important in the proof of Theorem 5.2, Section 5. 6
3. NORMAL CUBIC SURFACES Proof. C2:y0y2−y2 1= 0 and C3:y0y2(y0−(a+ 1)y1+ay2) = 0 intersect at six points Q0= (1 : 1 : 1), Q1= (1 : 0 : 0), Q2= (0 : 0 : 1), Q3= (a2:a: 1), and the infinitely near points Q0 k(k= 1,2) corresponding to the tangent line of C2to Qk(k= 1,2): Q1Q0 1:y2= 0 and Q2Q0 2:y0= 0. e Ycan be obtained by blowing up these six points. Apply the quadratic transformation centered at Q0, Q1, Q2,namely, first blow up σ1:Y0→P2at Q0, Q1, Q2,then blow down the strict transforms of Q0Q1, Q0Q2and Q1Q2to three points P2, P1and P0 3respectively. Denote the images of Q0 1, Q0 2, Q3by P0 2, P0 1, P3respectively. Then P1,...,P0 3satisfy the configuration above. Remark 3.2. From the equation of a 3A1-type cubic surface, we see that it has one parameter a. If a=−1,there are three lines x0=x3= 0, x2=x3= 0, x0−x2=x3= 0 containing a smooth point (0,1,0,0) of Y. Correspondingly, three lines PiP0 i’s pass through a common point in the configuration of P1,...,P0 3,and three (−1)-curves Γ1,Γ2,Γ3pass through a common point of e Y . See Figure 2. 3.2 D4(1)-type and D4(2)-type Cubic Surfaces Resolution of the D4(1)-type and the D4(2)-type cubic surfaces. Assume that Yis a D4(1)-type or a D4(2)-type cubic surface. Then e Y can be obtained as the blowup σ:e Y→P2of six points with the following configuration (see Figure 4 and Figure 5): P1, P2, P3are three distinct collinear points on P2,and P0 iis an infinitely near point lying over Pifor all i= 1,2,3. If Yis of D4(1)-type, we require the three lines PiP0 i’s to pass through a common point. If Yis of D4(2)-type, we require the three lines PiP0 i’s to form a triangle, with vertices Q1, Q2, Q3,where Qiis the intersection point of the lines Pi+1P0 i+1 and Pi+2P0 i+2. Rational curves on e Y.In both cases, e Yhas four (−2)-curves, Ni=Ei−E0 i, Z =L−E1−E2−E3,with Ni.Z = 1 and Ni.Ni+1 = 0, 13
3. NORMAL CUBIC SURFACES and six (−1)-curves E0 i,Γi:= L−Ei−E0 i,for i= 1,2,3. For each i= 1,2,3,e Yhas a pencil of rational curves Ciin the linear system |2L−Ei+1 −Ei+2 −E0 i+1 −E0 i+2|, so that Ci+ Γi≡ −Ke Y.The only singular element in the pencil is: Γi+1 + Γi+2. Proof. (1) Assume that Yis of D4(1)-type. C2:y2 0= 0 and C3:y3 1+y3 2= 0 intersect at six points: P1= (0 : 1 : −1), P2= (0 : 1 : −ζ), P3= (0 : 1 : −ζ2), where ζis a primitive cubic root of 1,and the infinitely near points P0 i’s corresponding to the lines P1P0 1:y1+y2= 0, P2P0 2:y1+ζ2y2= 0, P3P0 3:y1+ζy2= 0. Note that these three lines intersect at a common point (1 : 0 : 0). (2) Assume that Yis of D4(2)-type. C2:y2 0= 0 and C3:y3 1+y3 2+y0y1y2= 0 intersect at six points: P1= (0 : 1 : −1), P2= (0 : 1 : −ζ), P3= (0 : 1 : −ζ2), and the infinitely near points P0 i’s corresponding to the tangent lines of C3to Pi’s: P1P0 1:−y0+ 3y1+ 3y2= 0, P2P0 2:−ζy0+ 3y1+ 3ζ2y2= 0, P3P0 3:−ζ2y0+ 3y1+ 3ζy2= 0. Note that these three lines form a triangle with vertices Q1= (−3 : 1 : 1), Q2= (−3 : ζ:ζ2), Q3= (−3 : ζ2:ζ). Then the conclusion follows from Theorem 3.1 (3). 14
3. NORMAL CUBIC SURFACES 3.3 4A1-type Cubic Surface Resolution of the 4A1-type cubic surface. Assume that Yis a 4A1type cubic surface. Then e Ycan be obtained as the blowup σ:e Y→P2of six points with the following configuration (see Figure 6): P1, P2, P3are collinear, and Pi, P0 i+1, P 0 i+2 are collinear for all i= 1,2,3, i.e., P1,...,P0 3are vertices of a complete quadrilateral. Rational curves on e Y.e Yhas four disjoint (−2)-curves, Ni=L−Ei−E0 i+1 −E0 i+2, Z =L−E1−E2−E3, and nine (−1)-curves, Ei, E0 i,Γi:= L−Ei−E0 i,for i= 1,2,3. For each i= 1,2,3,e Yhas a pencil of rational curves Ciin the linear system |2L−Ei+1 −Ei+2 −E0 i+1 −E0 i+2|, so that Ci+ Γi≡ −Ke Y.The singular elements in the pencil are: Γi+1 + Γi+2, Ni+1 +Ni+2 + 2E0 i, Z +Ni+ 2Ei. Proof. C2:y0y2−y2 1= 0 and C3: (y0−y1)(y1−y2)y1= 0 intersect at six points Q1= (1 : 0 : 0), Q2= (0 : 0 : 1), Q3= (1 : 1 : 1), and infinitely near points Q0 icorresponding to the tangent line of C2to Qi: Q1Q0 1:y2= 0, Q2Q0 2:y0= 0, Q3Q0 3:y0−2y1+y2= 0. e Ycan be obtained by blowing up these six points. Apply the quadratic transformation centered at Q1, Q2, Q3,namely, first blow up σ1:Y0→P2at Q1, Q2, Q3,then blow down the strict transforms of Q1Q2, Q2Q3and Q3Q1to three points P0 3, P0 1and P0 2respectively. Denote the images of Q0 1, Q0 2, Q0 3by P1, P2, P3respectively. Then P1,...,P0 3satisfy the configuration above. 15
3. NORMAL CUBIC SURFACES The geometry of the 4A1-type cubic surface. We explain more about the geometry of the 4A1-type cubic surface Y. See the following figure. Yhas 4 nodes Q0, Q1, Q2, Q3,which do not lie in a plane. By B´ezout’s theorem, any line connecting two nodes is contained in Y. We can view Q0, Q1, Q2, Q3as the vertices of a tetrahedron. The edges of the tetrahedron correspond to six lines of Y. The (−1)-curves Eiand E0 ion e Ycorrespond to a pair of opposite edges of the tetrahedron, for i= 1,2,3. There are three more lines l1, l2, l3of Ywhich do not pass any nodes. They lie in a plane and form a triangle. Each one of them intersects exactly one of the three pairs of opposite edges. The three (−1)-curves Γ1,Γ2,Γ3on e Ycorrespond to these three lines. From this we see that the pencil of curves Cion e Ycorrespond to the residual conics cut by planes containing one of the li’s. Q0 Q1 Q2 Q3 l1 l3 l2 Figure 10: Singularities and lines of the 4A1-type cubic surface. 16
4. BURNIAT SURFACES AND EXTENDED BURNIAT SURFACES Part II The Irreducible Component Containing the Extended Burniat Surfaces 4 Burniat Surfaces and Extended Burniat Surfaces This section gives an introduction to the construction of the (extended) Burniat surfaces with K2= 3 and the main results on their moduli spaces obtained in [BC10-b]. Assume that Yis a 3A1-type cubic surface and e Yis its minimal resolution. Recall the notation introduced in Subsection 3.1. Assume that the lines PiP0 i’s do not pass through a common point. See Figure 1. Definition 4.1 ([Pet77], [BC10-b, Definition 1.1 and Definition 1.3]). (1) Define strictly extended Burniat divisors on e Yas follows: ∆1= Γ1+N2+C3,∆2= Γ2+N3+C1,∆3= Γ3+N1+C2,(4.1) where all Ci’s are irreducible smooth curves. (2) If one or two of the three Ci’s become reducible in the way Ci=Ni+Ei+|L−Ei+1 −Ei+2|, then we define three new divisors by subtracting from ∆i+1 the divisor Ni,and subtracting from ∆i−1the divisor Ni,and adding it to ∆i. These new divisors and the strictly extended Burniat divisors are all called extended Burniat divisors. 17
4. BURNIAT SURFACES AND EXTENDED BURNIAT SURFACES (3) If all three Ci’s become reducible in the way above, then we get three new divisors, called nodal Burniat divisors: D1=|L−E1−E2|+N1+ Γ1+E3, D2=|L−E2−E3|+N2+ Γ2+E1, D3=|L−E3−E1|+N3+ Γ3+E2. (4.2) Definition 4.2 ([BC10-b, Definition 1.4]).A (strictly) extended Burniat surface with K2= 3 is the minimal model Sof a bidouble cover π:e S→e Y associated to a (strictly) extended Burniat divisor. A nodal Burniat surface with K2= 3 is the minimal model Sof a bidouble cover π:e S→e Yassociated to a nodal Burniat divisor. Remark 4.1. (1) By Proposition 1.2, e Sin the definition is a smooth surface. However, it is not necessarily minimal. Whenever Niis a connected component in ∆, π−1Niis a disjoint union of two (−1)-curves. (2) In particular, for a strictly extended Burniat divisor ∆,all Ni’s are connected components in ∆.This implies that KSis ample for a strictly extended Burniat surface S. (3) Note that in Definition 4.1 (2), the procedure applied to the branch divisors is actually related to the procedure of normalization in the theory of bidouble covers (cf. [Cat99, Section 2, Remark 3]). Theorem 4.1. Let Sbe the minimal model of e Sin Definition 4.2. Then S is a surface of general type with K2 S= 3, pg(S) = q(S) = 0. Moreover, πtop 1(S)∼ =H8×Z/2Z,where H8is the quaternion group of order 8. For the first statement see [BC10-b], or apply Corollary 1.4. For the second statement see [BC11, Theorem 3.2]. See also [In94]. Corollary 4.2 ([BC10-b, Remark 1.5]).If Xis the canonical model of an extended Burniat surface or a nodal Burniat surface Swith K2 S= 3,then the bicanonical map of Xrealizes Xas a finite bidouble cover of a 3A1-type cubic surface Y. 18
5. ONE PARAMETER LIMITS In [BC10-b], Bauer and Catanese proved, among other things, the following theorem about the subset in the moduli space corresponding to extended Burniat surfaces and nodal Burniat surfaces with K2= 3. Theorem 4.3 ([BC10-b, Proposition 5.7, Theorem 0.1 and Theorem 0.2]). (1) The subset N EB3of the moduli space of canonical surfaces of general type Mcan 1,3corresponding to extended Burniat surfaces and nodal Burniat surfaces with K2= 3 is an irreducible open set, normal, unirational of dimension 4. (2) Let Sbe an extended Burniat surface or a nodal Burniat surface with K2 S= 3.Then h1(S, ΘS) = 4, h2(S, ΘS) = 0 and the base of the Kuranishi family of such a minimal model Sis smooth. (3) If Xis the canonical model of an extended Burniat surface or a nodal Burniat surface Swith K2 S= 3,then Def(X, (Z/2Z)2) = Def(X). Remark 4.2. (1) Here we give a geometric explanation of the dimension of NEB3: a 3A1-type cubic surface has one parameter (cf. Section 3), and each Cimoves in a pencil of curves. This gives the 4 dimensions. (2) Theorem 4.3 is obtained by a more careful study of deformations of the extended Burniat surfaces (cf. [BC10-b, Proposition 5.7]), using bidouble cover theory. We will follow this method in Part III. (3) Denote by SEB the subset of N EB3corresponding to the strictly extended Burniat surfaces. Then SEB is a proper open subset of N EB3of dimension 4. Theorem 4.3 (1) and (2) imply that NEB3is an irreducible component in Mcan 1,3.Here comes a natural question: is N EB3is closed in Mcan 1,3? Bauer and Catanese already showed that the answer is No (cf. [BC10-b, Section 7]). The aim of Part II is to complete the following task. Task : Determine the irreducible component N EB3in Mcan 1,3,i.e., describe all the surfaces corresponding to N EB3\ N EB3. 19
5. ONE PARAMETER LIMITS 5 One Parameter Limits of the Extended Burniat Surfaces This section is the first step to study limits of extended Burniat surfaces with K2= 3 in the moduli space. We need the following proposition concerning normal Del Pezzo surfaces. Let Ybe a normal Q-Gorenstein surface. Denote the dualizing sheaf of Yby ωY,and denote the associated Weil divisor by KY.Then there is a minimal positive integer msuch that ω⊗m Yis an invertible sheaf. So it makes sense to define KYto be ample or anti-ample. If KYis anti-ample, we call YaDel Pezzo surface. Also note that Yis Gorenstein if and only if m= 1. Proposition 5.1 ([HW81, Theorem 4.4 (ii)]).Let Ybe a normal Gorenstein Del Pezzo surface with K2 Y= 3.Then Yis a cubic surface in P3. The main result of this section is the following Theorem. Theorem 5.2. Let Tbe a smooth affine curve and o∈T, and let F:X → T be a flat family of canonical surfaces. Suppose that Xtis the canonical model of an extended Burniat surface or a nodal Burniat surface with K2 Xt= 3 for t6=o∈T. Then (after possibly shrinking T) there is a group action of G:= (Z/2Z)2on Xand the quotient map Π: X → Y := X/G yields a one parameter family of finite (Z/2Z)2-covers, XΠ// F @ @ @ @ @ @ @Y F0 T (i.e., Πt:Xt→ Ytis a finite (Z/2Z)2-cover), such that for each t6=o, Ytis a3A1-type cubic surface, and Yois a normal cubic surface. Remark 5.1. To study the limits of the extended Burniat surfaces with K2= 3,it suffices to require that Xtis a strictly extended Burniat surface for t6=oin Theorem 5.2. In fact, Remark 4.2 (3) implies that SEB =N EB3. Proof. Note that Xis Gorenstein, since the base Tis smooth and the fibres have only rational double points. 20
5. ONE PARAMETER LIMITS Since X\F−1(o)→T\{o}is a family of canonical models of extended Burniat surfaces or nodal Burniat surfaces with K2= 3,we have a (Z/2Z)2action on X \F−1(o).This is the Galois group action inducing the bicanonical morphism (the key point is that we work on the canonical models, cf. [BC10b, Theorem 0.2]). Hence, by [Cat83, Theorem 1.8], the (Z/2Z)2-action extends to X. Let Ybe the quotient of Xby the group action, and let Π: X → Y be the quotient map. Set Xt:= F−1(t) and Yt:= F0−1(t) for all t∈T. Then we have for all t∈T:KYt=KY|Yt, KXt=KX|Xt. Moreover, 2KX= Π∗(2KY+B),where Bis the branch divisor of Π: X → Y(cf. Theorem 1.3). Since for t6=o, we have 2KXt= Π∗ t(−KYt) (cf. Corollary 4.2), it follows that 2KX+ Π∗KY≡0 on X \Xo. Since Xois irreducible, we obtain (after possibly shrinking T) that 2KX+ Π∗KY≡0 on X.In particular, 2KXt= Π∗ t(−KYt) for all t∈T, (5.1) which implies that −KYtis ample and K2 Yt=K2 Xt= 3 for all t∈T. By construction, as the bicanonical image of Xt(cf. Corollary 4.2), Yt is a cubic surface with three A1-singularities for t6=o, and Yois a normal Q-Gorenstein surface. We claim that Yois Gorenstein. Then Yois a normal cubic surface by Proposition 5.1. We shall prove the claim by contradiction. Assume that Yois nonGorenstein. Recall that 2KXo≡Π∗ o(−KYo), K2 Yo= 3,(5.2) and −KYois ample. Step 1: All the possibilities of the non-Gorenstein locus of Yoare (cf. Theorem 2.3, Theorem 2.4 and Remark 2.1) (a) one B1-singularity. (b) one B1-singularity and one Y1-singularity. 21
5. ONE PARAMETER LIMITS (c) one Y2-singularity. (d) one Y1-singularity. (e) two Y1-singularity. In fact, Xohas at most rational double points. Hence by Remark 2.2, for a non-Gorenstein point qon Yo,Π−1 o(q) consists of two points p1, p2,and the stabilizers of p1and p2in Gare isomorphic to Z/2Z.By Theorem 2.3 and Theorem 2.4, either qis a Bk-singularity and both p1and p2 are A2k-singularities of Xo,or qis a Yk+1-singularity and both p1and p2are A2k+1-singularities of Xofor some k≥0. Hence an upper bound for the number of singularities of Xowould bound the number of non-Gorenstein singularities of Yo.Since the minimal resolution Soof Xohas Picard number 7, Sohas at most six (−2)- curves (cf. [BHPV, Page 272, Proposition 2.5]). An easy calculation shows that the list of the non-Gorenstein singularities of Yostated above is complete. Step 2: Let ˜ Yobe the minimal resolution of Yo.Then K2 ˜ Yois an integer. The resolution of a rational double point does not change K2,while the resolution of a B1-singularity (respectively, a Y1-singularity) contributes −1 3(respectively, −1) to K2(for example, cf. [Barlow99, Section 6]). Since K2 Yo= 3 is an integer, case (a) and case (b) cannot occur. Step 3: Assume that Yohas exactly one Y2-singularity q. The discussion in Step 1 shows that Π−1 o(q) consists of two A3-singularities p1, p2of Xoand p1, p2are the only singularities of Xo.Moreover, there is an involution g∈Gpermuting p1and p2. Lift gto the minimal resolution Soof Xo,and denote it by ˆg. Denote by Rthe divisorial part of the fix locus of ˆgand by tthe trace of ˆg∗:H2(So,C)→H2(So,C).Denote by N1, N2, N3(respectively, Z1, Z2, Z3) the (−2)-curves of Solying over p1(respectively, p2). Note that c1(KSo) and c1(N1),...,c1(Z3) are a basis of H2(So,C).Since gpermutes p1and p2on Xo, N1,...,Z3are disjoint from the fix lo22
6. EXCLUSION OF CERTAIN CUBIC SURFACES (1) An A5-type cubic surface x3x0x1−(x3 0+x3 1−x1x2 2) = 0 has three lines, l1:x0=x1= 0, l2:x0= 0, x1−x2= 0, l3:x0= 0, x1+x2= 0, which all pass through the A5-singularity P= (0 : 0 : 0 : 1). (2) A 3A2-type cubic surface x3x0x1−x3 2= 0 has three lines, l1:x0= x2= 0, l2:x1=x2= 0, l3:x2=x3= 0,which form a triangle with the three A2-singularities P1= (0 : 0 : 0 : 1), P2= (1 : 0 : 0 : 0), P3= (0 : 1 : 0 : 0) as the vertices. (3) An (A1+A4)-type cubic surface x3(x0x2−x2 1)−x2 0x1= 0 has four lines, l1:x0=x1= 0, l2:x0=x3= 0, l3:x1=x2= 0, l4:x1=x3= 0. l1, l3contain the A1-singularity P1= (0 : 0 : 0 : 1) and l2, l4contain the A4-singularity P2= (0 : 0 : 1 : 0). (4) A (2A1+A3)-type cubic surface x3(x0x2−x2 1)−x0x2 1= 0 has five lines, l1:x0=x1= 0, l2:x1=x2= 0, l3:x1=x3= 0, l4:x0=x3= 0, l5:x2=x0+x3= 0,and three singularities P1= (0 : 0 : 1 : 0)(A3), P2= (1 : 0 : 0 : 0)(A1), P3= (0 : 0 : 0 : 1)(A1). l1, l2, l3form a triangle with vertices P1, P2, P3,and l4contains P1.There is only one line l5 which does not contain any singularity. (5) An (A1+ 2A2)-type cubic surface x3(x0x2−x2 1)−x3 1= 0 has five lines, l1:x0=x1= 0, l2:x1=x2= 0, l3:x1=x3= 0, l4:x0=x1+x3= 0, l5:x2=x1+x3= 0,and it has three singularities P1= (0 : 0 : 0 : 1)(A1), P2= (0 : 0 : 1 : 0)(A2), P3= (1 : 0 : 0 : 0)(A2). l1, l2, l3form a triangle with vertices P1, P2, P3, l4contains P2and l5 contains P3. (6) An (A1+A3)-type cubic surface x3(x0x2−x2 1)−(x0−x1)(−x1+x2)(x0−2x1+x2) = 0 has two singularities P= (0 : 0 : 0 : 1)(A1), Q = (1 : 1 : 1 : 0)(A3). It has seven 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. Note that l1, l2, l3, l5meet at Q, and l4, l6meet at P. There is only one line l7which does not contain any singularity. 29
7. D4-GENERALIZED BURNIAT SURFACES (7) A (2A1+A2)-type normal cubic surface x3(x0x2−x2 1)−x2 1(x0−x1) = 0 has two A1-singularities P1= (0 : 0 : 0 : 1), P2= (1 : 0 : 0 : 0),and one A2-singularity Q= (0 : 0 : 1 : 0).It has eight 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. Note that l1, l3, l4, l6meet at Q, l1, l2, l5meet at P1, l2, l3, l7meet at P2. There is only one line l8which does not contain any singularity. Combining these three propositions with the classification of cubic surfaces (cf. Theorem 3.3), Theorem 6.1 follows. 7D4-generalized Burniat Surfaces By Theorem 6.1, Yocan be only one of the following types: 3A1, D4(1), D4(2) and 4A1.For each case we will either exclude it or find all the possible branch loci such that the associated bidouble cover Xocan be deformed to extended Burniat surfaces with K2= 3. In order to apply the theory of Section 1 to smooth surfaces, we make the following conventions for the remaining sections of Part II. Conventions Let Πo:Xo→ Yobe the bidouble cover as in Theorem 5.2. Let µ:e Y→ Yobe the minimal resolution of Yo.Denote by e Sthe normalization of the fiber product of Xoand e Yover Yo,and π:e S→e Ythe induced bidouble cover. Moreover, let ∆ be the branch locus of the bidouble cover π:e S→e Y . Write ∆ as ∆=∆1+ ∆2+ ∆3according to the group action (cf. Theorem 1.1, Section 1). In view of Corollary 5.4, ∆ has the following properties. Proposition 7.1. (1) Every irreducible component of ∆is a (−1)-curve, or a (−2)-curve or a 0-curve. (2) −Ke Y.∆i= 3 for i= 1,2,3. (3) µ∗(∆) ≡ −3KYo. 30
7. D4-GENERALIZED BURNIAT SURFACES Proof. By adjunction, for a smooth rational curve D, −Ke Y.D =D2+ 2. Hence a (−1)-curve on e Ycorresponds to a line on Yo,and a 0-curve corresponds to a smooth conic. Thus (1) follows from Corollary 5.4. Effective divisors in the linear system |−Ke Y|correspond to hyperplane sections of Yo. Note that OYo(KYo) is invertible, µ∗(Ke Y) = KYoand µ∗(KYo) = Ke Y.Since µ∗(∆i) = Bi(cf. Corollary 5.4), (2) follows from the projection formula and Corollary 5.4 (1), and (3) follows from Corollary 5.4 (4). Remark 7.1. By Proposition 1.2, ∆ = ∆1+ ∆2+ ∆3is reduced. By Theorem 1.1, ∆i’s are divisors such that for any i= 1,2,3,∆i+ ∆i+1 is even in Pic(e Y).See also Remark 1.1. We will use this remark frequently in the following sections. In this section we first deal with the case when Yohas a D4-singularity. 7.1 Configuration of Branch Divisors Assume that Yois of D4(1)-type or of D4(2)-type. Let yobe the D4-singularity and e Ybe its minimal resolution. Recall the notation introduced in Subsection 3.2. See Figure 4 and Figure 5. Lemma 7.2. Π−1 o(yo)consists of one point xoand xois an A1-singularity of Xo.Moreover, locally, Πo: (Xo, xo)→(Yo, yo)is isomorphic to (X0, x0) : z2+x2+y2= 0 →(Y0, y0) : w2+uv(u+v) = 0, (x, y, z)7→ (u, v, w) = (x2, y2, xyz), with the G-action on (X0, x0)given 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). Proof. Consider the family of bidouble covers Π: X → Y in Theorem 5.2. For t6=o, Ythas three nodes n1(t), n2(t), n3(t).Their limits in Yomust be the singularity yo.Thus their inverse images under Πtmust have limit points in Π−1 o(yo).By the construction (cf. Definition 4.1), for each ni(t),every point of Π−1 t(ni(t)) is fixed by gi.Note that for any i, giand gi+1 generates G. Since Π−1 o(yo) forms an orbit under the group action, the cardinality of Π−1 o(yo) can only be 4,2,or 1.The argument above shows that Π−1 o(yo) consists of one 31
7. D4-GENERALIZED BURNIAT SURFACES point xo.By looking at Theorem 2.5 Table 3 where the quotient (Y0, y0) is a D4-singularity, the conclusion follows. It is easy to see that u=x2, v =y2, w =xyz generate the ring of invariants for the action, and satisfy the equation w2+uv(u+v) = 0. Theorem 7.3. Assume that Yohas a D4-singularity. Then (1) Yomust be of D4(2)-type. (2) π:e S→e Yis isomorphic to the bidouble cover associated to the following branch divisors: ∆1= Γ1+N2+C3,∆2= Γ2+N3+C1,∆3= Γ3+N1+C2, where all Ci’s are irreducible smooth curves. Proof. First we consider the (−2)-curves. Lemma 7.2 and Example 2.1 show that one may assume that ∆1≥N2,∆2≥N3,∆3≥N1,∆6≥ Z, that N1, N2, N3are connected components of ∆,and that any irreducible component in ∆ −N1−N2−N3does not intersect any of the four (−2)-curves N1, N2, N3, Z. This shows that (∆−Ni).Ni= 0, i = 1,2,3 and (∆−N1−N2−N3).Z = 0. It follows that ∆ ≡ −3Ke Y+N1+N2+N3.In fact, by Proposition 7.1 (3) we may assume that ∆≡ −3Ke Y+x1N1+x2N2+x3N3+yZ, where x1, x2, x3, y are integers. The conditions above show that x1=x2=x3= 1, y = 0. Second, we consider the (−1)-curves. Recall that e Ycontains exactly six (−1)-curves: E0 1, E0 2, E0 3,Γ1,Γ2,Γ3.Since E0 i.Ni= 1,the discussion above shows that ∆ 6≥ Eifor i= 1,2,3.But ∆ contains at least three (−1)-curves, thus ∆ ≥Γ1+ Γ2+ Γ3. Let ∆0:= ∆ −N1−N2−N3−Γ1−Γ2−Γ3≡ −2Ke Y.Since we have considered all the (−2)-curves and all the (−1)-curves, ∆0consists of 0-curves. Note that ∆0is effective, reduced and is disjoint from all (−2)-curves. An easy argument using the following Lemma 7.4 shows that ∆0=C1+C2+C3. 32
7. D4-GENERALIZED BURNIAT SURFACES Lemma 7.4. Assume that Cis a smooth rational curve on e Ywith C2= 0. If C.N1=C.N2=C.N3=C.Z = 0,then Cbelongs to one of the following linear systems: |2L−Ei+1 −Ei+2 −E0 i+1 −E0 i+2|for i= 1,2,3. Proof. We may assume that C≡λL −P3 i=1(xiEi+yiE0 i) in Pic(e Y), λ and xi, yiare integers. C.N1=C.N2=C.N3=C.Z = 0 show that xi=yifor 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 effective and irreducible, the conclusion follows. We have seen ∆ = N1+N2+N3+ Γ1+ Γ2+ Γ3+C1+C2+C3 ≡9L−2E1−4E0 1−2E2−4E0 2−2E3−4E0 2. By Corollary 5.4 (1) and Proposition 7.1 (2), we have ∆1=N2+Cj+ Γα,∆2=N3+Ck+ Γβ,∆3=N1+Cl+ Γγ, where {j, k, l}={α, β, γ}={1,2,3}.By Remark 7.1, each ∆ihas even coefficients in E1, E0 1, E2, E0 2, E3, E0 3.So there are only two possibilities: (a) ∆i= Γi+Ni+1 +Ci+2,(b) ∆i= Γi+2 +Ni+1 +Ci,for each i= 1,2,3. If Yois of D4(1)-type, Γ1,Γ2,Γ3meet at a point Pon e Y . Note that any other irreducible component of ∆ does not pass through P. Then Example 1.1 shows that e Shas a 1 4(1,1)-singularity P0,which is not a rational double point. Since the Γi’s are disjoint from any (−2)-curves, e S→ Xois locally isomorphic at P0.This contradicts that Xois a canonical surface. Thus Yomust be of D4(2)-type. Note that there is an involution τ:P2→P2such that τ(P1) = P1, τ(P0 1) = P0 1, τ(P2) = P3, τ(P0 2) = P0 3, τ(P3) = P2, τ(P0 2) = P0 3(for example, in the notation of Subsection 3.2, τis defined by (y0:y1:y2)7→ (y0:y2:y1)). τ induces an involution on e Y . It maps the divisor classes of ∆1,∆2,∆3in case (a) to the ones of ∆2,∆1,∆3in case (b) respectively. Hence the bidouble covers associated to the two kinds of branch loci are essentially the same. 33
7. D4-GENERALIZED BURNIAT SURFACES Remark 7.2. If Yois of D4(1)-type, then we already see that e Shas a 1 4(1,1)- singularity. If we resolve this singularity and blow down the (−1)-curves π−1Ni,we get a family of minimal smooth surfaces of general type with K2= 2, pg=q= 0.We remark that the fundamental group of such a surface is isomorphic to (Z/2Z)3. 7.2 D4-generalized Burniat Surfaces Assume that Yois the D4(2)-type cubic surface, and e Yis its minimal resolution. Recall the notation introduced in Subsection 3.2 and Figure 5. We define three effective divisors on e Y , ∆i= Γi+Ni+1 +Ci+2 ≡3L−2Ei−2E0 i−2E0 i+1, i = 1,2,3,(7.1) where all Ci’s are irreducible smooth curves. And define three divisors Li=−Ke Y+Ei−E0 i+2, i = 1,2,3.(7.2) Theorem 7.5 ([BC10-b, Section 7]).Let π:e S→e Ybe the bidouble cover associated to the above data ∆1,∆2,∆3,L1,L2,L3.Then e Sis a smooth surface with K2 e S=−3, pg(e S) = q(e S) = 0. Moreover, |2Ke S|=π∗| − Ke Y|+π∗(N1+N2+N3)and P2(e S) = 4. Proof. First note that ∆i’s and Li’s satisfy the equations (1.1) and (1.2). Since the total branch divisor ∆ is normal crossing and each ∆iis smooth, e Sis smooth by Proposition 1.2 (2). Note that L2 i= 1, Ke Y.Li=−3.By Corollary 1.4, K2 e S=−3 and χ(Oe S) = 1.From (7.2), one sees that Ke Y+Liis not effective for all i= 1,2,3. Hence by Corollary 1.4, pg(e S) = pg(e Y) = 0.It follows that q(e S) = 0. From (7.2), one sees that 2Ke S+Li+Li+1 is not effective for all iand L1+L2+L3≡ −3Ke Y+N1+N2+N3.By Theorem 1.3 (2) and Corollary 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. It follows that |2Ke S|=π∗|−Ke Y+N1+N2+N3|=π∗|−Ke Y|+π∗(N1+N2+N3), since N1+N2+N3is the fixed part of | − Ke Y+N1+N2+N3|. 34
8. 4A1-GENERALIZED BURNIAT SURFACES Definition 7.1. The minimal model of e Sin the Theorem 7.5 is called a D4-generalized Burniat surface. Corollary 7.6 ([BC10-b, Section 7]).Let f:e S→Sbe the blow down of the six (−1)-curves π−1Nifor i= 1,2,3.Then Sis a smooth minimal surface of general type with K2 S= 3, pg(S) = q(S) = 0 and P2(S) = 4. S has exactly one (−2)-curve Z0.Moreover, f∗|2KS|=π∗|−Ke Y|and the bicanonical linear system of Sis base-point-free. Proof. Since each Ni, i = 1,2,3,forms a connected component of the branch locus, each π−1Niis a disjoint union of two (−1)-curves. Note that Zis not in the branch locus, and Z.Ni= 1, i = 1,2,3.Then Hurwitz’s Theorem shows that π∗Zis a smooth rational curve with self-intersection number −8. Let f:e S→Sbe the blow down of the six (−1)-curves. Then K2 S= 3 and the image of π∗Zis a (−2)-curve Z0. Since pg, q, P2are birational invariants, pg(S) = 0 and P2(S) = 4.Moreover, since |2Ke S|=f∗|2KS|+π∗(N1+N2+N3) by the Theorem 7.5, we have f∗|2KS|=π∗|−Ke Y|.|−Ke Y|is base-point-free, thus |2KS|is base-point-free. Moreover, −Ke Yis nef and big, so is KS.Thus Sis minimal and of general type. Corollary 7.7 ([BC10-b, Section 7]).Let ϕ:S→Xbe the contraction of the (-2)-curve Z0,i.e., Xis the canonical model of S. Then Xis a bidouble cover of the D4(2)-type cubic surface Yoby the bicanonical morphism. Moreover, Xhas an A1-singularity, lying over the D4-singularity of Yo, where the bicanonical morphism is totally ramified. Proof. It follows from Corollary 7.6 and Lemma 7.2. 84A1-generalized Burniat Surfaces 8.1 Configuration of Branch Divisors Assume that Yois the 4A1-type cubic surface. Let µ:e Y→ Yobe its minimal resolution. Recall the notation introduced in Subsection 3.3 and Figure 6. 35
8. 4A1-GENERALIZED BURNIAT SURFACES Theorem 8.1. π:e S→e Yis isomorphic to the bidouble cover associated to the following branch divisors: ∆1= Γ1+N2+C3,∆2= Γ2+N3+C1,∆3= Γ3+N1+C2, where all Ci’s are irreducible smooth curves. Before giving the proof, we make the following remark. Remark 8.1. (1) By Theorem 3.3, up to an isomorphism, there is exactly one 4A1-type cubic surface. (2) It well known (cf. [Sak10, Theorem 3]), the automorphism group of a 4A1-type cubic surface is isomorphic to the symmetry group of four letters, which permutes the 4 nodes of the surface. Proof. First we consider the (−1)-curves. All the (−1)-curves except Γ1, Γ2,Γ3intersect at least one (−2)-curve, which correspond to the lines in Yo passing through singularities. By Corollary 5.4 (2) and Lemma 6.4, ∆ ≥ Γ1+ Γ2+ Γ3. Next we consider 0-curves. Lemma 8.2. Fix k∈ {1,2,3}.Assume that Cis a reduced curve of e Ysuch that C6≥ Γi, Ni,for i= 1,2,3.If µ(C+ Γk)is a hyperplane section of Yo, then Cis a smooth irreducible curve in the linear system |2L−Ek+1 −Ek+2 − E0 k+1 −E0 k+2|.It follows that Cis disjoint from all Ni’s and Z. Proof. Without loss of generality, assume that k= 1.Note that elements in | − Ke Y|correspond to hyperplane sections of Yoand Γ1+ (2L−E2−E3− E0 2−E0 3)≡ −Ke Y.If Cis a singular element in |2L−E2−E3−E0 2−E0 3|, then C=N1+Z+ 2E1,or C=N2+N3+ 2E0 1or C= Γ2+ Γ3.Thus the first conclusion follows. The second conclusion follows from the calculation of intersection numbers. By Corollary 5.4 (3) and Lemma 8.2, one sees that ∆ must contain a smooth curve Ciin the linear system |2L−Ei+1 −Ei+2 −E0 i+1 −E0 i+2|for each i. 36
8. 4A1-GENERALIZED BURNIAT SURFACES We have shown that ∆ ≥Γ1+ Γ2+ Γ3+C1+C2+C3.By Corollary 5.4 (1), (2) and (3), up to a permutation of 1,2,3,one of the following two holds: (a) ∆1≥Γ1+C3,∆2≥Γ2+C1,∆3≥Γ3+C2, (b) ∆1≥Γ3+C1,∆2≥Γ1+C2,∆3≥Γ2+C3. Since the 3 nodes on Ytare in the branch locus of πt,thus at least 3 nodes of Yoare in the branch locus of πo.Equivalently, ∆ contains at least three (−2)-curves. We distinguish two cases. Case I: One of the four (−2)-curves is not in ∆. Without loss of generality (cf. Remark 8.1 (2)), assume that ∆ 6≥ Z. Then ∆ = 3 X i=1 (Γi+Ci+Ni)≡12L− 3 X i=1 (4Ei+ 5E0 i). Thus ∆ihas even coefficients in E1, E2, E3and odd coefficients in E0 1, E0 2, E0 3 by Remark 7.1. It follows that if (a) holds then ∆i= Γi+Ni+1 +Ci+2,and if (b) holds then ∆i= Γi+2 +Ni+1 +Ci. Take an involution τof P2such that τ(P1) = P1, τ(P0 1) = P0 1, τ(P2) = P3, τ(P0 2) = P0 3.Then it follows that τ(P3) = P2, τ(P0 3) = P0 2.It induces an involution on e Ywhich maps the divisor classes of ∆1,∆2,∆3in case (a) to the ones of ∆2,∆1,∆3in case (b). Hence the bidouble covers associated to the two kinds of branch loci are essentially the same. Case II: All the 4 nodes are contained in the branch locus, i.e, ∆≥N1+N2+N3+Z. We intend to exclude this case. Assume that (a) holds. Then we may assume that ∆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 for each k= 1,2,3,4,exactly one of the ak, bk, ckis 1 and the other two is 0,since ∆ is effective and reduced. The following table gives the coefficients (up to sign) of E1, E2, E3in the branch divisors. ∆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 Remark 7.1, a1+a4+ 2, b1+b4, c1+c4+ 1 must be of the same parity. Since their sum is 5,they must be all odd integers. Thus either (a1, b1, c1) = (1,0,0) and (a4, b4, c4) = (0,1,0),or (a1, b1, c1) = (0,1,0) and (a4, b4, c4) = (1,0,0). If the former holds, then the coefficients a3, b3+ 2, c3+ 2 of E3cannot have the same parity. If the latter holds, then the coefficients a2+2, b2+2, c2 of E2cannot have the same parity. So this case is excluded. If (b) holds, a similar argument shows that Case II can be excluded. Remark 8.2. In the course of excluding Case II, we find another family of surfaces of general type which are also bidouble covers of the 4A1-type cubic surface, but branched on all the nodes. First construct the bidouble cover π:e S→e Yassociated to the following data, ∆1=C1+ Γ2+N1+N2,∆2=C2+ Γ1+N3+Z, ∆3=C3+ Γ3. Then blow down the eight (−1)-curves π−1Niand π−1Z, f :e S→S. S is of general type with K2 S= 3 and pg(S) = 0. S has 4 nodes coming from the nodes of the curve ∆3.However, note that ∆3≡ −Ke Y,we can deform Sto smooth surfaces by deforming ∆3to smooth curves. For details, see Section 14 in Part IV. 8.2 4A1-generalized Burniat Surfaces Assume that Yois the 4A1-type cubic surface, and e Yis its minimal resolution. Recall the notation introduced in Subsection 3.3 and Figure 6. 38
10. KEY TOOLS Moreover, let Mbe a divisor on e Ysuch that (Ke Y+ 2C+M).C < 0.Then H0(Ω1 e Y(log(∆ −C))(C+M)) ∼ =H0(Ω1 e Y(log ∆)(M)). Proof. Since Cis a connected component of a smooth divisor ∆,we have the following exact sequence, 0→Ω1 e Y(log ∆) →Ω1 e Y(log(∆ −C))(C)→Ω1 C(C)→0 Tensor it with the invertible sheaf Oe Y(M) and use the adjunction formula Ω1 C=OC(Ke Y+C),to get the exact 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,the associated exact sequence of cohomology groups shows that H0(Ω1 e Y(log ∆)(M)) ∼ = H0(Ω1 e Y(log(∆ −C))(C+M)). Lemma 10.3 ([Cat84, Lemma 3.7], [CHKS06, Lemma 3, page 675]).Let ∆ = ∪i∆ibe a union of smooth divisors ∆1,...,∆kon a smooth surface e Y , such that ∆has only normal crossing singularities. Then (1) there is an exact sequence 0→Ω1 e Y→Ω1 e Y(log ∆1,...,log ∆k)→ ⊕k i=1O∆i→0. (2) In the cohomology exact sequence associated to the above exact sequence ∂:⊕k i=1 H0(O∆i)→H1(Ω1 e Y),if 1∆iis the function which is ≡1on ∆i and 0elsewhere, then ∂(1∆i) = c1(∆i). The next theorem studies how the dimensions of the cohomology groups of tangent sheaf change when blowing down a (−1)-curve. Theorem 10.4 (cf. [Cat88, Lemma 9.22]).Let Sbe a smooth surface, and f:e S→Sbe the blowup of Sat a point p. Then R1f∗Θe S= 0. Moreover, if Sis of general type, then 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 Proof. Let Ebe the exceptional curve of f. The sheaf R1f∗Θe Sis supported on the point p, by formal function theorem (cf. [Har77, Theorem 11.1]), it suffices to show H1(En,Θe S⊗ OEn) = 0,where Enis the closed subscheme of e Sdefined by In,where Iis the ideal sheaf of E. There is an exact sequence 0→In In+1 → OEn+1 → OEn→0. for all n≥0.Tensor the exact sequence by Θe S,it remains exact since Θe Sis a locally free sheaf. Note that E1=Eand In In+1 ∼ =OE(n),it suffices to show H1(E, Θe S⊗ OE(n)) = 0 for all n≥0. We have a normal exact sequence 0→ΘE→Θe S⊗ OE→ OE(E)→0. Tensor it with OE(n),we get 0→ OE(n+ 2) →Θe S⊗ OE(n)→ OE(n−1) →0. Since H1(E, OE(n−1)) = 0 for n≥0,one sees that H1(E, Θe S⊗OE(n)) = 0. Hence we have shown that R1f∗(Θe S) = 0.(See [Cat88, Lemma 9.22] for another proof). There is an exact sequence (cf. [Ser06, page 73]), 0→Θe S→f∗ΘS→ OE(−E)→0. By [Har77, Proposition 3.4, Chapter V] Rkf∗Oe S= 0 for k≥1,then the projection formula shows that Rkf∗(f∗ΘS) = Rkf∗Oe S⊗ΘS= 0.Thus we have an exact sequence 0→f∗Θe S→ΘS→f∗OE(−E)→0. If Sis of general type, h0(e S, Θe S) = h0(S, ΘS) = 0 (cf. [Mats63]). Note that f∗OE(−E) is supported on p, thus Hk(S, f∗OE(−E)) = 0 for k≥1.By the long exact sequence of cohomology associated to the last exact sequence above, and OE(−E)∼ =OE(1),we have h1(S, f∗Θe S) = h1(S, ΘS) + 2 and h2(S, f∗Θe S) = h2(S, ΘS). Finally since Rkf∗Θe S= 0 for k≥1,Leray spectral sequence shows that h1(e S, Θe S) = h1(S, f∗Θe S) and h2(e S, Θe S) = h2(S, f∗Θe S).Hence the conclusion follows. 46
11. DEFORMATIONS OF THE D4-GENERALIZED BURNIAT SURFACES The last theorem describes how the dimensions of the cohomology groups of the tangent sheaf change when contracting a (−2)-curve to a node. Theorem 10.5 ([BW74, Proposition 1.10, Theorem 2.14]).Let Sbe a minimal surface of general type, and let ϕ:S→Xbe a morphism contracting a (−2)-curve Nof Sto an A1-singularity on X. Then ϕ∗ΘS= ΘX, H1(S, ΘS)∼ =H1(X, ΘX)⊕H1 N(ΘS), H2(S, ΘS)∼ =H2(X, ΘX). Moreover, dim H1 N(ΘS) = 1. 11 Deformations of the D4-generalized Burniat Surfaces Throughout this section, we use the notation introduced in Subsection 3.2 and Subsection 7.2. See Figure 5. We start to study the local deformations of the D4-generalized Burniat surfaces. Let Xbe the canonical model of a D4-generalized Burniat surface S. We intend to calculate the dimension of the tangent space to the base of the Kuranishi family of X, i.e., dim Ext1 OX(Ω1 X,OX).For this we first calculate hi(e S, Θe S) (cf. Theorem 7.5), using the bidouble cover structure as described in Theorem 10.1. Then we pass from e Sto the minimal model S, calculate hi(S, ΘS) by Theorem 10.4. Finally, we pass from Sto the canonical model Xby Theorem 10.5, and use the spectral sequence Epq 2=Hp(X, Extq OX(Ω1 X,OX)) ⇒Extp+q OX(Ω1 X,OX). By Serre Duality and Theorem 10.1, Hk(e S, Θe S)inv =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) for k= 0,1,2 and i= 1,2,3. Since e Sis a surface of general type, H0(e S, Θe S) = 0.Therefore the righthand sides of the equations equal 0 when k= 0. Proposition 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 Proof. By Lemma 10.3 (1), we have an exact 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) Note that H0(e Y , Ω1 e Y) = 0 and −Ke Yis effective, thus H0(e Y , Ω1 e Y(Ke Y)) = 0. To prove the first equality, it suffices to show the boundary map δ:H0(e Y , ⊕3 i=1O∆i(Ke Y)) →H1(e Y , Ω1 e Y(Ke Y)) is injective. Since ∆iis a disjoint union of three smooth rational curves Γi, Ni+1, Ci+2, H0(e Y , O∆i(Ke Y)) ∼ =H0(e Y , ONi+1 )∼ =C. | − Ke Y|is base-point-free, therefore there is a morphism Oe Y(Ke Y)→ Oe Y, which is not identically zero on any component of ∆i’s, in particular on Ni’s. Now consider the commutative diagram coming from the above morphism 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. It gives a commutative diagram of cohomology groups, 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), the image of the function identically equal to 1 on Ni maps under ψ1to the first Chern class of Ni.Because Ni’s are disjoint (−2)- curves, their Chern classes are linearly independent in H1(e Y , Ω1 e Y).Thus the composite map ψis injective. Hence δis also injective 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,to calculate the dimension of H1(e Y , Ω1 e Y(log ∆1,log ∆2,log ∆3)⊗Ω2 e Y) is the same as to calculate χ(Ω1 e Y(log ∆1,log ∆2,log ∆3)⊗Ω2 e Y).By the exact 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 Serre’s Duality and Riemann-Roch theorem show that, 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. Note that ∆iis a disjoint union of three smooth rational curves Γi, Ni+1, Ci+2. It follows that χ(O∆i(Ke Y)) = 0 for i= 1,2,3. Hence χ(Ω1 e Y(log ∆1,log ∆2,log ∆3)⊗Ω2 e Y) = −4 and it follows that h1(e Y , Ω1 e Y(log ∆1,log ∆2,log ∆3)⊗Ω2 e Y) = 4. In order to calculate h0(e Y , Ω1 e Y(log ∆i)(Ke Y+Li)) for i= 1,2,3, we need the following lemmas. Lemma 11.2. Let p1:W→C2be the blowup of C2at (0,0),and let p2: Σ → Wbe the blowup of Wat the intersection point O0of the strict transform of the line l:y= 0 with the exceptional curve Eof p1. Denoted by E0the exceptional curve of p2and by Γthe strict transform of the line lunder the morphism p=p2◦p1: Σ →W→C2.Then (1) p∗Ω1 Σ(−E0)⊆Ω1 C2is the subsheaf of forms {ω∈Ω1 C2|ω=α(x, y)dx +β(x, y)dy, α(0,0) = 0}. (2) p∗Ω1 Σ(−2E0)⊆Ω1 C2is the subsheaf of forms {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 the subsheaf of forms {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}. Proof. Wcan be covered by two affine coordinate charts V1∼ =C2(x, t) and V2∼ =C2(s, y),such that p1is given by V1→C2,(x, t)7→ (x, tx), V2→C2,(s, y)7→ (sy, y). 49
11. DEFORMATIONS OF THE D4-GENERALIZED BURNIAT SURFACES p−1 2(V1) can be covered by two affine coordinate charts U11 ∼ =C2(x, u) and U12 ∼ =C2(v, t) such that the morphism p: Σ →C2is given by U11 →V1→C2,(x, u)7→ (x, ux)7→ (x, x2u), U12 →V1→C2,(v, t)7→ (vt, t)7→ (vt, vt2). And similarly for p−1 2(V2) = U21 ∪U22.Note that both E0and Γ are contained in U11 ∪U12. First use the coordinate chart U11.Locally E0is defined by x= 0 and Γ is defined by u= 0. (1) By Riemann’s extension theorem, p∗Ω1 Σ(−mE0)⊆Ω1 C2for all m≥0. Assume that ω=α(x, y)dx +β(x, y)dy for some holomorphic function 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 to the OΣ-module generated by xdx, xdu if and only if α(0,0) = 0. (2) By the calculation above, locally p∗ωbelongs to the OΣ-module generated by x2dx, x2du if and only if α(x, x2u) + 2xuβ(x, x2u) is divisible by x2.Assume that α(x, y) = a+bx +cy +higher degree terms, (11.4) β(x, y) = A+Bx +Cy +higher degree terms, (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), then α(x, x2u) + 2xuβ(x, x2u) = a+bx + 2Axu +x2h(x, u), for some holomorphic function h(x, u).Thus p∗ωbelongs to the OΣmodule generated by x2dx, x2du, if and only if a=b=A= 0. 50
11. DEFORMATIONS OF THE D4-GENERALIZED BURNIAT SURFACES (3) Observe that p∗Ω1 Σ(log Γ)(−E0) consists of rational differential 1-forms ωwhich, when restricted to C2\{(0,0)},yield sections of Ω1 C2(log l).In particular, yω is a regular 1-form on C2\{(0,0)},which can be extended to a regular 1-form on C2.Assume that ω=α1(x, y)dx y+β(x, y)dy yfor some holomorphic function α1(x, y) and β(x, y),then 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 to the OΣ-module generated by xdx, xdu uif and only if α1(x, x2u)+2xuβ(x, x2u) is divisible by x3uand β(x, x2u) is divisible by x. If α1(x, x2u) + 2xuβ(x, x2u) is divisible by x3u, then α1(x, x2u) is divisible by u. This implies α1(x, y) = yα(x, y) for some holomorphic function 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. If we write α(x, y), β(x, y) as (11.4) and (11.5), then one sees that p∗ωbelongs to the OΣ-module generated by xdx, xdu uif and only if A= 0, a + 2B= 0. Hence we see that (1),(2),(3) hold locally. Similar calculation with other coordinate charts show the same results. Lemma 11.3. Let ldenote the line on the projective plane P2defined by x1= 0.Then any ω∈H0(Ω1 P2(log l)(2)) is of the form ω= (−Ax1x2−Cx1x3+Dx2 2+Ex2 3+Fx2x3)dx1 x1 + (Ax1−Dx2−Bx3)dx2+ (Cx1+Bx2−Fx2−Ex3)dx3,(11.6) where A, B, C, D, E, F ∈C. 51
11. DEFORMATIONS OF THE D4-GENERALIZED BURNIAT SURFACES Proof. By [BC10-b, Lemma 5.2 (1)], the vector space H0(Ω1 P2(2)) is 3-dimensional with a basis: −x2dx1+x1dx2,−x3dx2+x2dx3,−x3dx1+x1dx3. By the exact 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 that h0(Ω1 P2(log l)(2)) = 6. Moreover, it is easy to show that the following forms x2 2 dx1 x1 −x2dx2, x2 3 dx1 x1 −x3dx3, x2x3 dx1 x1 −x2dx3 in the vector space H0(Ω1 P2(log l)(2)),are mapped to a basis of H0(Ol(2)). Hence these forms and the above basis of H0(Ω1 P2(2)) are linearly independent in H0(Ω1 P2(log l)(2)).Then their linear combination 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 of the form (11.6). Proposition 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,for i= 1,2,3. Proof. To prove the first equality for i= 3,note that 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 to the curve C2and then to 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)). Without loss of generality, we may assume that P1= (1:0:0),P2= (1:1:0),Q1= (0:1:1),Q2= (0:0:1). It follows that P3= (0:1:0)and Q3= (1:0:−1).See Figure 7. Note that σ∗(Ω1 e Y(log Γ3)(2L−2E0 1−E0 2−E0 3)) is a subsheaf of Ω1 P2(log l)(2), thus we can apply Lemma 11.3. Any ω∈H0(e Y , Ω1 e Y(log ∆3)(Ke Y+L3)),considered as an element of H0(P2,Ω1 P2(log l)(2)),is of the form (11.6). 52
11. DEFORMATIONS OF THE D4-GENERALIZED BURNIAT SURFACES Locally around the point P1= (1:0:0), x1= 1 and the line P1P0 1is defined by x2= 0.So locally we may write 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 then ω= (Dx2 2+Fx2x3)dx1 x1 + (−Dx2−Bx3)dx2+ (B−F)x2dx3. Locally around the point P3= (0:1:0), x2= 1 and the line P3P0 3is defined by x1= 0.So locally we may write ω= (D+Fx3)dx1 x1 + (B−F)dx3. Then by Lemma 11.2 (3), D= 0, B +F= 0,and then ω=F(x2x3 dx1 x1 +x3dx2−2x2dx3). Locally around the point P2= (1:1:0), x1= 1. P2is the intersection point of the line x3= 0,and the line P2P0 2: 1−x2+x3= 0.Let 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. Note that H2(e Y , Ω1 e Y(log ∆3)(Ke Y+L3)) = 0,so to calculate the dimension of H1(e Y , Ω1 e Y(log ∆3)(Ke Y+L3)) is equivalent to calculate χ(Ω1 e Y(log ∆3)(Ke Y+L3)).Twist the following exact sequence with the invertible sheaf associated to the divisor 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 get χ(Ω1 e Y(log ∆3)(Ke Y+L3)) = χ(Ω1 e Y(F)) + χ(O∆3(F)). For the second summand, since ∆3is the disjoint union of rational curves N1, C2,Γ3, and F.N1= 0,F.C2= 1,F.Γ3= 1,we have χ(O∆3(F)) = χ(ON1) + χ(OC2(1)) + χ(OΓ3(1)) = 5. For the first summand, using the splitting principle, formally write Ω1 e Y=Oe Y(A1)⊕ Oe Y(A2),and A1+A2=Ke Y, A1.A2=c2(Y) = 9. Note that F2=−2 and F.Ke Y= 0,Riemann-Roch Theorem gives 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. Similarly, the statement also holds for i= 1,2. Theorem 11.5. Let π:e S→e Ybe the bidouble cover as in Subsection 7.2. Let Sbe the minimal model of e Sand Xthe canonical model of e S(cf. Subsection 8.1). The respective dimensions of the cohomology groups of the tangent sheaves Θe S,ΘS,ΘXare as follows. 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. Proof. By (11.1), (11.2), Proposition 11.1 and Proposition 11.4, h1(e S, Θe S) = 16 and h2(e S, Θe S) = 0. Since Sis obtained by blowing down six (−1)-curves (cf. Corollary 7.6) on e S, then by Theorem 10.4, h1(S, ΘS) = 4 and h2(S, ΘS) = 0. Xis obtained by contracting the (−2)-curve Z0on S(cf. Corollary 7.7), then by Theorem 10.5, h1(X, ΘX) = 3 and h2(X, ΘX) = 0. Corollary 11.6. The base of the Kuranishi family of Sis smooth. 54
12. DEFORMATIONS OF THE 4A1-GENERALIZED BURNIAT SURFACES Xis obtained by contracting four (−2)-curves on S(cf. Corollary 8.5). The group Gacts on the set of the four (−2)-curves transitively. Thus the conclusion about Xfollows by Theorem 10.5. Lemma 12.5. The sheaf Ext1 OX(Ω1 X,OX)has support on the 4nodes of X, such that every stalk over a node has length 1.Moreover, we have a decomposition of the global section group of Ext1 OX(Ω1 X,OX),according to the group action, H0(X, Ext1 OX(Ω1 X,OX)) = H0(X, Ext1 OX(Ω1 X,OX))inv⊕ ⊕3 i=1 H0(X, Ext1 OX(Ω1 X,OX))χi, and each direct summand has dimension 1. Proof. Since the group acts transitively on four A1-singularities, it induces the regular representation on H0(X, Ext1 OX(Ω1 X,OX)).Hence the conclusion follows. Corollary 12.6. dim Ext1 OX(Ω1 X,OX)inv = 4 and dim Ext2 OX(Ω1 X,OX)inv = 0. Proof. We have an exact sequence 0→H1(X, ΘX)→Ext1 OX(Ω1 X,OX)→H0(X, Ext2 OX(Ω1 X,OX)) →H2(X, ΘX)→Ext2 OX(Ω1 X,OX)→0, associated to the spectral sequence, Epq 2=Hp(X, Extq OX(Ω1 X,OX)) ⇒Extp+q OX(Ω1 X,OX). The exact sequence is a G-equivariant sequence of C-vector spaces, since all sheaves have a natural G-linearization. Then the conclusion follows by Theorem 12.4 and Lemma 12.5. Unlike the case of the D4-generalized surfaces, we cannot determine the deformations of the 4A1-generalized surfaces completely by using the bidouble cover structure to the 4A1-type cubic surface. 61
13. KEUM-NAIE-MENDES LOPES-PARDINI SURFACES Part IV The Irreducible Component containing the Keum-NaieMendes Lopes-Pardini Surfaces 13 Keum-Naie-Mendes Lopes-Pardini Surfaces J. H. Keum and later D. Naie ([Ke88], [Na94]) constructed a family of surfaces of general type with K2= 3 and pg(S) = 0.These surfaces are double covers of nodal Enriques surfaces with 8 nodes (cf. [Na94, Th´eor`eme 2.10]). Also these surfaces are different from the (extended) Burniat surfaces with K2= 3, since they have different fundamental groups. Theorem 13.1 ([Na94, Th´eor`eme 3.1]).If Sis a Keum-Naie surface with K2= 3,then πtop 1(S)∼ =(Z/2Z)2×Z/4Z. Another property of Keum-Naie surfaces is that their bicanonical map factors through the covering map to the nodal Enriques surface and is of degree 4.Later, in the article [MP04], Mendes Lopes and Pardini gave an explicit construction of surfaces of general type whose bicanonical map is a morphism of degree 2.They proved the following theorem about the corresponding subset in the moduli space. Theorem 13.2 ([MP04, Theorem 2.1, Theorem 7.1]).Let Mcan 1,3be the moduli space of canonical models of surfaces of general type with χ= 1 and K2= 3.Let Ebe the subset of Mcan 1,3consisting of the canonical surfaces with pg= 0 whose bicanonical map is composed with an involution such that the quotient surface is birational to an Enriques surface. (1) If Xbelongs to Eand τis the involution satisfying the property above, then X/τ is a nodal Enriques surface with 7nodes. 62
14. A SUBFAMILY OF KNMP SURFACES (2) The set Eis constructible. (3) The closure Ein Mcan 1,3is irreducible and uniruled of dimension 6. (4) Econtains the Keum-Naie surfaces with K2= 3. As pointed out in [MP04, Remark 7.2], there is a question left open: whether Eis an irreducible component of Mcan 1,3or not. We will reconstruct a subset E0in Ethrough bidouble covers of a 4A1-type cubic surface. Then by studying the deformations of the surfaces in E0,we give an affirmative answer to this question. 14 A Subfamily of KNMP Surfaces In this section we will construct the family of surfaces of general type already mentioned in Remark 8.2. The construction here is similar to (but different from) the one in [MP04, Example 3.6]. Assume that Yis a 4A1-type cubic surface, and e Yis its minimal resolution. Recall the notation introduced in Subsection 3.3 and Figure 6. Especially recall that e Yhas a pencil of rational curves Ciin the linear system |2L−Ei+1 −Ei+2 −E0 i+1 −E0 i+2|for i= 1,2,3. We define three effective divisors 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) where C1, C2, H are irreducible smooth curves. And define three divisors 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) Thoughout the following sections, we will assume that the divisor ∆ := ∆1+ ∆2+ ∆3has only normal crossing singularities. 63
14. A SUBFAMILY OF KNMP SURFACES Theorem 14.1. Let π:e S→e Ybe the bidouble cover associated to the above data ∆1,∆2,∆3,L1,L2,L3.Then e Sis a smooth surface with K2 e S=−5and pg(e S) = q(e S) = 0. Moreover, |2Ke Y| ≡ π∗| − Ke Y|+π∗(N1+N2+N3+Z)and P2(e S) = 4. Proof. Note that ∆i’s and Li’s satisfy the equations (1.1) and (1.2). Since the total branch divisor ∆ has normal crossings and each ∆iis smooth, e Sis smooth by Proposition 1.2 (2). Note that L1+L2+L3≡ −3Ke Y+N1+N2+N3+Z, L2 i= 1, Ke Y.Li=−3. By Corollary 1.4, K2 e S=−5 and χ(Oe S) = 1.From (14.2) one sees that Ke Y+Li is not effective, hence by Corollary 1.4, pg(e S) = pg(e Y) = 0.It follows that q(e S) = 0. By Theorem 1.3, 2Ke S≡π∗(−Ke Y+N1+N2+N3+Z).Moreover, from (14.2) 2Ke Y+Li+Li+1 is not effective for all i. Take i= 2 for example, assume that |2Ke Y+L2+L3|contains an effective divisor D. Then D.N1=−2, (D−N1).N2=−2 and (D−N1−N2).Z =−1 show that D≥N1+N2+Z. But D−N1−N2−Z≡E1−E0 2,which is not effective. This gives a contradiction. Hence 2Ke Y+L2+L3is not effective. It follows that 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 the fixed part of | − Ke Y+N1+N2+N3+Z|. Corollary 14.2. Let f:e S→Sbe the blow down of the eight (−1)-curves π−1Nk(k= 1,2,3) and π−1Z. Then Sis a smooth minimal surface of general type with K2 S= 3, pg(S) = 0 and P2(S) = 4. Moreover, KSis ample and |2KS|is base-point-free. Sis a bidouble cover of the 4A1-type cubic surface Ythrough the bicanonical morphism. Proof. Since each Nk, k = 1,2,3,or Zforms a connected component of the branch locus, each π−1Nkor π−1Zis a disjoint union of two (−1)-curves. Let f:e S→Sbe the blow down of these eight (−1)-curves. Then K2 S= 3. 64
14. A SUBFAMILY OF KNMP SURFACES Since pg, q, P2are birational invariants, pg(S) = 0 and P2(S) = 4.Moreover, since |2Ke S|=f∗|2KS|+π∗(N1+N2+N3+Z),by Theorem 14.1, we have f∗|2KS|=π∗|−Ke Y|.Since |−Ke Y|is base-point-free, |2KS|is base-point-free. The minimal resolution µ:e Y→Ycontracts exactly the (−2)-curves N1, N2, N3, Z. From the construction, we have a finite bidouble cover p:S→ Ysuch that the following diagram commutes: e Sf// π S p e Yµ//Y and 2KS≡p∗(−KY).Since −KYis ample and pis finite, KSis ample and thus Sis minimal. It also shows that the bicanonical morphism of Sis the composition of pand the anticanonical embedding of Yinto P3. We denote by E0the corresponding subset of smooth surfaces constructed above in the moduli space Mcan 1,3. Proposition 14.3. E0is contained in E. Proof. Given a surface Sin E0,consider the intermediate double cover ˆπ:ˆ S→e Y associated to the data 2L3≡∆1+ ∆2. Standard formulae for double covers (for example, see [BHPV, Page 236237]) show that 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, where ˆ Ek:= ˆπ−1Nkand ˆ E4:= ˆπ−1Zare (−1)-curves. Moreover, ˆ Shas 7 nodes lying over the nodes of the curve C1+C2+ Γ1+ Γ2. 65
15. LOCAL DEFORMATIONS AND IRREDUCIBLE COMPONENT Let ˆ f:ˆ S→S0be the blow down of the four (−1)-curves. We obtain a nodal Enriques surface S0with 7 nodes. The following diagram commutes: e Sf// ˆπ = = = = = = = = π S > > > > > > > > p ˆ S// S0 e Yµ//Y Thus the bicanonical morphism S→Y ,→P3of Sfactors through S0. By the definition of E(cf. Theorem 13.2), Sbelongs to E. 15 Local Deformations and Irreducible Component In this section we will prove the following theorem. Theorem 15.1. (1) For a general surface Sin E0, h1(S, ΘS) = 6, h2(S, ΘS) = 2 and the base of the Kuranishi family of Sis smooth. (2) Eis an irreducible component of the moduli space Mcan 1,3. The key point is to prove the following proposition. Proposition 15.2. For a general surface Sin E0, h2(S, ΘS)≤2. Proof of Theorem 15.1 assuming Proposition 15.2. Since −h1(S, ΘS) + h2(S, ΘS) = 2K2 S−10χ(S) = −4,by Proposition 15.2 h1(S, ΘS)≤6.Since Sis smooth and KSis ample (cf. Corollary 14.2), the minimal model and the canonical model of Scoincide. We have the following inequalities, 6≥h1(S, ΘS)≥the dimension of the base of the Kuranishi family of S = the dimension of Mcan 1,3at the point [S] ≥the dimension of E. 66
15. LOCAL DEFORMATIONS AND IRREDUCIBLE COMPONENT Since the dimension of Eis 6 by Theorem 13.2, we see that all the equalities hold. The second equality shows that the base of the Kuranishi family of Sis smooth. Since locally the germ of the complex space (Mcan 1,3,[S]) is analytically isomorphic to the quotient of the base of the Kuranishi family by the finite group Aut(S),it follows that (Mcan 1,3,[S]) is irreducible. Since E is irreducible by Mendes Lopes and Pardini’s Theorem 13.2, the last equality shows that Ecoincides with Mcan 1,3locally at [S].It follows that Eis an irreducible component of Mcan 1,3. By Theorem 10.4, to prove Proposition 15.2, it suffices to show h2(e S, Θe S)≤2.By Serre Duality and Theorem 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 it suffices to calculate the dimension of each summand. Lemma 15.3. H0(e Y , Ω1 e Y(log ∆1,log ∆2,log ∆3)⊗Ω2 e Y) = 0. Proof. By Lemma 10.3, we have an exact 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) Note that H0(e Y , Ω1 e Y) = 0 and −Ke Yis effective, thus H0(e Y , Ω1 e Y(Ke Y)) = 0. To prove the claimed equality, it suffices to show the boundary map δ:H0(e Y , ⊕3 i=1O∆i(Ke Y)) →H1(e Y , Ω1 e Y(Ke Y)) is injective. 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-point-free, there is a morphism Oe Y(Ke Y)→ Oe Y,which is not identically zero on any component of ∆i’s. Now consider the commutative 67
15. LOCAL DEFORMATIONS AND IRREDUCIBLE COMPONENT diagram coming from the morphism 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. It gives a commutative diagram of cohomology groups, 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, the image of the function identically equal to 1 on Nk (k= 1,2,3),respectively on Zmaps under ψ1to the first Chern class of Nk, respectively of Z. Because the Nk’s and Zare 4 disjoint (−2)-curves, their Chern classes are independent in H1(e Y , Ω1 e Y). Thus the composite map ψis injective. It follows that δis also injective and H0(e Y , Ω1 e Y(log ∆1,log ∆2,log ∆3)⊗Ω2 e Y) = 0. To calculate other summands, we fix the coordinates of Piand P0 i.Without loss of generality, assume that 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 Figure 9. Lemma 15.4. H0(e Y , Ω1 e Y(log ∆3)(Ke Y+L3)) = 0 for a general H∈ | − Ke Y|. Proof. Let M:= Ke Y+L3= 2L−E1−E2−E3−E0 1−E0 2−E0 3(cf. (14.2)). Recall that ∆3=H∈ | − Ke Y|.Then Ke Y.M = ∆3.M = 0.For a general H, H is a smooth elliptic curve and O∆3(M) is a 2-torsion element, thus H0(O∆3(M)) = 0. In fact, note that 2M≡N1+N2+N3+Z. Take the double cover ˜q:˜ Σ→e Y associated to the data 2M≡N1+N2+N3+Z, and blow down the (−1)- curves ˜q−1Ni, i = 1,2,3 and ˜q−1Z, η :˜ Σ→Σ.We have a morphism q: Σ →Y 68
15. LOCAL DEFORMATIONS AND IRREDUCIBLE COMPONENT and the following commutative diagram ˜ Ση// ˜q Σ q e Yµ//Y. qonly ramifies over the 4 nodes of Yand q∗(−KY)≡ −KΣ.Σ is a smooth Del Pezzo surface of degree 6,i.e., K2 Σ= 6 and −KΣis very ample. By Bertini’s theorem, a general curve Cof |−KY|is smooth and irreducible and q−1Cis an irreducible smooth curve in | − KΣ|.Since | −Ke Y|=µ∗| − KY|and a general element H∈ | − Ke Y|is disjoint from the (−2)-curves, the commutative diagram shows that ˜q−1His an irreducible smooth curve. Hence OH(M) is a 2-torsion element. Tensor the following exact sequence with Oe Y(M), 0→Ω1 e Y→Ω1 e Y(log ∆3)→ O∆3→0, we see that h0(e Y , Ω1 e Y(log ∆3)(M)) = h0(e Y , Ω1 e Y(M)). Since σ∗Ω1 e Y(M) is a subsheaf of Ω1 P2(2),one can view H0(e Y , Ω1 e Y(M)) as a subspace of H0(P2,Ω1 P2(2)).By [BC10-b, Lemma 5.2], any form of H0(P2,Ω1 P2(2)) can be written as ω=A(x1dx2−x2dx1) + B(x2dx3−x3dx2) + C(x1dx3−x3dx1). Evaluating at P2= (0 : 1 : 0),by Lemma 12.2 (1), we get A=B= 0.Then evaluate at P3= (1 : 0 : 0) and get C= 0. Thus we see that H0(e Y , Ω1 e Y(log ∆3)(Ke Y+L3)) = 0. Proposition 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. First we prove the following 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 Proof. H0(e Y , Ω1 e Y(log ∆1)(Ke Y+L1)) = H0(e Y , Ω1 e Y(log ∆1)(Γ1−E3)) by (14.2). Note that ∆1is the disjoint union of 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 to C1and then to 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 to N3and then to 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 the first equality holds. A similar argument shows that the second equality also holds. Proof of Proposition 15.5. There is an automorphism τ:P2→P2such that τ(P1) = P0 2, τ(P2) = P1, τ(P0 1) = P2, τ(P0 2) = P0 1. It follows that τ(P3) = P0 3, τ(P0 3) = P3.This automorphism induces an automorphism of e Yand shows that 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). Together with Lemma 15.6, it suffices to 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. Pardini, A connected component of the moduli space of surfaces with pg= 0, Topology 40 (5) (2001), 977-991. [MP04] ,A new family of surfaces with pg= 0 and K2= 3, Ann. Scient. Ec. Norm. Sup. 4es´erie, 37 (2004), 507-531. [Miy77] Y. Miyaoka, On numerical Campedelli surfaces, Complex analysis and algebraic geometry, Iwanami Shoten, Tokyo, 1977, 113-118. [Na94] D. Naie, Surfaces d’Enriques et une construction de surfaces de type g´en´eral avec pg= 0,, Math. Z. 215 (2) (1994), 269-280. [NP11] J. Neves and R. Pignatelli, Unprojection and deformations of tertiary Burniat surfaces. [PR04] S. Papadakis and M. Reid, Kustin-Miller unprojection without complexes, J. Algebraic Geometry 13 (2004), 563-577. [Par91] R. Pardini, Abelian covers of algebraic varieties, J. Reine Angew. Math. 417 (1991), 191-213. [Pet77] C. A. M. Peters, On certain examples of surfaces with pg= 0 due to Burniat, Nagoya Math. J. 66 (1977), 109-119. [Reid79] M. Reid, Surfaces with pg= 0, K2= 2, preprint (1979). [Reid85] ,Young person’s guide to canonical singularities, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), Proc. Sympos. Pure Math., 46, Part 1, Amer. Math. Soc., Providence, RI, (1987), 345-414. [Sc64] L. Schl¨afli, On the distribution of surfaces of the third order into species, 193247. [Ser06] E. Sernesi, Deformations of Algebraic Schemes, Springer, Grundlehren 334, 2006. [Weng95] L. Weng, A result on bicanonical maps of surfaces of general type, Osaka J. Math., 32 (1995), no. 2, 467-473. [Sak10] Yoshiyuki Sakamaki, Automorphism groups on normal singular cubic surfaces with no parameters, Transactions of the American Mathematical Society Volume 362, Number 5, May 2010, 2641-2666. [Zar58] O. Zariski, On the purity of the branch locus of algebraic functions, Proc. Nat. Acad. Sci. U. S. A. 44 (1958), 791-796. 77