Surfaces of general type with $p_g=1$ and $q=0$

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by  Heesang Park

A CONSTRUCTION OF SURFACES OF GENERAL TYPE
WITH p =1 AND q =0
g
9
0 HEESANGPARK,JONGILPARK,ANDDONGSOOSHIN
0
2
Abstract. Inthispaperwereconstructminimalcomplexsurfacesofgeneral
n type with pg = 1, q = 0, and 1 ≤ K2 ≤ 2 using a Q-Gorenstein smoothing
u
theory. Furthermore, we also construct a new family of simply connected
J surfaceswithpg=1,q=0,and3≤K2≤6usingthesamemethod.
9
2

]
G 1. Introduction
A
Complex surfaces of general type with p = 1 and q = 0 have been drawn
g
. attention because they provide counterexamples to the Torelli problems. All such
h
t surfaces are constructed by classical methods: Quotient and covering. Kynev [6]
a constructed a surface with K2 = 1 as a quotient of the Fermat sextic in P3 by
m
a suitable action of a group of order 6. According to Catanese [2], all minimal
[ surfaces of general type with pg = 1 and K2 = 1 are diffeomorphic and simply
1 connected. Catanese and Debarre [3] constructed surfaces with K2 = 2 by double
v coverings of the projective plane P2 or smooth minimal K3 surfaces. In fact, they
5
classified such surfaces into five classes according to the degree and the image of
9
the bicanonical map. Four of them are simply connected and the other one has a
1
5 torsion Z/2Z. Todorov [13] constructed surfaces with 2 ≤ K2 ≤ 8 by considering
. doublecoversofK3surfaces. His exampleswith3≤K2 ≤8havebigfundamental
6
groups.
0
9 Inthispaperweconstructminimalcomplexsurfacesofgeneraltypewithpg =1,
0 q = 0, and 1 ≤ K2 ≤ 6 by a Q-Gorenstein smoothing theory. In particular, we
: constructsimplyconnected surfaceswith1≤K2 ≤6andasurfacewithK2 =2and
v
i H1 =Z/2Z. Weuseasimilarmethodin[8]: Weblowupanelliptic K3surfaceina
X
suitable set of points so that we obtain a surface with a very special configuration
r of rational curves. Inside this configuration we find some disjoint chains which
a
can be contracted to special quotient singularities. These singularities admit a
localQ-Gorensteinsmoothing, which is a smoothing whose relative canonicalclass
is Q-Cartier. Finally, we prove that these local smoothings can be glued to a
global Q-Gorenstein smoothing of the whole singular surface by showing that the
obstructionspaceofaglobalsmoothingiszero. Thenitisnotdifficulttoshowthat
a general fiber of the smoothing is the desired surface. The key ingredient of this
paper is to develop a new method for proving that the obstruction space is zero
because the Lee-Park’s method in [8] for a computation of the obstruction space
cannot be applied to our cases; Section 3.
Date:June1,2009.
2000 Mathematics Subject Classification. Primary14J29; Secondary14J10,14J17, 53D05.
Keywords and phrases. Q-Gorensteinsmoothing,rationalblow-downsurgery,surfaceofgen-
eraltype.
1
2 HEESANGPARK,JONGILPARK,ANDDONGSOOSHIN
InSection2wereconstructcomplexsurfaceswithp =1,q =0,andK2 =2: A
g
simply connected surface and a surface with H = Z/2Z by using a (generalized)
1
rational blow-down surgery. We prove in Section 3 that the obstruction space of a
globalsmoothingofthesingularsurfaceconstructedinSection2iszero. Inthefinal
section, we also construct various examples of simply connected complex surfaces
satisfying p =1, q =0, and 1≤K2 ≤6.
g
Acknowledgement. The authors would like to thank Professor Yongnam Lee for
somevaluablecommentsonthefirstdraftofthisarticle. JongilParkwassupported
bytheKoreaResearchFoundationGrantfundedbytheKoreanGovernment(KRF-
2008-341-C00004). He alsoholdsa jointappointmentatKIASandinthe Research
Institute of Mathematics, SNU.
2. A Construction of surfaces with p =1, q =0, and K2 =2
g
In this section we construct a simply connected minimal surface of generaltype
with p = 1, q = 0, and K2 = 2. We start with a special rational elliptic surface
g
E(1). Let L , L , L , and A be lines in P2 and let B be a smooth conic in P2
1 2 3
intersecting as in Figure 1(A). We consider a pencil of cubics {λ(L +L +L )+
1 2 3
µ(A+B) | [λ : µ] ∈ P1} in P2 generated by two cubic curves L +L +L and
1 2 3
A+B, which has 4 base points, say, p, q, r and s. In order to obtain an elliptic
fibration over P1 from the pencil, we blow up three times at p and r, respectively,
andtwice at s, including infinitely near base-pointsat eachpoint. We performone
furtherblowing-upatthebasepointq. Byblowing-uptotallyninetimes,weresolve
all base points (including infinitely near base-points) of the pencil and we then get
a rational elliptic surface E(1) with an I -singular fiber, an I -singular fiber, two
8 2
nodal singular fibers, and four sections; Figure 1(B).
A
p
b
B q
L b L3
1
r b L sb
2
(a)Apencil (b)E(1)
Figure 1: A rational elliptic surface E(1)
Let Y be a double coverof the rationalelliptic surfaceE(1) branchedalongtwo
general fibers. Then the surface Y is an elliptic K3 surface with two I -singular
8
fibers, two I -singular fibers, four nodal singular fibers, and four sections; Fig-
2
ure2(A). In the following constructionwe use only oneI -singularfiber, one nodal
8
singular fiber, and three sections; Figure 2(B).
Let τ : V → Y be the blowing-up at the node of the nodal singular fiber and
let E be the exceptional divisor of τ. Since K = 0, we have K = E. Let
Y V
A CONSTRUCTION OF SURFACES OF GENERAL TYPE WITH pg=1 AND q=0 3
(a)ThefibrationstructureofY
(b)ApartofY
Figure 2: An elliptic K3 surface Y
D = D +···+D be the part of the I -singular fiber. Let S (i = 1,2,3) be the
1 6 8 i
sections of the fibration V → P1 and set S = S +S +S . Let F be the proper
1 2 3
transform of the nodal fiber of the K3 surface Y; Figure 3.
S1 ×2
b
b D4b bD3 D2 S2
−4F
D5 D1 E
−1
D6
S3
2
Figure 3: A surface V =Y♯P
We blow up the surface V three times totally at the three marked points • and
blow up twice at the marked point . We then get a surface Z; Figure 4. There
exist three disjoint linear chains of PJ1’s in Z:
−3 −6 −2 −3 −2 −4 −2 −2 −3 −2 −4
◦ − ◦ − ◦ − ◦ − ◦, ◦ − ◦ − ◦ − ◦ − ◦, ◦.
Main construction. By applying Q-Gorenstein smoothing theory to the surface
Z as in [8, 11, 12], we construct a complex surface of general type with p = 1,
g
4 HEESANGPARK,JONGILPARK,ANDDONGSOOSHIN
−3
−1 −2
−4 −−14 −1
−2
−1−3 −3
−6
−2
−2 −1
−2
2
Figure 4: A surface Z =Y♯6P
q = 0, and K2 = 2. That is, we first contract the three chains of P1’s from the
surface Z so that it produces a normalprojective surfaceX with three permissible
singular points. In Section 3 we will show that the singular surface X has a global
Q-Gorensteinsmoothing. Let X be a generalfiber ofthe Q-Gorensteinsmoothing
t
of X. Since X is a (singular) surface with p =1, q =0, and K2 =2, by applying
g
general results of complex surface theory and Q-Gorenstein smoothing theory, one
may conclude that a general fiber X is a complex surface of general type with
t
p = 1, q = 0, and K2 = 2. Furthermore, it is not difficult to show that a general
g
fiberX isminimalandsimplyconnectedbyusingasimilartechniquein[8,11,12].
t
Remark. CataneseandDebarre[3]provedthatsurfacesofgeneraltypewithp =1,
g
q =0,andK2 =2aredividedintofivetypesaccordingtothedegreeandtheimage
of the bicanonical map. Four of them are simply connected and the other one has
atorsionZ/2Z. It is aninteresting problemto determine inwhichclass thesimply
connected example constructed in this section is contained. We leave this question
for the future research.
2.1. An example with K2 = 2 and H = Z/2Z. We construct a minimal com-
1
plex surface of general type with p = 1, q = 0, K2 = 2, and H = Z/2Z. Let A,
g 1
L (i = 1,2,3) be lines on the projective plane P2 and B a nonsingular conic on
i
P2 which intersect as in Figure 5(A). Consider a pencil of cubics generated by the
two cubics A+B and L +L +L . By resolving the base points of the pencil
1 2 3
of cubics including infinitely near base-points, we obtain a rational elliptic surface
E(1) with an I -singular fiber, an I -singular fiber, an cusp singular fiber, a nodal
7 2
singular fiber, and five sections as in Figure 5(B).
Let Y be a double cover of the rational elliptic surface E(1) branched along
two general fibers near the cusp singular fiber. Then Y is an elliptic K3 surface
with two I -singular fibers, two I -singular fibers, two cusp singular fibers, two
7 2
nodal singular fibers, and five sections. We use only two I -singular fibers, two
7
I -singularfibers,andtwosections; Figure6(A). We blow upeighttimes totally at
2
the eight marked points •. We then get a surface Z; Figure 6(B). There exist six
disjoint linear chains of P1’s in Z:
−5 −3 −2 −2 −5 −3 −2 −2 −3 −2 −3 −3 −2 −3 −4 −4
◦ − ◦ − ◦ − ◦, ◦ − ◦ − ◦ − ◦, ◦ − ◦ − ◦, ◦ − ◦ − ◦, ◦, ◦
We now briefly explain how to prove that the complex surface obtained by the
above configuration has H = Z/2Z. Let L (i = 1,...,6), M (i = 1,...,6), N
1 i j k
(k =1,2) be the parts of the I -singular fibers and the I -singular fibers of the K3
7 2
A CONSTRUCTION OF SURFACES OF GENERAL TYPE WITH pg=1 AND q=0 5
A
B
L
L 3
1
L
2
(a)Apencil (b)E(1)
Figure 5: A rational elliptic surface E(1) for K2 =2 and H =Z/2Z
1
S1 b b
N1 N2
b L6 bM3
L5 M4 M2
L4 M5 M1
L1 L3 b M6
L2 b
S2 S4
b b
S3
(a)Y
−1 −1 −5
−1
−3
−3 −2 −−31 −2
−2 −3 −2
−4 −4
−1 −2
−2
−3 −3
−1
−5
−1 −1
(b)Z=Y♯8P2
Figure 6: An example with H =Z/2Z
1
surface Y and let S (l =1,...,4) the four sections of Y; Figure 6(A). The matrix
l
whose entries are the intersection numbers of L , M , N , and S is invertible.
i j k l
Therefore the Picard number of the K3 surface Y is at least 18. Furthermore,
during the fiber sum surgery, four new homology elements, say T ,...,T , are
1 4
constructed. Twoelementsamongthefourelementsarerepresentedbytoriandthe
other two elements are represented by spheres. Therefore the homology H (Y,Z)
2
aregeneratedbytheeighteen(−2)-curvesL ,M ,N ,S ,andthefournewelements
i j k l
T . Hence the homology H (Z,Z) are generated by the proper transforms of L ,
m 2 i
6 HEESANGPARK,JONGILPARK,ANDDONGSOOSHIN
M ,N ,S ,T ,andthe eightexceptionalcurves. Then,byusingasimilarmethod
j k l m
in[7, 10],it is notdifficult toshow thatthe complex surfaceobtainedbythe above
configuration has H =Z/2Z.
1
3. The Existence of Q-Gorenstein smoothings
This section is devoted to the proof of the following theorem.
Theorem 3.1. The singular surface X constructed in the main construction in
Section 2 has a global Q-Gorenstein smoothing.
For this, we first briefly review some basic facts on Q-Gorenstein smoothing
theoryfornormalprojectivesurfaceswithspecialquotientsingularities(referto[8]
for details).
Definition. Let X be a normal projective surface with quotient singularities. Let
X → ∆ (or X/∆) be a flat family of projective surfaces over a small disk ∆. The
one-parameter family of surfaces X →∆ is called a Q-Gorenstein smoothing of X
if it satisfies the following three conditions;
(i) the general fiber X is a smooth projective surface,
t
(ii) the central fiber X is X,
0
(iii) the canonical divisor KX/∆ is Q-Cartier.
AQ-Gorensteinsmoothingforagermofaquotientsingularity(X ,0)isdefined
0
similarly. A quotient singularity which admits a Q-Gorenstein smoothing is called
a singularity of class T.
Proposition 3.2 ([5, 9]). Let (X ,0) be a germ of two dimensional quotient sin-
0
gularity. If (X ,0) admits a Q-Gorenstein smoothing over the disk, then (X ,0) is
0 0
either a rational double point or a cyclic quotient singularity of type 1 (1,dna−1)
dn2
for some integers a,n,d with a and n relatively prime.
−4 −3 −2 −2 −2 −3
Proposition 3.3 ([5, 9]). (1) The singularities ◦ and ◦ − ◦ − ◦ −···− ◦ − ◦
are of class T.
(2) If the singularity −◦b1−···−−◦br is of class T, then so are
−◦2−−◦b1−···−−b◦r−1−−b◦r−1 and −b◦1−1−−◦b2−···−−◦br−−◦2.
(3) Every singularity of class T that is not a rational double point can be obtained
by starting with one of the singularities described in (1) and iterating the steps
described in (2).
Let X be a normal projective surface with singularities of class T. Due to the
resultofKoll´arandShepherd-Barron[5],thereisaQ-Gorensteinsmoothinglocally
foreachsingularityofclassT onX. The naturalquestionariseswhether this local
Q-GorensteinsmoothingcanbeextendedovertheglobalsurfaceX ornot. Roughly
geometric interpretationis the following: Let ∪ V be an open coveringof X such
α α
that each V has at most one singularity of class T. By the existence of a local Q-
α
Gorensteinsmoothing,thereisaQ-GorensteinsmoothingV /∆. Thequestionisif
α
these families glueto a globalone. The answercanbe obtainedby figuringoutthe
obstructionmapofthe sheavesofdeformationTi =Exti (Ω ,O )for i=0,1,2.
X X X X
For example, if X is a smooth surface, then T0 is the usual holomorphic tangent
X
A CONSTRUCTION OF SURFACES OF GENERAL TYPE WITH pg=1 AND q=0 7
sheaf T and T1 =T2 =0. By applying the standard result of deformations to a
X X X
normal projective surface with quotient singularities, we get the following
Proposition 3.4 ([14, §4]). Let X be a normal projective surface with quotient
singularities. Then
(1) The first order deformation space of X is represented by the global Ext 1-group
T1 =Ext1 (Ω ,O ).
X X X X
(2) The obstruction lies in the global Ext 2-group T2 =Ext2 (Ω ,O ).
X X X X
Furthermore, by applying the general result of local-global spectral sequence of
ext sheaves to deformation theory of surfaces with quotient singularities so that
Ep,q =Hp(Tq)⇒Tp+q, and by Hj(Ti )=0 for i,j ≥1, we also get
2 X X X
Proposition 3.5 ([9, 14]). Let X be a normal projective surface with quotient
singularities. Then
(1) We have the exact sequence
0→H1(T0)→T1 →ker[H0(T1)→H2(T0)]→0
X X X X
where H1(T0) represents the first order deformations of X for which the sin-
X
gularities remain locally a product.
(2) If H2(T0)=0, every local deformation of the singularities may be globalized.
X
Asmentionedabove,thereisalocalQ-Gorensteinsmoothingforeachsingularity
ofX duetotheresultofKoll´arandShepherd-Barron[5]. Henceitremainstoshow
that every local deformation of the singularities can be globalized. Note that the
following proposition tells us a sufficient condition for the existence of a global
Q-Gorenstein smoothing of X.
Proposition 3.6 ([8]). Let X be a normal projective surface with singularities
of class T. Let π : V → X be the minimal resolution and let A be the reduced
exceptional divisor. Suppose that H2(T (−logA)) = 0. Then H2(T0) = 0 and
V X
there is a Q-Gorenstein smoothing of X.
Furthermore, the proposition above can be easily generalized to any log resolu-
tionofX bykeepingthevanishingofcohomologiesunderblowingupatthepoints.
It is obtained by the following well-known result.
Proposition 3.7 ([4, §1]). Let V be a nonsingular surface and let A be a simple
′
normal crossing divisor in V. Let f :V →V be a blowing up of V at a point p of
A. Set A′ =f−1(A)red. Then h2(TV′(−logA′))=h2(TV(−logA)).
Therefore Theorem 3.1 follows from Proposition 3.6 and the following proposi-
tion:
Proposition 3.8. H2(T (−log(D+S+F)))=H0(Ω (log(D+S+F))(E))=0.
V V
The idea of the proof is as follows: There is an exact sequence of locally free
sheaves
6 3
0→Ω →Ω (log(D+S+F))→ O ⊕ O ⊕O →0.
V V Di Si F
Mi=1 Mi=1
By tensoring O (E), we have an exact sequence
V
6 3
0→Ω (E)→Ω (log(D+S+F))(E)→ O ⊕ O ⊕O (E)→0.
V V Di Si F
Mi=1 Mi=1
8 HEESANGPARK,JONGILPARK,ANDDONGSOOSHIN
Since H0(Ω (E)) = 0, the proof of Proposition 3.8 is done if we show that the
V
connecting homomorphism
6 3
H0(O )⊕ H0(O )⊕H0(O (E))→H1(Ω (E))
Di Si F V
Mi=1 Mi=1
is injective.
The proof of Proposition 3.8 begins with the following Lemma. We have a
commutative diagram
6 3
H0(O )⊕ H0(O )⊕H0(O ) c1 // H1(Ω ) (3.1)
Di Si F V
Mi=1 Mi=1
β1
(cid:15)(cid:15)
6 3 (cid:15)(cid:15)
H0(O )⊕ H0(O )⊕H0(O (E)) d1 //H1(Ω (E))
Di Si F V
Mi=1 Mi=1
Lemma 3.9. The composition map
6 3
β ◦c : H0(O )⊕ H0(O )⊕H0(O )→H0(Ω (E))
1 1 Di Si F V
Mi=1 Mi=1
is injective.
Proof. Note that the map c is the first Chern class map. Since the intersection
1
matrix, whose entries are the intersection numbers of D (i = 1,...,6), S (j =
i j
1,2,3), and F, is invertible, their images by the map c are linearly independent.
1
Therefore the map c is injective.
1
Consider the commutative diagram
0
(cid:15)(cid:15)
H0(Ω ⊗O (E))
V E
δ
6 3 (cid:15)(cid:15)
H0(O )⊕ H0(O )⊕H0(O ) c1 //H1(Ω )
Di Si F V
Mi=1 Mi=1
β1
(cid:15)(cid:15)
6 3 (cid:15)(cid:15)
H0(O )⊕ H0(O )⊕H0(O (E)) d1 // H1(Ω (E))
Di Si F V
Mi=1 Mi=1
where the vertical sequence is induced from the exact sequence
0→H0(Ω ⊗O (E))−→δ H1(Ω )−β→1 H1(Ω (E))−→γ H1(Ω ⊗O (E))→0.
V E V V V E
A CONSTRUCTION OF SURFACES OF GENERAL TYPE WITH pg=1 AND q=0 9
Claim: Theconnectinghomomorphismδ :H0(Ω ⊗O (E))→H1(Ω )isthefirst
V E V
Chern class map of O (E): Since H0(Ω ⊗O (E))=0, we have an isomorphism
V E E
∼
H0(O (−E)⊗O (E))−=→H0(Ω ⊗O (E)).
E E V E
Here the above map is given by z ⊗ 1 7→ dzα , where z is a local equation of
α zα n zα o α
E. Therefore the connecting homomorphism δ : H0(Ω ⊗O (E)) → H1(Ω ) is
V E V
given by
dz dz dz z z
α α β α α
7→ − = d / ,
(cid:26) z (cid:27) (cid:26) z z (cid:27) (cid:26) (cid:18)z (cid:19) (cid:18)z (cid:19)(cid:27)
α α β β β
which is the first Chern class map of O (E). This proves the claim.
V
Since E is the exceptional divisor, it is independent of the other divisors in
H1(Ω ). Thereforeimc ∩imδ =0;hencethe compositionmapβ ◦c isinjective.
V 1 1 1
(cid:3)
′
We now concentrate on the following restriction map d of the map d in (3.1):
1 1
d′ :H0(O (E))→H1(Ω (E)),
1 F V
which is also regarded as a connecting homomorphism induced from the exact
sequence
0→Ω →Ω (logF)→O →0
V V F
tensored by O (E).
V
Lemma 3.10. We have H0(Ω (logF)(E))=0.
V
Proof. Let C be a general fiber of the elliptic fibration φ : Y → P1. By the
projection formula we have
H0(Ω (logF)(E))⊆H0(Ω (F +E))=H0(Ω (τ∗C−E))
V V V
⊆H0(Ω (τ∗C))=H0(Y,Ω (C)).
V Y
OntheotherhanditisnotdifficulttoshowthatH0(Y,ΩY(C))=H0(P1,ΩP1(1))=
0 by a similar method in [8, Lemma 2]. Therefore the assertion follows. (cid:3)
By the above lemma, we have the following commutative diagram of exact se-
quences:
0 0 (3.2)
(cid:15)(cid:15) (cid:15)(cid:15)
H0(ΩV ⊗OE(E)) H0(ΩV(logF)⊗OE(E))
δ
(cid:15)(cid:15) (cid:15)(cid:15)
0 //H0(OF) // H1(ΩV) α1 //H1(ΩV(logF)) //0
β1 β2
0 //H0(OF(cid:15)(cid:15) (E)P)PPPγP◦dPd′1P′1PPPP// HP((1(ΩV(cid:15)(cid:15)(cid:15)(cid:15)γ(E)) α2 // H1(ΩV(lo(cid:15)(cid:15)(cid:15)(cid:15)gF)(E)) //0
H1(ΩV ⊗OE(E)) H1(ΩV(logF)⊗OE(E))
(cid:15)(cid:15) (cid:15)(cid:15)
0 0
10 HEESANGPARK,JONGILPARK,ANDDONGSOOSHIN
Lemma 3.11. h0(Ω (logF)⊗O (E))=1.
V E
Proof. Let
r :Ω (logF)⊗O →Ω ⊗O (F)
V E E E
be the restriction to the subsheaf Ω (logF) of the restriction map r⊗id:Ω | ⊗
V V E
O (F)→Ω ⊗O (F).
E E E
Claim. kerr =O (−E): Let z , z be the local equations of F, E, respectively.
E 1 2
Note that the restriction map r :Ω | →Ω is defined by
V E E
r :adz +bdz =adz .
1 2 1
Hence the map r is defined by
dz dz
1 1
r a +bdz =a .
2
(cid:18) z (cid:19) z
1 1
It follows that kerr =kerr =O (−E).
E
By the claim we have
ker[r⊗id:Ω (logF)⊗O (E)→Ω ⊗O (F +E)]=O (−E)⊗O (E)=O .
V E E E E E E
Hence we have the exact sequence
0→H0(O )→H0(Ω (logF)⊗O (E))→H0(im(r⊗id)).
E V E
Since H0(im(r⊗id))⊂H0(Ω ⊗O (F +E))=0, the result follows. (cid:3)
E E
Lemma 3.12. The composition map
γ◦d′ :H0(O (E))→H1(Ω ⊗O (E))
1 F V E
is surjective.
Proof. Onthesecondcolumnin(3.2),wehaveh0(Ω ⊗O (E))=1,h1(Ω )=21,
V E V
h1(Ω ⊗O (E))=2. Henceh1(Ω (E))=22. Sinceh0(O )=1andh0(O (E))=
V E V F F
3, it follows from the first and the second columns that h1(Ω (logF)) = 20 and
V
h1(Ω (logF)(E)) = 19. Therefore, on the third column, we have h1(Ω (logF)⊗
V V
O (E)) = 0 because h0(Ω (logF)⊗O (E)) = 1 by Lemma 3.11. Hence β is
E V E 2
surjective.
Note that H0(O ) ⊆ β (H1(Ω ))∩kerα in H1(Ω (E)). On the other hand,
F 1 V 2 V
since α and β are surjective, we have
1 2
(α ◦β )(H1(Ω ))=(β ◦α )(H1(Ω ))=H1(Ω (logF)(E)).
2 1 V 2 1 V V
Therefore dim[β (H1(Ω ))∩kerα ]=1; hence H0(O )=β (H1(Ω ))∩kerα in
1 V 2 F 1 V 2
H1(Ω (E)). Thus
V
dim[β (H1(Ω ))∩H0(O (E))]=1.
1 V F
Since kerγ = imβ , we have dim[kerγ∩H0(O (E))] = 1; hence the composition
1 F
γ◦d′ is surjective because h0(O (E))=3 but h1(Ω ⊗O (E))=2. (cid:3)
1 F V E