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arXiv:0708.3319v3 [math.AG] 15 Jun 2008
A CONSTRUCTION OF HORIKAWA SURFACE VIA
QGORENSTEIN SMOOTHINGS
YONGNAM LEE AND JONGIL PARK
Abstract. In this article we prove that FintushelStern’s construction of Horikawa
surface, which is obtained from an elliptic surface via a rational blowdown surgery in
smooth category, can be performed in complex category. The main technique involved
is QGorenstein smoothings.
1. Introduction
As an application of a rational blowdown surgery on 4manifolds, R. Fintushel and
R. Stern showed that Horikawa surface H(n) can be obtained from an elliptic surface
E(n) via a rational blowdown surgery in smooth category [3]. Note that Horikawa
surface H(n) is defined as a double cover of a Hirzebruch surface Fn−3branched over
6C0+ (4n − 8)f, where C0is a negative section and f is a fiber of Fn−3.
In this article we show that a rational blowdown surgery to obtain Horikawa surface
can be performed in fact in complex category.
FintushelStern’s topological construction [3] of Horikawa surface H(n) to give a complex
structure on it. The main technique we use in this paper is QGorenstein smoothings.
Note that QGorenstein smoothing theory developed in deformation theory in last thirty
years is a very powerful tool to construct a nonsingular surface of general type. The
basic scheme is the following: Suppose that a projective surface contains several disjoint
chains of curves representing the resolution graphs of special quotient singularities. Then,
by contracting these chains of curves, we get a singular surface X with special quotient
singularities. And then we investigate the existence of a QGorenstein smoothing of X. It
is known that the cohomology H2(T0
X) contains the obstruction space of a QGorenstein
smoothing of X. That is, if H2(T0
X) = 0, then there is a QGorenstein smoothing of X.
For example, we recently constructed a simply connected minimal surface of general type
with pg= 0 and K2= 2 by proving the cohomology H2(T0
cohomology H2(T0
X) is not zero and it is a very difficult problem to determine whether
there exists a QGorenstein smoothing of X. In this article we also give a family of
examples which admit QGorenstein smoothings even though the cohomology H2(T0
does not vanish. Our main technique is a QGorenstein smoothing theory with a cyclic
group action. It is briefly reviewed and developed in Section 2.
The sketch of our construction whose details are given in Section 3 is as follows: We
first construct a simply connected relatively minimal elliptic surface E(n) (n ≥ 5) with
That is, we reinterpret algebraically
X) = 0 [9]. But, in general, the
X)
Date: June 1, 2008.
2000 Mathematics Subject Classification. Primary 14J29; Secondary 14J10, 14J17.
Key words and phrases. Horikawa surface, QGorenstein smoothing, rational blowdown.
1
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2 YONGNAM LEE AND JONGIL PARK
a special fiber, which contains two linear chains of configurations of P1’s
−n
◦
Un−3−
−2
◦
Un−4−
−2
◦
Un−5− ··· −
−2
◦
U1.
We construct this kind of an elliptic surface E(n) explicitly by using a double cover of
a blowingup of Hirzebruch surface Fnbranched over a special curve. The double cover
of Fnhas two rational double points A1and A2n−9. Then its minimal resolution is an
elliptic surface E(n) which has an I2n−6as a special fiber. Now we contract these two
linear chains of configurations of P1’s to produce a normal projective surface Xnwith two
special quotient singularities, both singularities are of type
apply QGorenstein smoothing theory with a cyclic group action developed in Section 2
for Xnin order to get our main result which is following.
1
(n−2)2(1,n − 3). Finally, we
Theorem 1.1. The projective surface Xn obtained by contracting two disjoint config
urations Cn−2 from an elliptic surface E(n) admits a QGorenstein smoothing of two
quotient singularities all together, and a general fiber of the QGorenstein smoothing is
Horikawa surface H(n).
Acknowledgements. The authors would like to thank Ronald Fintushel and Ronald Stern
for helpful comments on Horikawa surfaces during the Conference of Algebraic Surfaces
and 4manifolds held at KIAS. The authors also wish to thank Roberto Pignatelli for
valuable discussions to prove Proposition 3.1. Yongnam Lee was supported by Korea
Research Foundation Grant funded by the Korean Government (KRF2005070C00005).
He would like to thank Meng Chen for his generous hospitality during his visit Fudan
University, where some of the paper was completed. Jongil Park was supported by SBS
Foundation Grant in 2007 and he also holds a joint appointment in the Research Institute
of Mathematics, Seoul National University.
2. QGorenstein smoothing
In this section we briefly review a theory of QGorenstein smoothing for projective
surfaces with special quotient singularities, which is a key technical ingredient in our
main construction.
Definition.
X → ∆ (or X/∆) be a flat family of projective surfaces over a small disk ∆. The
oneparameter family of surfaces X → ∆ is called a QGorenstein smoothing of X if it
satisfies the following three conditions;
(i) the general fiber Xtis a smooth projective surface,
(ii) the central fiber X0is X,
(iii) the canonical divisor KX/∆is QCartier.
Let X be a normal projective surface with quotient singularities.Let
A QGorenstein smoothing for a germ of a quotient singularity (X0,0) is defined
similarly. A quotient singularity which admits a QGorenstein smoothing is called a
singularity of class T.
Proposition 2.1 ([8, 12, 16]). Let (X0,0) be a germ of two dimensional quotient singu
larity. If (X0,0) admits a QGorenstein smoothing over the disk, then (X0,0) is either
a rational double point or a cyclic quotient singularity of type
integers a,n,d with a and n relatively prime.
1
dn2(1,dna − 1) for some
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A CONSTRUCTION OF HORIKAWA SURFACE VIA QGORENSTEIN SMOOTHINGS3
Proposition 2.2 ([8, 12, 17]).
are of class T.
(1) The singularities
−4
◦ and
−3
◦ −−2
◦ −−2
◦ −···−−2
◦ −−3
◦
(2) If the singularity
−b1
◦ − ··· −
−br
◦ is of class T, then so are
−2
◦ −
−b1
◦ − ··· −
−br−1
◦−
−br−1
◦ and
−b1−1
◦−
−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 result
of Koll´ ar and ShepherdBarron [8], there is a QGorenstein smoothing locally for each
singularity of class T on X (see Proposition 2.5). The natural question arises whether
this local QGorenstein smoothing can be extended over the global surface X or not.
Roughly geometric interpretation is the following: Let ∪αVαbe an open covering of X
such that each Vαhas at most one singularity of class T. By the existence of a local
QGorenstein smoothing, there is a QGorenstein smoothing Vα/∆. The question is if
these families glue to a global one. The answer can be obtained by figuring out the
obstruction map of the sheaves of deformation Ti
example, if X is a smooth surface, then T0
Xis the usual holomorphic tangent sheaf TX
and T1
X= 0. By applying the standard result of deformations [10, 14] to a normal
projective surface with quotient singularities, we get the following
X= Exti
X(ΩX,OX) for i = 0,1,2. For
X= T2
Proposition 2.3 ([16], §4). Let X be a normal projective surface with quotient singu
larities. Then
(1) The first order deformation space of X is represented by the global Ext 1group
T1
X(ΩX,OX).
(2) The obstruction lies in the global Ext 2group T2
X= Ext1
X= Ext2
X(ΩX,OX).
Furthermore, by applying the general result of localglobal spectral sequence of ext
sheaves ([14], §3) to deformation theory of surfaces with quotient singularities so that
Ep,q
2
= Hp(Tq
X) ⇒ Tp+q
X, and by Hj(Ti
X) = 0 for i,j ≥ 1, we also get
Proposition 2.4 ([12, 16]). Let X be a normal projective surface with quotient singu
larities. Then
(1) We have the exact sequence
0 → H1(T0
where H1(T0
X) represents the first order deformations of X for which the singu
larities remain locally a product.
(2) If H2(T0
X) = 0, every local deformation of the singularities may be globalized.
The vanishing H2(T0
where V is the minimal resolution of X and E is the reduced exceptional divisors. Note
that every singularity of class T has a local QGorenstein smoothing by Proposition 2.5
below.
Let X be a normal projective surface with singularities of class T. Our concern is
to understand QGorenstein smoothings in T1
These special deformations can be constructed via local index one cover. Let U ⊂ X
be an analytic neighborhood with an index one cover U′. For the case of the field C,
X) → T1
X→ ker[H0(T1
X) → H2(T0
X)] → 0
X) = 0 can be obtained via the vanishing of H2(TV(−log E)),
X, not the whole first order deformations.
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4 YONGNAM LEE AND JONGIL PARK
this index one cover is unique up to isomorphism. The first order deformations which
associate QGorenstein smoothings can be realized as the invariant part of T1
sheaves˜T1
Xare defined by the index one covering stack and by the ´ etale sites [5]. The
first order deformation of a QGorenstein smoothing of singularities of class T is expressed
by the cohomology H0(˜T1
X) [5, 7, 8]. By the help of the birational geometry in threefolds
and their applications to deformations of surface singularities, the following proposition
is obtained. Note that the cohomology H0(˜T1
U′. The
X) is given explicitly as follows.
Proposition 2.5 ([8, 12]).
and consider a map π : Y/µn→ Cd, where Y ⊂ C3× Cdis the hypersurface of
equation uv − ydn=?d−1
acts on Y by
µn∋ ξ : (u,v,y,t0,...,td−1) → (ξu,ξ−1v,ξay,t0,...,td−1)
and π is the factorization to the quotient of the projection Y → Cd. Then π
is a QGorenstein smoothing of the cyclic singularity of a germ (X0,0) of type
1
dn2(1,dna−1). Moreover every QGorenstein smoothing of (X0,0) is isomorphic
to the pullback of π for some germ of holomorphic map (C,0) → (Cd,0).
(2) Let X be a normal projective surface with singularities of class T. Then
(1) Let a,d,n > 0 be integers with a,n relatively prime
k=0tkykn; t0,...,td−1are linear coordinates over Cd, µn
H0(˜T1
X) =
?
p∈singular points of X
C⊕dp
p
where a singular point p is of type
1
dpn2(1,dpan − 1) with (a,n) = 1.
Theorem 2.1 ([9]). Let X be a normal projective surface with singularities of class T.
Let π : V → X be the minimal resolution and let E be the reduced exceptional divisors.
Suppose that H2(TV(−log E)) = 0. Then H2(T0
smoothing of X.
X) = 0 and there is a QGorenstein
As we see in Theorem 2.1 above, if H2(T0
smoothing of X. For example, we constructed a simply connected minimal surface of
general type with pg = 0 and K2= 2 by proving the cohomology H2(T0
But, in general, the cohomology H2(T0
X) is not zero and it is a very difficult problem to
determine whether there exists a QGorenstein smoothing of X. Hence, in the case that
H2(T0
X) ?= 0, we have to develop another technique in order to investigate the existence
of QGorenstein smoothings. Even though we do not know whether such a technique
exists in general, if X is a normal projective surface with singularities of class T which
admits a cyclic group with some nice properties, then we are able to show that it admits
a QGorenstein smoothing. Explicitly, we get the following theorem.
X) = 0, then there is a QGorenstein
X) = 0 [9].
Theorem 2.2. Let X be a normal projective surface with singularities of class T. As
sume that a cyclic group G acts on X such that
(1) Y = X/G is a normal projective surface with singularities of T,
(2) pg(Y ) = q(Y ) = 0,
(3) Y has a QGorenstein smoothing,
(4) the map σ : X → Y induced by a cyclic covering is flat, and the branch locus D
(resp. the ramification locus) of the map σ : X → Y is an irreducible nonsingular
curve lying outside the singular locus of Y (resp. of X), and
(5) H1(Y,OY(D)) = 0.
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A CONSTRUCTION OF HORIKAWA SURFACE VIA QGORENSTEIN SMOOTHINGS5
Then there exists a QGorenstein smoothing of X that is compatible with a QGorenstein
smoothing of Y . And the cyclic covering extends to the QGorenstein smoothing.
Proof. Let Y → ∆ be a QGorenstein smoothing of Y , and let Ytbe a general fiber of the
QGorenstein smoothing. By the semicontinuity, we have pg(Yt) = q(Yt) = 0. The base
change theorem and Leray spectral sequence imply that H1(Y,OY) = H2(Y,OY) = 0. It
gives an isomorphism r0: Pic(Y) → Pic(Y ) and an injective map rt: Pic(Y) → Pic(Yt)
(Lemma 2 in [12]). The vanishing H1(Y,OY(D)) = 0 ensures that the deformation
of Y can be lifted to the deformation of the pair (Y,D), i.e. the branch divisor D is
extended to Dtin Yt. Since the divisor D is nonsingular, Dtis also nonsingular. And
the flatness of the map ensures that the divisor L which is the data of the cyclic cover,
i.e. L⊗G ∼= D, is extended to Ltwith L⊗G
t
to the QGorenstein smoothing of Y .
∼= Dt. Hence, the cyclic covering extends
?
3. A construction of Horikawa surface
Let E(n) be a simply connected relatively minimal elliptic surface with a section and
with c2= 12n. Then there is only one up to diffeomorphism such an elliptic surface and
the canonical class is given by KE(n)= (n−2)C, where C is a general fiber of an elliptic
fibration. Hence each section is a nonsingular rational curve whose selfintersection
number is −n. Assume that n ≥ 4 and let Cn−2 denote a simply connected smooth
4manifold obtained by pluming the (n − 3) disk bundles over the 2sphere according to
the linear diagram
−n
◦
Un−3−
Assume that an elliptic surface E(n) has two configurations Cn−2such that all em
bedded 2spheres Ui are holomorphic curves (We show the existence of such an E(n)
later). Let Y′
nbe a normal projective surface obtained by contracting one configura
tion Cn−2from E(n). Then Y′
ndoes not admit a QGorenstein smoothing because it
violates Noether inequality (Corollary 7.5 in [3]). In fact, it does not satisfy the vanish
ing condition in the hypothesis of Theorem 2.1, that is, we have H2(E(n),TE(n)) ?= 0:
Let h : E(n) → P1be an elliptic fibration. Assume that C is a general fiber of the
map h. We have an injective map 0 → h∗ΩP1 → ΩE(n)and the map induces an in
jection H0(P1,ΩP1(n − 2)) ֒→ H0(E(n),ΩE(n)((n − 2)C)) by tensoring (n − 2)C on
0 → h∗ΩP1 → ΩE(n). Since KE(n)= (n − 2)C, the cohomology H0(E(n),ΩE(n)(KE(n)))
is not zero. Hence the Serre duality implies that H2(E(n),TE(n)) is not zero.
Next, let Xnbe a normal projective surface obtained by contracting two disjoint con
figurations Cn−2from E(n), and we want to investigate the existence of a QGorenstein
smoothing of Xn. As a warmingup, we first investigate n = 4 case.
−2
◦
Un−4−
−2
◦
Un−5− ··· −
−2
◦
U1.
Example. R. Gompf constructed a family of symplectic 4manifolds by taking a fiber
sum of other symplectic 4manifolds [4]. To recall Gompf’s example briefly, we start
with a simply connected relatively minimal elliptic surface E(4) with a section and with
c2 = 48. It is known that E(4) admits nine rational (−4)curves as disjoint sections.
Rationally blowingdown n (−4)curves of E(4) is the same as the normal connected
sum of E(4) with n copies of P2by identifying a conic in each P2with one (−4)curve in
E(4). Let us denote this 4manifold by W4,n. Then the manifold W4,1does not admit
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6 YONGNAM LEE AND JONGIL PARK
any complex structure because it violates the Noether inequality. But we will show that
W4,2admits a complex structure using a QGorenstein smoothing theory. For this, let us
first denote the singular projective surface obtained by contracting n (−4)sections from
E(4) by W′
4,2has a QGorenstein smoothing. The reason
is following: Consider E(4) as a double cover of Hirzebruch surface F4branched over an
irreducible nonsingular curve D in the linear system 4(C0+4f), where C0is a negative
section and f is a fiber of F4. Then H1(F4,OF4(D)) = 0: Since pg(F4) = q(F4) = 0,
4,n. And then we claim that W′
H1(F4,OF4(D)) ≃ H1(D,OD(D)) ≃ H0(D,OD(KD− D))∨.
And degKD− D2= 4(C0+ 4f)(2C0+ 10f) − 16(C0+ 4f)2= −24 < 0 implies that
H0(D,OD(KD− D)) = 0. Since D does not intersect C0, W′
coneˆF4which is a contraction of C0from F4. This implies that the map σ induced by a
double cover is flat and H1(Y,OY(D)) = 0. Note thatˆF4has a QGorenstein smoothing
whose general fiber is P2. It is obtained by a pencil of hyperplane section of the cone
of the Veronese surface imbedded in P5. Hence W′
by Theorem 2.2. Finally, since the rational blowdown manifold W4,2is diffeomorphic
to the general fiber of the QGorenstein smoothing of W′
structure. Furthermore, using a triple cover of F4branched over D in the linear system
3(C0+ 4f), we can also prove that W′
4,3has a QGorenstein smoothing by the similar
proof as above. And, by extending Theorem 2.2 to a finite abelian group, it is possible
to show that some other manifolds W′
4,nhas a QGorenstein smoothing, too. We leave
it for a future research.
4,2is a double cover of a
4,2has a QGorenstein smoothing
4,2, W4,2 admits a complex
Now we investigate the general case. Assume that n ≥ 5 and let Fnbe a Hirzebruch
surface. Let C0be a negative section with C2
linear system 4(C0+nf). The surface Fncan be obtained from the cone over a rational
normal curve of degree n by blowing up the vertex. And a curve in the linear system
4(C0+ nf) is the strict transform of the hyperplane section of the cone. By Bertini’s
theorem, there is an irreducible nonsingular curve in the linear system 4(C0+nf). The
double cover of Fnbranched over an irreducible nonsingular member in 4(C0+nf) is an
elliptic surface E(n): Let σ :ˆ Xn→ Fnbe a double covering branched over an irreducible
nonsingular member in the linear system 4(C0+nf). Then, by the invariants of a double
covering ([2], Chapter V), we have pg(ˆ Xn) = pg(Fn) + h0(Fn,KFn+ L) = h0(Fn,(n −
2)f) = n−1 and χ(Oˆ Xn) = 2χ(OFn)+1
Therefore we have q(ˆ Xn) = 0 and K2
0= −n and f be a fiber of Fn. Consider the
2(L·KFn)+1
ˆ Xn= 2(σ∗(KFn+ L))2= 2((n − 2)f)2= 0.
2(L·L) = n, where L = 2(C0+nf).
In this article we want to choose a special irreducible (singular) curve D in the linear
system 4(C0+nf), which has a special intersection with one special fiber f: Note that
D·f = 4. We want D to intersect with f at two distinct points p and q that are not in C0.
Let x = 0 be the local equation of f and x,y be a coordinate at p (resp. at q). We require
that the local equation of D at p (resp. at q) is (y −x)(y +x) = 0 (resp. (y −xn−4)(y +
xn−4) = 0). These are 3(n−4)+3conditions: 1,x,x2,...,x2n−9,y,yx,...,yxn−5terms
should vanish to have the local analytic equation (y − xn−4)(y + xn−4) = 0. By next
lemmas and proposition, we have such a curve D satisfying the conditions above.
Lemma 3.1. We have h0(Fn,OFn(D)) = 10n + 5, where D is a member in the linear
system 4(C0+ nf).
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A CONSTRUCTION OF HORIKAWA SURFACE VIA QGORENSTEIN SMOOTHINGS7
Proof. Let C = C0+ nf. Then, by the following two exact sequences
0 → OFn→ OFn(f) → OP1 → 0
0 → OFn(kf) → OFn((k + 1)f) → OP1 → 0,
we have h1(Fn,OFn(kf)) = 0 for all nonnegative integers k. And from the exact sequence
0 → OFn(nf) → OFn(C) → OC0(C) → 0,
we also have h1(Fn,OFn(C)) = 0 and h0(Fn,OFn(C)) = n + 2. Hence, by considering
the exact sequences similarly
0 → OFn(kC) → OFn((k + 1)C) → OC((k + 1)C) → 0,
we finally get h0(Fn,OFn(D)) = 10n + 5.
?
First, we assume that D is nonsingular at every point except the two points p and
q. Let σ :˜ Xn → Fnbe a double covering branched over the curve D chosen above.
Then˜ Xnis a singular elliptic surface with pg= n −1 and χ = n which has two rational
double points by the local equations of D at p and q  one is A1(z2= y2− x2) and
the other one is A2n−9(z2= y2− x2n−8). Therefore its minimal resolution is also an
elliptic surface E(n). First we blow up at p and q in Fn. Then we have an exceptional
curve coming from a blowing up at p which intersects with the proper transform of D
transversally at two points, and we also have an exceptional curve coming from a blowing
up at q which intersects with the proper transform of D at one point, say q1. Let x = 0
be the local equation of the (−1)exceptional curve at q1. Then the local equation of
the proper transform of D at q1is (y − xn−5)(y + xn−5) = 0. We blow up again at q1.
By the continuation of blowing up at infinitely near points of q, we have the following
configuration of smooth rational curves
−n
◦
Un−3−
−2
◦
Un−4

◦
E1,−1
−
−2
◦
Un−5−···−
−2
◦
U1

◦
E2,−1
where the proper transform of D intersects with Ei, i = 1,2 at two points transversally.
We denote this surface by Zn, which is obtained by (n − 3) times blowingups of Fn.
Let π : Zn→ Fnbe a map and ∆ = π∗(4C0+4nf)−2E1−2Un−5−4Un−6−6Un−7−
··· − 2(n − 5)U1− 2(n − 4)E2. For a simple computation, we write it as
∆ = π∗(4C0+ 4nf) − 2F − 2
n−4
?
i=1
Fi,
where F = E1, F1= Un−5+ Un−6+ ··· + E2, F2= Un−6+ ··· + E2,...,Fn−4= E2.
Note that Fiis not necessarily irreducible and F2= F2
all i = 1,...,n − 4. Let f0= Un−4, which is a proper transform of the fiber, and let
L = ∆ − (π∗C0+ f0) − KZn.
i= KZn· F = KZn· Fi= −1 for
In Proposition 3.1 below, we prove that the linear system ∆ is base point free. Then,
by Bertini’s theorem, we conclude that D is nonsingular except the two points p and q.
Lemma 3.2. L2≥ 5 and L is nef.
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8 YONGNAM LEE AND JONGIL PARK
Proof. Since π∗f = f0+ F +?n−4
3?n−4
not in C0, we also have C0· f = C0· f0= 1 and f2
i=1Fi, we have L = π∗(5C0+ (5n + 2)f) − f0− 3F −
i=1Fi= π∗(5C0+ (5n − 1)f) + 2f0. Furthermore, since the two points p and q are
0= −2. Therefore
L2= −25n + (5n − 1)10 + 20 − 8 = 25n + 2 ≥ 5.
Let G be an irreducible curve which is neither f0nor C0. Note that L · f0= 5 − 4 = 1
and L · C0= −5n + 5n − 1 + 2 = 1. Write π∗G = aC0+ bf. Then we have
G · L ≥ π∗G · (5C0+ (5n − 1)f) = (aC0+ bf) · (5C0+ (5n − 1)f) = −a + 5b.
We note that the linear system aC0+bf contains an irreducible curve in Fnif and only
if a = 0,b = 1; or a = 1,b = 0; or a > 0,b > an; or a > 0,b = an with n > 0 (refer to
Corollary 2.18, Chapter V in [6]). Therefore it is impossible that G · L < 0. It implies
that L is nef.
?
Lemma 3.3. The linear system ∆ − (π∗C0+ f0) on Znis base point free.
Proof. By Lemma 3.2 above, L is nef and L2≥ 5. Hence, applying to Reider’s theorem
[15], if the adjoint linear series ∆−(π∗C0+f0) = KZn+L has a base point at x then
there is an effective divisor G in Znpassing through x such that either G · L = 0 and
G2= −1; or G · L = 1 and G2= 0.
Assume that G · L = 0 and G2= −1. Write G = G1+ ··· + Gk, where Gkis an
irreducible curve. Since L is nef, G · L = 0 implies that Gi· L = 0 for all i = 1,...,k.
Then we get a contradiction by a similar argument to show that −a+5b ≤ 0 in the proof
Lemma 3.2 above.
Assume that G · L = 1 and G2= 0. By the same argument in the case G · L = 0,
G = G1. Then we get a contradiction by a similar argument to show that −a + 5b ≤ 1
unless G = C0or f0. Furthermore, since C2
0= −n and f2
0= −2, it also contradicts.
?
Proposition 3.1. The linear system ∆ on Znis base point free.
Proof. Note that ∆ · π∗C0= ∆ · f0= 0. Therefore we have Oπ∗C0+f0(∆) = Oπ∗C0+f0.
Furthermore, by Lemma 3.2 above and by the vanishing theorem, we also have
H1(Zn,∆ − (π∗C0+ f0)) = H1(Zn,KZn+ L) = 0.
Hence, using Lemma 3.3 above and the short exact sequence
0 → OZn(∆ − (π∗C0+ f0)) → OZn(∆) → Oπ∗C0+f0→ 0,
we conclude that the linear system ∆ is base point free.
?
Next, by Artin’s criterion of contraction [1], we can contract a configuration Cn−2,
which is a linear chain of P1’s
−n
◦
Un−3−
−2
◦
Un−4−
−2
◦
Un−5− ··· −
−2
◦
U1,
so that it produces a singular normal projective surface. We denote this surface by Yn.
We note that ∆ is the proper transform of D in Znand that Ynhas a cyclic quotient
singularity of type
(n−2)2(1,n − 3), which is a singularity of class T.
1
Lemma 3.4. H1(Zn,OZn(∆)) = 0.
Page 9
A CONSTRUCTION OF HORIKAWA SURFACE VIA QGORENSTEIN SMOOTHINGS9
Proof. Since pg(Zn) = q(Zn) = 0, we have
H1(Zn,OZn(∆)) ≃ H1(∆,O∆(∆)) ≃ H0(∆,O∆(K∆− ∆))∨.
We also have ∆2= D2− 4(n − 3) = 12n + 12 and degK∆ = degKD− 2(n − 3) =
12n − 8 − 2(n − 3) = 10n − 2. Therefore it satisfies
degK∆− ∆2= 10n − 2 − 12n − 12 < 0,
and it implies that H0(∆,O∆(K∆− ∆)) = 0.
?
Proposition 3.2. The singular surface Ynadmits a QGorenstein smoothing.
Proof. It is enough to show that −KYnis effective (Theorem 21 in [12]). Let π : Zn→ Fn
be a composition of blowingups, and ψ : Zn→ Ynbe a contraction. Then we have
KZn= π∗KFn+ E1+ Un−5+ 2Un−6+ ··· + (n − 5)U1+ (n − 4)E2.
Since KFn= −2C0− (n + 2)f, −KZnis effective. Furthermore, since h0(−KYn) =
h0(ψ∗(−KZn)) = h0(−KZn) (§3.9.2 in [16]), −KYnis also effective.
?
Now we are in a position to prove our main theorem mentioned in the Introduction.
First remind that Horikawa surface H(n) is a double cover of Fn−3 branched over a
smooth curve Dn−2in the linear system 6C0+ (4n − 8)f. R. Fintushel and R. Stern
showed that Horikawa surface can be decomposed into
H(n) = Bn−2∪ Dn−2∪ Bn−2,
where Bn−2is the complement of a neighborhood of the pair of 2spheres (C0+(n−2)f)
and C0 in Fn−3 (Lemma 2.1 in [3]), and they proved that an elliptic surface E(n) is
obtained from H(n) by replacing two rational balls Bn−2with two configurations Cn−2
(Lemma 7.3 in [3]). In other words, R. Fintushel and R. Stern proved that Horikawa
surface H(n) can be obtained from an elliptic surface E(n) by rationally blowingdown
two disjoint configurations Cn−2lying in E(n) in smooth category. The aim of this article
is to prove that the rational blowdown surgery above can be performed in complex
category, which is following.
Proof of Theorem 1.1. Note that˜ Xnis a double covering of Fnbranched over D, and
the minimal resolution of two rational double points of type A1 and A2n−9 in˜ Xn is
E(n), which is also a double cover of Znbranched over the proper transform of D. Since
the proper transform of D does not meet the contracted linear chain of P1’s, we have a
double cover of Ynbranched over the image of the proper transform of D by the map ψ.
We denote this surface by Xn. Then Xnis the singular surface obtained by contracting
two disjoint configurations Cn−2from an elliptic surface E(n) and it has two quotient
singularities of class T, both are of type
(n−2)2(1,n − 3). By the fact that the proper
transform of D is disjoint from the contracted liner chain of P1’s and Lemma 3.4, the
map from Xnto Ynis flat and H1(Yn,OYn(¯DYn)) = 0, where¯DYnis the image of ∆ in
Ynunder the contraction Cn−2of the liner chain of P1’s. Therefore we have the following
commutative diagram of maps
1
˜ Xn
↓
←E(n)
↓
→Xn
↓
Fn
π
←Zn
ψ
→Yn
Page 10
10 YONGNAM LEE AND JONGIL PARK
where all vertical maps are double coverings. Then, by Theorem 2.2 and Proposition 3.2
above, the singular surface Xnhas a QGorenstein smoothing of two quotient singularities
all together.
Finally, by applying the standard arguments about Milnor fibers (§5 in [11] or §1
in [13]), we know that a general fiber of a QGorenstein smoothing of Xnis diffeomorphic
to the 4manifold obtained by rational blowdown of E(n). And we also know that H(n)
has one deformation class ([2], Chapter VII). Therefore a general fiber of a QGorenstein
smoothing of Xnis a Horikawa surface H(n) in complex category.
?
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Department of Mathematics, Sogang University, Sinsudong, Mapogu, Seoul 121742,
Korea
Email address: ynlee@sogang.ac.kr
Department of Mathematical Sciences, Seoul National University, San 561, Sillim
dong, Gwanakgu, Seoul 151747, Korea
Email address: jipark@math.snu.ac.kr