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arXiv:0708.3319v3 [math.AG] 15 Jun 2008
A CONSTRUCTION OF HORIKAWA SURFACE VIA
Q-GORENSTEIN SMOOTHINGS
YONGNAM LEE AND JONGIL PARK
Abstract. In this article we prove that Fintushel-Stern’s construction of Horikawa
surface, which is obtained from an elliptic surface via a rational blow-down surgery in
smooth category, can be performed in complex category. The main technique involved
is Q-Gorenstein smoothings.
1. Introduction
As an application of a rational blow-down surgery on 4-manifolds, R. Fintushel and
R. Stern showed that Horikawa surface H(n) can be obtained from an elliptic surface
E(n) via a rational blow-down 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 blow-down surgery to obtain Horikawa surface
can be performed in fact in complex category.
Fintushel-Stern’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 Q-Gorenstein smoothings.
Note that Q-Gorenstein smoothing theory developed in deformation theory in last thirty
years is a very powerful tool to construct a non-singular 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 Q-Gorenstein smoothing of X. It
is known that the cohomology H2(T0
X) contains the obstruction space of a Q-Gorenstein
smoothing of X. That is, if H2(T0
X) = 0, then there is a Q-Gorenstein 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 Q-Gorenstein smoothing of X. In this article we also give a family of
examples which admit Q-Gorenstein smoothings even though the cohomology H2(T0
does not vanish. Our main technique is a Q-Gorenstein 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, Q-Gorenstein smoothing, rational blow-down.
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 blowing-up 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 Q-Gorenstein 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 Q-Gorenstein smoothing of two
quotient singularities all together, and a general fiber of the Q-Gorenstein 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 4-manifolds 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 (KRF-2005-070-C00005).
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. Q-Gorenstein smoothing
In this section we briefly review a theory of Q-Gorenstein 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
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 Xtis a smooth projective surface,
(ii) the central fiber X0is X,
(iii) the canonical divisor KX/∆is Q-Cartier.
Let X be a normal projective surface with quotient singularities.Let
A Q-Gorenstein smoothing for a germ of a quotient singularity (X0,0) is defined
similarly. A quotient singularity which admits a Q-Gorenstein 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 Q-Gorenstein 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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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 Q-Gorenstein 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 Q-Gorenstein smoothing of Xnis diffeomorphic
to the 4-manifold obtained by rational blow-down of E(n). And we also know that H(n)
has one deformation class ([2], Chapter VII). Therefore a general fiber of a Q-Gorenstein
smoothing of Xnis a Horikawa surface H(n) in complex category.
?
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Department of Mathematics, Sogang University, Sinsu-dong, Mapo-gu, Seoul 121-742,
Korea
E-mail address: ynlee@sogang.ac.kr
Department of Mathematical Sciences, Seoul National University, San 56-1, Sillim-
dong, Gwanak-gu, Seoul 151-747, Korea
E-mail address: jipark@math.snu.ac.kr