arXiv:0708.3319v3 [math.AG] 15 Jun 2008
A CONSTRUCTION OF HORIKAWA SURFACE VIA
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.
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 . 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  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
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 . But, in general, the
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.
2 YONGNAM LEE AND JONGIL PARK
a special fiber, which contains two linear chains of configurations of P1’s
Un−5− ··· −
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.
(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
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.
dn2(1,dna − 1) for some
A CONSTRUCTION OF HORIKAWA SURFACE VIA Q-GORENSTEIN SMOOTHINGS3
Proposition 2.2 ([8, 12, 17]).
are of class T.
(1) The singularities
(2) If the singularity
◦ − ··· −
◦ is of class T, then so are
◦ − ··· −
◦ − ··· −
(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 Shepherd-Barron , there is a Q-Gorenstein smoothing locally for each
singularity of class T on X (see Proposition 2.5). The natural question arises whether
this local Q-Gorenstein 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
Q-Gorenstein smoothing, there is a Q-Gorenstein 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
X= 0. By applying the standard result of deformations [10, 14] to a normal
projective surface with quotient singularities, we get the following
X(ΩX,OX) for i = 0,1,2. For
Proposition 2.3 (, §4). Let X be a normal projective surface with quotient singu-
(1) The first order deformation space of X is represented by the global Ext 1-group
(2) The obstruction lies in the global Ext 2-group T2
Furthermore, by applying the general result of local-global spectral sequence of ext
sheaves (, §3) to deformation theory of surfaces with quotient singularities so that
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-
(1) We have the exact sequence
0 → 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 Q-Gorenstein smoothing by Proposition 2.5
Let X be a normal projective surface with singularities of class T. Our concern is
to understand Q-Gorenstein 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) → H2(T0
X)] → 0
X) = 0 can be obtained via the vanishing of H2(TV(−log E)),
X, not the whole first order deformations.
4 YONGNAM LEE AND JONGIL PARK
this index one cover is unique up to isomorphism. The first order deformations which
associate Q-Gorenstein smoothings can be realized as the invariant part of T1
Xare defined by the index one covering stack and by the ´ etale sites . The
first order deformation of a Q-Gorenstein 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
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 Q-Gorenstein smoothing of the cyclic singularity of a germ (X0,0) of type
dn2(1,dna−1). Moreover every Q-Gorenstein smoothing of (X0,0) is isomorphic
to the pull-back 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
p∈singular points of X
where a singular point p is of type
dpn2(1,dpan − 1) with (a,n) = 1.
Theorem 2.1 (). 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 Q-Gorenstein
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 Q-Gorenstein smoothing of X. Hence, in the case that
X) ?= 0, we have to develop another technique in order to investigate the existence
of Q-Gorenstein 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 Q-Gorenstein smoothing. Explicitly, we get the following theorem.
X) = 0, then there is a Q-Gorenstein
X) = 0 .
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 Q-Gorenstein 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.
A CONSTRUCTION OF HORIKAWA SURFACE VIA Q-GORENSTEIN SMOOTHINGS5
Then there exists a Q-Gorenstein smoothing of X that is compatible with a Q-Gorenstein
smoothing of Y . And the cyclic covering extends to the Q-Gorenstein smoothing.
Proof. Let Y → ∆ be a Q-Gorenstein smoothing of Y , and let Ytbe a general fiber of the
Q-Gorenstein smoothing. By the semi-continuity, 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 ). 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|
to the Q-Gorenstein 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 self-intersection
number is −n. Assume that n ≥ 4 and let Cn−2 denote a simply connected smooth
4-manifold obtained by pluming the (n − 3) disk bundles over the 2-sphere according to
the linear diagram
Assume that an elliptic surface E(n) has two configurations Cn−2such that all em-
bedded 2-spheres 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 Q-Gorenstein smoothing because it
violates Noether inequality (Corollary 7.5 in ). 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 Q-Gorenstein
smoothing of Xn. As a warming-up, we first investigate n = 4 case.
Un−5− ··· −
Example. R. Gompf constructed a family of symplectic 4-manifolds by taking a fiber
sum of other symplectic 4-manifolds . 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 blowing-down 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 4-manifold by W4,n. Then the manifold W4,1does not admit