Page 1

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

Page 2

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

Page 3

A CONSTRUCTION OF HORIKAWA SURFACE VIA Q-GORENSTEIN 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 Shepherd-Barron [8], 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

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 1-group

T1

X(ΩX,OX).

(2) The obstruction lies in the global Ext 2-group T2

X= Ext1

X= Ext2

X(ΩX,OX).

Furthermore, by applying the general result of local-global 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 Q-Gorenstein smoothing by Proposition 2.5

below.

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→ 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.

Page 4

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

sheaves˜T1

Xare defined by the index one covering stack and by the ´ etale sites [5]. 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

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 Q-Gorenstein smoothing of the cyclic singularity of a germ (X0,0) of type

1

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

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 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

H2(T0

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 [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 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.

Page 5

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 [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 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

−n

◦

Un−3−

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 [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 Q-Gorenstein

smoothing of Xn. As a warming-up, we first investigate n = 4 case.

−2

◦

Un−4−

−2

◦

Un−5− ··· −

−2

◦

U1.

Example. R. Gompf constructed a family of symplectic 4-manifolds by taking a fiber

sum of other symplectic 4-manifolds [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 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