On 4-fold covers over

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On 4-fold covers over

smooth algebraic surfaces

2011

Taketo Shirane

Department of Mathematics and Information Sciences

Tokyo Metropolitan University

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Introduction

In this thesis, all varieties are defined over the field of complex numbers,C.We call a finite surjective morphism between normal varieties a cover. In addition, if degπ= 4, πis called a 4-fold cover. A coverπ:X →Y induces the extension of fields π^∗ : C(Y) ,→ C(X), where C(X) and C(Y) are the rational function fields ofX andY,respectively. A coverπis called aGalois cover if the extension C(X)/C(Y) is a Galois extension. Given a finite groupG,we simply callπaG- cover if it is a Galois cover and Gal(C(X)/C(Y))∼=G.For a coverπ:X →Y, we denote thebranch locus ofπby ∆(π), namely

∆(π) :={P ∈Y |π^∗:OY,P → OX,Q is not isomorphism for^∃Q∈π⁻¹(P)}. It is well-known that ∆(π) is an algebraic subset of pure codimension one if Y is smooth ([30]). In this case, we can regard the branch locus ∆(π) as a reduced divisor on Y. Covers have been studied from various points of view such as algebraic geometry, topology and Galois theory, which make covers interesting objects.

In algebraic geometry, we often make use of covers in order to give new varieties from well-known varieties. For example, if there is a cover over a given variety, then we obtain a new variety, which is the “bottom up” approach.

Conversely, if a ﬁnite group acts a given variety, then we obtain a new variety as the quotient, which is the “top down” approach. When we study covers for the “bottom up” construction of algebraic varieties, the following questions are fundamental.

Question 1.0.1. Suppose that a smooth varietyY is given.

(1) Given a reduced divisorB onY and a ﬁnite groupG,give a criterion for the existence ofG-cover overY whose branch locus is B.

(2) What properties doesX have for a given coverπ:X →Y? Question 1.0.1 (1) is equivalent to the following question ([5]).

5

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6 CHAPTER 1. INTRODUCTION Question 1.0.2. LetY be a smooth variety,B a reduced divisor onY,andG a ﬁnite group. Give a criterion of existence of a surjective homomorphism from the fundamental group ofY \B toG:

π1(Y \B)³G

Therefore we can make use of results in covers in order to study the topology of the complements to reduced divisors on Y. For example, there are a lot of results of Zariski pairs as one of applications of covers (cf. [1], [8], [24], [25], [26]). An answer of Question 1.0.1 (1) for double covers is well-known such as:

There exists a double cover over Y whose branch locus is B if and only if OY(B) is 2-divisible in Pic(Y).

However this question for a general group (eg. the symmetric group Si of degreei≥3) seems to be diﬃcult (cf. [25], [26]).

In this thesis, we mainly discuss Question 1.0.1 (2) for 4-fold covers over smooth surfaces. For a cover π : X → Y over a smooth surface Y, X has singularities in general. These singularities make the study ofX complicated.

To avoid this diﬃculty, we often investigate its resolutionX⁰ such as its exceptional divisor, its numerical invariants, and so on. For a double or triple cover π:X→Y,there is a nice method of resolving singularities ofX which is called thecanonical resolution as follows:

There exists a composition of blowing-ups,σ:=σ⁽¹⁾◦σ⁽²⁾◦· · ·◦σ^(r):Y^(r)→ Y,such that the induced cover π^(r): X⁰ :=X^(r) →Y^(r) (cf. Section 4.2) is a cover between smooth surfaces. In particular,X⁰ is a resolution of X.We also callX⁰ the canonical resolution of X.

X

Y Y⁽¹⁾

X⁽¹⁾ X⁽²⁾

Y⁽²⁾ Y^(r)

X^(r)=:X⁰

σ⁽¹⁾ σ⁽²⁾

π π⁽¹⁾ π⁽²⁾ π^(r)

σ⁽³⁾

σ^(r)

In [6], Horikawa introduced the canonical resolution for double covers. It is easy to compute various numerical invariants and the exceptional divisor of the canonical resolutionX⁰ for a double coverπ:X →Y.Thus this method is useful for the global or local study of double covers over smooth surfaces, and it is an eﬀective tool to answer Question 1.0.1 (2). Hence double covers have been intensively used to study surfaces, in particular surfaces of general type (cf. [3], [6], [17] and so on).

Triple covers are also studied by many authors (cf. [2], [8], [11], [22], [23]).

Among them, Tan’s canonical resolution [22] is useful for the study of triple covers over smooth surfaces likewise Horikawa’s canonical resolution. Triple covers and the canonical resolution are also used to study the topology of the complements to plane curves (cf. [8]).

In [24], Tokunaga studied Galois covers for the symmetric group of degree 4, S₄,and the alternating group of degree 4,A₄,which correspond to 4-fold covers

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1.1. NOTATION AND RESULTS 7 (see Chapter 2 and 3), and applied them to studying the topology of the complements to plane curves. However there seem to be not so many results on 4-fold covers. The main aim of this thesis is to give a method of resolving singularities of 4-fold covers over smooth surfaces likewise the canonical resolution for double and triple covers. Unfortunately, this method is not so simple as the canonical resolution for double and triple covers (Figure 1.1.2). However, we compute the Chern numbers of our resolutions of certain 4-fold covers (Theorem 1.1.3 and 1.1.5).

As its application, we also consider “families of Galois closure curves” for plane quintic curves. In 1996, Yoshihara introduced a new notion in algebraic geometry, which is called aGalois point(see Deﬁnition 1.1.7). The motivation of Galois point seems to be based on the following fact (cf. [15, Theorem 5.3.17]).

The gonality for a smooth plane curve of degreed(see [12], [13] or [27] for the deﬁnition) is equal to d−1, and it is given by the projection from a point in the curve. In [12], [13], [27] and [28], Miura and Yoshihara have studied it in several points of view, and raised the following question.

Question 1.0.3. Let Γ be a smooth plane curve. How does the Galois closure curve Γ_P at P ∈Γ (see Deﬁnition 1.1.7) varies whenP moves on Γ?

In [29], Yoshihara answered this question for smooth plane quartic curves by constructing a family of Galois closure curves, and he raised the following problem.

Problem 1.0.4. (1) Do the similar study as in [29, Theorem 2.1] in the case where Γ is a smooth curve of degree d≥5.

(2) Let Γ be a smooth plane curve of degreed≥3 andP a point inP².For the projection π_P : Γ →l consider the Galois closure curve Γ_P at P.Then we will obtain similarly a smooth threefoldV and a morphismρ:V →P², whose fiber overP is isomorphic to Γ_P for a general pointP.Study the structure ofV and singular fibers of ρ.In this case, if P ∈Γ,then ρ^∗(P) becomes a singular fiber. Are these singular fibers semi-stable, too?

We answer Problem 1.0.4 (1) in the cased= 5 applying our resolution for 4-fold covers (Theorem 1.1.10).

1.1 Notation and Results

Let Y be a smooth variety, and let π : X → Y be a 4-fold cover. Then, since [C(X) :C(Y)] = 4, there is an element z ofC(X) such that its minimal polynomial over C(Y) is

f :=z⁴+g₁z²+g₂z+g₃,

where g_i (i = 1,2,3) are elements of C(Y). Let Ke be the Galois closure of C(X)/C(Y). By [24], we obtain the diagram of the ﬁeld extensions and the normalizations ofY in Figure 1.1.1 (see Section 2.1 and 3.1 for detail).

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8 CHAPTER 1. INTRODUCTION

Gal(K/e C(Y)) S4 A4 D8 V₄ Z4

˜

π S4-cover A4-cover D8-cover V4-cover Z4-cover ψ1 Z2-cover X1∼=Y Z2-cover X1∼=Y Z2-cover ψ2 Z3-cover Z3-cover X2∼=X1 X2∼=X1 X2∼=X1

ψ3 V4-cover V4-cover V4-cover V4-cover Z2-cover ϕ S3-cover Z3-cover Z2-cover Xe ∼=X Xe ∼=X

Table 1.1: The covers ˜π, ψ_i andϕ.

The possibilities of Gal(C(X)/C(Y)) are the symmetric group of degree 4, S4, the alternating group of degree 4, A4, the dihedral group of order 8, D8, the Klein group, V4={(1 2)(3 4),(1 3)(2 4),(1 4)(2 3)} ∼=Z^⊕2²,andZ4,where Zn := Z/nZ. In each case, the Galois groups of the Galois covers ˜π, ψi and ϕ in Figure 1.1.1 are as in Table 1.1. For a prime divisor E on Y, we say that ˜π is ramiﬁed at just ψ1 over E if E ⊂ ∆(ψ1), ψ⁻₁¹(E) 6⊂ ∆(ψ2) and (ψ1◦ψ2)⁻¹(E)6⊂∆(ψ3).We use similar terminology for other cases. We put

Ω_A:={prime divisors onY over which ˜πis ramiﬁed at justψ₁},

ΩB:={prime divisors onY over which ˜πis ramified at justψ1 andψ3}, ΩC:={prime divisors onY over which ˜πis ramified at justψ2}, and Ω_D:={prime divisors onY over which ˜πis ramified at justψ₃}.

We deﬁne reduced divisors ∆A(π), ∆B(π), ∆C(π) and ∆D(π) onY as follows:

∆A(π) := X

E∈ΩA

E; ∆B(π) := X

E∈ΩB

E;

∆C(π) := X

E∈Ω_C

E; ∆D(π) := X

E∈Ω_D

E.

Then we can see ∆(π) = ∆_A(π) + ∆_B(π) + ∆_C(π) + ∆_D(π) (Lemma 3.2.1 and 3.2.2).

1.1.1 4-fold covers over smooth surfaces

We consider a resolution of singularities ofX in Chapter 4. LetY be a smooth surface, and π : X → Y a 4-fold cover. Put Y⁽⁰⁾ := Y. Let σ⁽ⁱ⁾ : Y⁽ⁱ⁾ →

Y

X1

X2

Xe

X ^π^˜ ^ψ²

ψ₃

ψ1

π ϕ

C(Y)

K1

K2

Ke

C(X) Normalization

Figure 1.1.1: The ﬁeld extensions and the normalizations of Y.

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1.1. NOTATION AND RESULTS 9

Y X

Y^(r) X^(r)

X₁^(r) X₂^(r) Xe^(r)

X₂⁰

Xe⁰

X⁰⁰

π

σ

ψ^(r)₂

ψ₁^(r) ψ^(r)₃ ϕ^(r)

π^(r)

σ2

ψ₃⁰

ϕ⁰⁰

ν⁰⁰ ν^(r)

Xe⁰⁰

σ3

Figure 1.1.2: The resolving method of singularities ofX.

Y⁽ⁱ⁻¹⁾be a blowing-up ofY⁽ⁱ⁻¹⁾at a singular point of the branch locus. Since C(Y⁽ⁱ⁾)∼=C(Y),we can construct the coversπ⁽ⁱ⁾,π˜⁽ⁱ⁾, ϕ⁽ⁱ⁾andψ⁽ⁱ⁾_i overY⁽ⁱ⁾in the same way to construct ˜π. In general, it is impossible to modify the branch locus of a 4-fold cover into smooth one by blowing-ups (Example 4.2.1). To avoid this diﬃculty, we will show the following property.

Proposition 1.1.1. Let π:X →Y be a4-fold cover over a smooth surfaceY.

Then there is a succession of blowing-ups, σ=σ⁽¹⁾◦ · · · ◦σ^(r), such that (i) ∆A(π^(r)) + ∆B(π^(r)) + ∆C(π^(r)) + ∆D(π^(r))is a simple normal crossing

(SNC) divisor;

(ii) ∆A(π^(r)) + ∆B(π^(r)) + ∆C(π^(r))is smooth, and

(iii) ifσ^(r+1) is a blowing-up at a singular point of ∆_D(π^(r)),then the excep- tional divisor is an irreducible component of∆_D(π^(r+1)).

Now we deﬁne a 4-fold cover “of good type”.

Deﬁnition 1.1.2. Let π:X →Y be a 4-fold cover over a smooth surface Y.

We call π a 4-fold cover of good type if it satisﬁes the conditions (i), (ii) and (iii) in Proposition 1.1.1.

We can construct a resolution,ν :=ν⁰⁰◦ν^(r):X⁰⁰→X,of a 4-fold coverX over a smooth surfaceY by the diagram in Figure 1.1.2. Hereσis a composition of blowing-ups such thatπ^(r)is of good type,σ2andσ3are certain blowing-ups, andϕ⁰⁰ is a certain quotient ofXe⁰⁰(see Chapter 4).

For a 4-fold cover of good type π : X → Y, we can compute the Chern numbers of X⁰⁰, c²₁(X⁰⁰) andc₂(X⁰⁰),as follows.

Theorem 1.1.3. Let π:X →Y be a4-fold cover of good type. LetX⁰⁰ be the

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10 CHAPTER 1. INTRODUCTION resolution ofX as above. We put ∆_•:= ∆_•(π)for•=A, B, C, D. Then

c²₁(X⁰⁰) = 4c²₁(Y) +¡

2∆_A+ 6∆_B+ 4∆_C+ 4∆_D¢ .K_Y +1

2∆²_A+9

4∆²_B+4

3∆²_C+ ∆²_D+ ∆A.∆D+ 3∆B.∆D, c₂(X⁰⁰) = 4c₂(Y) + (∆_A+ 3∆_B+ 2∆_C+ 2∆_D).K_Y

+ ∆²_A+ 3∆²_B+ 2∆²_C+ 2∆²_D+ 2∆A.∆D+ 3∆B.∆D−3s, whereKY is a canonical divisor onY, andsis the number of nodes of ∆D.

1.1.2 Generic 4-fold covers

We introduce a notion of “generic” 4-fold covers as follows.

Deﬁnition 1.1.4. LetY be a smooth surface. A 4-fold cover π :X → Y is said to begeneric if ∆(π) = ∆_A(π) (i.e. ∆_B(π) = ∆_C(π) = ∆_D(π) = 0).

The above definition is different from the one of generic covers in Kulikov and Kulikov [9]. “Generic” in both definitions means “simple”. However, ours is based on anS4-cover ˜π, but the other is based on a generic projection. In par- ticular, ifπis a generic 4-fold cover in [9], then it is our one (see Remark 5.2.1).

We consider a generic 4-fold cover whose branch locus is a reduced curve with at worst simple singularities. Letπ:X →Y be a generic 4-fold cover as above. We denote the minimal resolution ofX byX.We introduce two integers δ1andδ2; both of them are determined by the conﬁguration of singularities and the ramiﬁcation of ˜πover them (see Section 5.2 for details).

Theorem 1.1.5. Let π:X →Y be a generic 4-fold cover whose branch locus is a reduced curve with at worst simple singularities. Then

c²₁(X) = 4c²₁(Y) + 2∆(π).KY +1

2∆(π)²−δ1−δ2, c₂(X) = 4c₂(Y) + ∆(π).K_Y +A²−3δ₁−2δ₂,

whereδ₁andδ₂are integers determined by the conﬁguration of singularities and the ramiﬁcation ofπ˜ over them (see Section 5.2 for detail).

Remark 1.1.6. If π is a generic 4-fold cover in [9], then the above formulas coincide with the formulas of [9, Corollary 4.1 and Lemma 4.4].

1.1.3 Families of Galois closure curves for plane quintic curves

Let Γ be a smooth curve of degreed(≥4) in the projective planeP²,andP a point in Γ. Consider πP : Γ→l with a center P,where l is a line not passing throughP. This projection induces the extension of ﬁeldsπ_P^∗ :C(l),→C(Γ) =:

K with [K :C(l)] =d−1.The structure of this extension does not depend on the choices of l,but onP. So we denote the image ofπ_P^∗ : C(l),→K byK_P. LetL_P be the Galois closure of this extensionK/K_P.

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1.1. NOTATION AND RESULTS 11 Deﬁnition 1.1.7. Let Γ_P be the nonsingular projective model ofL_P. We call Γ_P theGalois closure curveatP ∈Γ,and letg(P) be the genus of Γ_P.We call P aGalois point for Γ ifK/K_P is a Galois extension.

Remark 1.1.8. ForP ∈Γ, letl^∨_P be the dual line on the dual space (P²)^∨ of P² corresponding toP.We can regardπP as the map Γ3Q7→lP Q∈l^∨_P,where lP Qis the line onP² passing throughP andQ ifP 6=Q, and the tangent line T_P of Γ atP ifP =Q.

In the case whered= 4,Yoshihara constructed a familyρ:S→Γ satisfying that ρ^∗(P) ∼= ΓP for a general point P ∈ Γ, and determined the types of its singular ﬁbers ([29]). As its corollary, he showed that ΓQ is not isomorphic to ΓP ifQ6=P is close toP.The above study is based onS3-covers. We answer Problem 1.0.4 (1) in the case where d = 5 based on our resolution of 4-fold covers.

We assume that Γ is a smooth plane quintic curve. Let I(Γ₁,Γ₂;Q) the intersection number of plane curves Γ₁ and Γ₂ at Q.A pointP ∈Γ is said to be an inﬂection point of orderi (i= 1,2,3) if I(Γ, T_P;P) =i+ 2. We have the fact P

Q∈Γ{I(Γ, T_Q;Q)−2} = 45 (cf. [7]). Let N_A =N_A(P) (resp. N_B, N_C and N_D) be the number of points in ∆_A(π_P) (resp. ∆_B(π_P), ∆_C(π_P) and

∆D(πP)). For example, ifP is an inﬂection point of order 1 and](TP∩Γ) = 3, then NA = NA+ 1 since πP(P) = TP ∈ ∆A(πP), where NA is the number of non-inﬂection points Q 6= P with P ∈ TQ and ](TQ∩Γ) = 4, and so on.

Applying the Riemann-Hurwitz formula to the cover πP : Γ → l^∨_P, we get NA+ 3NB+ 2NC+ 2ND = 18. We put Σ :={P ∈ Γ| NA <18}. Then Σ is ﬁnite (Lemma 6.1.2). Note thatGP ∼=S4ifP ∈Γ\Σ,that there exist at most one Galois point, and that the Galois point is an inﬂection point of order 3 (but not the converse) (see [27]).

Note that if GP ∼= D8, then NC = 0 [19, Remark 4.1]. Moreover we will show the following proposition.

Proposition 1.1.9. GP is isomorphic toA4 if and only if NA=NB = 0.

Now we state our results. We put

Λ_B= Λ_B(P) :={l_P ∈l^∨_P |l_P ∈∆_B(π_P), l_P 6=T_P}.

For • = B, C, D, we put k_• = k_•(P) := 1 if T_P ∈ ∆_•(π_P) and k_• := 0 if T_P 6∈∆_•(π_P).We put

n_i:=]{P ∈Γ|I(Γ, T_P;P) =i+ 2} (i= 1,2,3),

n1,s:=]{P ∈Γ|I(Γ, TP;P) = 3, I(Γ, TP;Q)≤1 for anyQ6=P}, n_b:=]{l: a bitangent line of Γ|I(Γ, l;Q)≤2 for anyQ∈Γ}, n1,b:=n1−n1,s.

Theorem 1.1.10. There exist a smooth projective minimal surface of general type S with

c²₁(S) = 5880−24n3,

c2(S) = 4920−90n3−18n2−36n1,b−18nb,

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12 CHAPTER 1. INTRODUCTION and a morphism ρ : S → Γ such that ρ^∗(P)_red is a divisor with only normal crossings for anyP ∈Γ and satisﬁes the following properties:

(I) If P ∈Γ\Σ, thenρ^∗(P)∼= ΓP andg(P) = 85.

(II) If P ∈ΣandGP ∼=S4,then g(P) = 31 + 3NA+ 2NC and ρ^∗(P) =Γ +b X

l_P∈Λ_B

3EB,l_P +kBEB+kC

X4 i=1

EC,i+kD

X6 i=1

ED,i,

where bΓ, E_B,l_P, E_B, E_C,i and E_D,i are irreducible curves with the fol- lowing properties:

(i) g(bΓ) =g(P),Γb²=−18N_B+ 12k_B−8k_C−12k_D,and the number of nodes of Γb is4(N_C−k_C) + 6(N_D−k_D).

(ii) E_B is a smooth curve of genus4 withE²_B=−6.

(iii) E_B,l_P, E_C,i and E_D,i are (−2)-curves (i.e. smooth rational curves with the self-intersection number −2).

(iv) ρ^∗(P)−Γb is smooth (i.e. E_B,l_P, E_B, E_C,i and E_D,i are disjoint), Γ.Eb B,l_P =bΓ.EB = 6andΓ.Eb C,i=Γ.Eb D,i= 2.

(III) If GP ∼=A4, theng(P) = 16 +NC and ρ^∗(P) =Γb₁+Γb₂+k_C

X4 i=1

E_C,i+k_D X6 i=1

E_D,i,

whereΓbj(j= 1,2),EC,iandED,iare irreducible curves with the following properties:

(i) Γbj∼= ΓP forj= 1,2,Γb²_j =−4NC−6ND, and Γb1.Γb2= 4(NC−kC) + 6(ND−kD).

(ii) EC,i andED,i are(−2)-curves.

(iii) ρ^∗(P)−Γb₁−bΓ₂ is smooth, and bΓ_j.E_C,i =Γb_j.E_D,i= 1.

(IV) If GP ∼=D8,then g(P) = 11 +NA and ρ^∗(P) =Γb1+Γb2+bΓ3+ X

l_P∈Λ_B

3EB,l_P +kBEB+kD

X6 i=1

ED,i,

whereΓb_j (j= 1,2,3),E_B,l_P, E_B andE_D,iare irreducible curves with the following properties:

(i) g(bΓ_j) =g(P),Γb²_j =−6N_B+ 4k_B+ 4k_D,bΓ_j.bΓ_j0 = 0 (j6=j⁰), and the number of nodes of bΓ_j is2(N_D−k_D).

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1.1. NOTATION AND RESULTS 13 (ii) E_B is a smooth curve of genus 4with E_B² =−6.

(iii) EB,l_P,andED,i are (−2)-curves.

(iv) ρ^∗(P)−P3

j=1Γbj is smooth, bΓj.EB,l_P = Γbj.EB = 2 for any i, j.

bΓj.ED,i= 2 ifj≡i (mod 3)andΓbj.ED,i= 0otherwise.

(V) IfP is the Galois point (hence GP ∼=Z4 and]ΛB=NB−1), then ρ^∗(P) =

X6 j=1

bΓ_j+

NXB−1 i=1

3E_B,i+E_B,

wherebΓj, EBandEB,iare irreducible curves with the following properties:

(i) g(Γbj) = 6,Γb²_j =−3NB+ 2, and the number of nodes ofΓbj isND. (ii) EB is a smooth curve of genus 4with E_B² =−6.

(iii) EB,Q⁰ are (−2)-curves.

(iv) PN_B−1

i=1 EB,i +EB is smooth, bΓj.bΓj⁰ = 0 (j 6= j⁰) and Γbj.EB,i = bΓ_j.E_B = 1.

Example 1.1.11. (1) IfGP ∼=S4and TP ∈∆B(πP),then ρ^∗(P) is described as in the following ﬁgure:

S⁰

∪

the blow-up at the nodes ofbΓ

EB

EB,l_P

ρ^∗(P)

∪

˜

π◦σ˜|ρ∗(P)

S

T_P ΛB

∆A(πP) ∆_C(π_P) ∆_D(π_P)

l_P^∨

−1 −1

Here S⁰ is the blow-up ofS at the nodes ofΓ,b thick lines describe (−1)-curves, the horizontal curve describe the strict transform of Γ,b and ˜π and ˜σ are as in Section 6.1. Note that the horizontal curve is isomorphic to the Galois closure curve Γ_P,that it meets transversally withE_B, E_B,l_P and (−1)-curves, and that