Proof of Theorem 1.1.5

44 CHAPTER 5. PROOFS OF THEOREM 1.1.3 AND 1.1.5

5.2. PROOF OF THEOREM 1.1.5 45 Let ¯ν :X →X be the minimal resolution ofX,andι:X⁰⁰→Xthe contraction of (−1)-curves. We put ²(P, π) be the number of (−1)-curves contracted by ι overπ⁻¹(P).Then we obtainc²₁(X) =c²₁(X⁰⁰) +P

P∈∆(π)²(P, π) and c₂(X) = c2(X⁰⁰)−P

P∈∆(π)²(P, π).Hence we obtain c²₁(X) =q₁(π)− X

P∈Sing(∆(π))

(κ₁(P, π)−²(P, π)), c2(X) =q2(π)− X

P∈Sing(∆(π))

(κ2(P, π) +²(P, π)).

Now letπ:X →Y be a generic 4-fold cover whose branch locus is a reduced curve with at worst simple singularities. Note that ∆A(π) = 0 if Gal(K/e C(Y)) is isomorphic to A4, V4 or Z4, hence the assertion of Theorem 1.1.5 is clear.

Here we show it in only the case Gal(K/e C(Y))∼=S4,but, in the other case, it is shown by the same way. So we assume that

Gal(K/e C(Y))∼=S4.

Let γ : Z₁ → X₁ be the minimal resolution of X₁, ψˆ₂ : Z₂ → Z₁ the K₂ -normalization of Z1, and ˆψ3 : Ze → Z2 the K-normalization ofe Z2. We call a reduced divisorH on a smooth surface anA2-configurationifH consists of two (−2)-curvesH1 andH2 such thatH1.H2= 1.

We will show the following proposition.

Proposition 5.2.2. Let π:X →Y be a generic 4-fold cover as above. Then π satisfies(♠),and ∆( ˆψ₂) (resp. ψˆ₂(∆( ˆψ₃))) consists of disjoint union of A2 -configurations (resp. (−2)-curves) on Z₁.

Letδ1 (resp. δ2) be the number ofA2-configurations (resp. (−2)-curves) in

∆( ˆψ₂) (resp. ˆψ₂(∆( ˆψ₃))).

To prove the above proposition and Theorem 1.1.5, we investigate each sin-gularity of ∆(π),so we consider local onY.We putR3⊂∆(π) the set of points over which ψ2 is unramified and ψ3 is ramified, and R2,3 ⊂ ∆(π) the set of points over whichψ2andψ3 are ramified.

If ∆( ˆψ2)∩(ψ1◦γ)⁻¹(P) (resp. ˆψ2(∆( ˆψ3))∩(ψ1◦γ)⁻¹(P)) consists of disjoint union ofA2-configurations (resp. (−2)-curves) forP ∈Sing(∆(π)),then we put δ1(P, π) (resp. δ2(P, π)) the number of such divisors. For P ∈ Sing(∆(π)), we say that π satisfies (♠) at P if there is a finite number of blowing-ups σ = σ⁽¹⁾◦ · · · ◦σ^(r) such that π^(r) is of good type and ∆D(π^(r))∩σ⁻¹(P) is smooth. Throughout this section, we describe the divisors ∆A(π⁽ⁱ⁾),∆B(π⁽ⁱ⁾),

∆C(π⁽ⁱ⁾) and ∆D(π⁽ⁱ⁾) onY⁽ⁱ⁾in each figure as follows:

: the divisor ∆_A(π⁽ⁱ⁾) : the divisor ∆_B(π⁽ⁱ⁾) : the divisor ∆C(π⁽ⁱ⁾) : the divisor ∆_D(π⁽ⁱ⁾) : unbranched divisors

Furthermore, the ramification loci ofϕandϕ⁰⁰ are described by thick lines.

46 CHAPTER 5. PROOFS OF THEOREM 1.1.3 AND 1.1.5 Proposition 5.2.3. Let P ∈ Sing(∆(π))\(R₃∪R_2,3). Then ∆( ˆψ₂)∩(ψ₁◦ γ)⁻¹(P) consists of disjoint union of A2-configurations, and π satisfies (♠) at P. Moreover, the following hold:

(i) δ1(P, π) =κ1(P, π)−²(P, π);

(ii) 3δ₁(P, π) =κ₂(P, π) +²(P, π); and (iii) δ₂(P, π) = 0.

Proof. The first assertion follows from [8, Remark 3.1]. Sinceψ3 is unramified overP, πsatisfies (♠) atP.Moreover (iii) is clear.

Let W be a quotient of X₂ by an involution of S3, and α : W → Y the induced triple cover. Similarly, we have the triple coversα⁽ⁱ⁾:W⁽ⁱ⁾→Y⁽ⁱ⁾.In particular, if π⁽ⁱ⁾ is of good type, then W⁽ⁱ⁾ is smooth. For the triple covers α⁽ⁱ⁾,let

q1(α⁽ⁱ⁾) := 3c²₁(Y⁽ⁱ⁾) + 2∆⁽ⁱ⁾_A.K_Y(i)+ 4∆⁽ⁱ⁾_C .K_Y(i)+1

2(∆⁽ⁱ⁾_A )²+4

3(∆⁽ⁱ⁾_C)², q₂(α⁽ⁱ⁾) := 3c₂(Y⁽ⁱ⁾) + ∆⁽ⁱ⁾_A .K_Y(i)+ 2∆⁽ⁱ⁾_C.K_Y(i)+ (∆⁽ⁱ⁾_A)²+ 2(∆⁽ⁱ⁾_C )², where ∆⁽ⁱ⁾_A := ∆A(π⁽ⁱ⁾) and ∆⁽ⁱ⁾_C := ∆C(π⁽ⁱ⁾).Let²(P, α) be the number forα as²(P, π) forπ.Since ∆B(π^(nP)) = ∆D(π^(nP)) = 0,we have

q1(π^(nP))−q1(α^(nP)) =c²₁(Y^(nP)) =c²₁(Y)−nP, q₂(π^(nP))−q₂(α^(nP)) =c₂(Y^(nP)) =c₂(Y) +n_P.

On the other hand, by [8, Proposition 3.1, 3.2, 3.3 and 3.4], we can see that q₁(α^(nP)) = 3c²₁(Y) + 2∆_A(π).K_Y +1

2∆_A(π)²−δ₁(P, π)−²(P, α), q2(α^(nP)) = 3c2(Y) + ∆A(π).KY + ∆A(π)²−3δ1(P, π) +²(P, α).

Hence, since²(P, π) =²(P, α) +nP,we have q1(π^(nP)) = 4c²₁(Y) + 2∆A(π).KY +1

2∆A(π)²−δ1(P, π)−²(P, π), q2(π^(nP)) = 4c2(Y) + ∆A(π).KY + ∆A(π)²−3δ1(P, π) +²(P, π).

Therefore we obtain κ₁(P, π) = δ₁(P, π) +²(P, π) and κ₂(P, π) = 3δ₁(P, π)−

²(P, π).

Proposition 5.2.4. The following hold:

(i) If R_2,36=∅, thenR_2,3 consists ofe₆-singularities.

(ii) If R36=∅, thenR3 consists ofa2k−1-,dk- and e7-singularities.

5.2. PROOF OF THEOREM 1.1.5 47 Proof. By [24, Lemma 9.1], R_2,3 consists of a_6k−1- and e₆-singularities. For P ∈ R_2,3, since ψ₂◦ψ₃ is not a cyclic cover, the local fundamental group at ψ₁⁻¹(P) is not cyclic, soa_6k−1-singularities are not inR_2,3.

(ii) is clear from [24, Lemma 9.1].

LetE⁽ⁱ⁾be the exceptional divisor ofσ⁽ⁱ⁾,and, by abuse of notation, we also use the same notation to describe its strict transform under σ⁽ⁱ⁺¹⁾◦ · · · ◦σ^(j): Y^(j)→Y⁽ⁱ⁾forj > i.

Proposition 5.2.5. Let P ∈ R23. Then π satisfies (♠) at P, and ∆( ˆψ2)∩ (ψ1◦γ)⁻¹(P) (resp. ψˆ2(∆( ˆψ3))∩(ψ1◦γ)⁻¹(P)) consists of disjoint union of A2-configurations (resp. (−2)-curves). Moreover,

κ1(P, π) = 11, κ2(P, π) = 0, ²(P, π) = 8, δ1(P, π) = 2, δ2(P, π) = 1.

Proof. By Proposition 5.2.4, P is an e6-singularity. Let σ⁽¹⁾ be the blowing-up at P. Then, since ψ⁽¹⁾ is ramified over E⁽¹⁾, the singularity of ∆(π⁽¹⁾) is of type a5, say P⁽¹⁾, and ψ₂⁽¹⁾ and ψ⁽¹⁾₃ are ramified over P⁽¹⁾. So E⁽¹⁾ is in B⁽¹⁾ from Proposition 5.2.4. Since ψ1◦ψ2 is anS3-cover, by the proof of [8, Proposition 3.4], the branch locus ofπ⁽⁵⁾ is as follows:

−4 −3 −3

−1

−1

σ⁽¹⁾ σ⁽²⁾ σ⁽³⁾ σ⁽⁴⁾ σ⁽⁵⁾

Here E⁽⁴⁾ andE⁽⁵⁾ is not in D⁽⁵⁾ by Lemma 4.2.2. Henceπsatisfies (♠) at P.

We infer that the local structure of ˜π⁽⁵⁾ over P is as Figure 5.2.1. We obtain

²(P, π) = 8.And we have

∆A(π⁽⁵⁾).K_Y(5) = ∆A(π).KY + 6, (∆A(π⁽⁵⁾))²= ∆A(π)²−12,

∆B(π⁽⁵⁾).K_Y(5) = ∆B(π).KY + 2, (∆B(π⁽⁵⁾))²= ∆B(π)²−4,

∆_C(π⁽⁵⁾).K_Y(5) = ∆_C(π).K_Y + 2, (∆_C(π⁽⁵⁾))²= ∆_C(π)²−6.

Hence we obtain

q1(π⁽⁵⁾) = 4c²₁(Y) + 2∆A(π).KY +1

2∆A(π)²−11, q2(π⁽⁵⁾) = 4c2(Y) + ∆A(π).KY + ∆A(π)²

Thereforeκ1(P, π) = 11 andκ2(P, π) = 0.

The second assertion is clear from Figure 5.2.1, and we obtain δ1(P, π) = 2, δ2(P, π) = 1.

Letη(k) := (1−(−1)^k)/2 (i.e. η(k) = 0 if k is even, andη(k) = 1 if k is odd).

48 CHAPTER 5. PROOFS OF THEOREM 1.1.3 AND 1.1.5

−1 −1 −3

−4

−1

−3

−1

−3

ψ⁽⁵⁾₁

ψ⁽⁵⁾₂ ψ₃⁽⁵⁾

ϕ⁽⁵⁾

π⁽⁵⁾ the triple cover

the double cover

Figure 5.2.1: The local structure of ˜π⁽⁵⁾ overe6-singularity.

Proposition 5.2.6. Let P ∈R3 be an a2k−1-singularity. Then πsatisfies(♠) atP,ψˆ2(∆( ˆψ3))∩(ψ1◦γ)⁻¹(P)consists of disjoint union of (−2)-curves, and

∆( ˆψ2)∩(ψ1◦γ)⁻¹(P) =∅.Moreover,

κ₁(P, π) = 3k+ 2η(k), κ₂(P, π) =−2η(k), ²(P, π) = 2k+ 2η(k), δ1(P, π) = 0, δ2(P, π) =k.

Proof. Letσ⁽¹⁾ be the blowing-up atP.ThenE⁽¹⁾ is not in ∆(ψ⁽¹⁾₁ ).

First we supposek = 1.Since ψ₂⁽¹⁾ is unramified over E⁽¹⁾, X₂⁽¹⁾ is smooth over E⁽¹⁾. So, if ψ⁽¹⁾₃ is unramified over E⁽¹⁾, then ψ₃⁽¹⁾ is unramified over all points ofE⁽¹⁾,which is a contradiction. HenceE⁽¹⁾is in ∆_D(π⁽¹⁾),andn_P = 1.

Next we supposek= 2.Then there is a singular point of ∆(π⁽¹⁾) onE⁽¹⁾,say P⁽¹⁾.Let σ⁽²⁾ be the blowing-up at P⁽¹⁾. Suppose thatE⁽¹⁾ is not in ∆(π⁽¹⁾).

Then the singularity is of typea1. So E⁽²⁾ is in ∆D(π⁽²⁾) by the casek = 1.

However, since π⁽²⁾ is of good type and E⁽¹⁾ ∼=P¹, this is in contradiction to Lemma 4.3.2. Hence, E⁽¹⁾ is in ∆D(π⁽¹⁾).Since E⁽²⁾ is not in ∆(ψ⁽²⁾₁ ), E⁽²⁾ is either in ∆D(π⁽²⁾) or not in ∆(π⁽²⁾). IfE⁽²⁾ is in ∆D(π⁽²⁾),thenπ⁽²⁾ is not of good type by Lemma 4.3.2. So letσ⁽³⁾ be the blowing-up at the intersection of E⁽¹⁾andE⁽²⁾.ThenE⁽³⁾is not in ∆(π⁽³⁾), andXe⁽³⁾is smooth overE⁽¹⁾.Hence the self-intersection number of an irreducible component of (ψ⁽³⁾₁ ◦ψ⁽³⁾₂ )^∗(E⁽¹⁾) must be even sinceE⁽¹⁾ is in ∆D(π⁽³⁾). However, since (E⁽¹⁾)² =−3 onY⁽³⁾, that number is odd, which is a contradiction. ThereforeE⁽²⁾ is not in ∆(π⁽²⁾), andπ⁽²⁾ is of good type.

For k > 2, by induction on k and the above argument, we can see that E⁽²ⁱ⁻¹⁾ is in ∆_D(π⁽²ⁱ⁻¹⁾),and thatE⁽²ⁱ⁾is not in the branch locus ofπ⁽²ⁱ⁾ for

5.2. PROOF OF THEOREM 1.1.5 49

×3

−4

−1

−1

−1

−4

−2

−4

−1

−1

−1

−4

−2

×3

ψ₂^(k)◦ψ^(k)₃

ψ^(k)₁ ϕ^(k)

π^(k)

Figure 5.2.2: The local structure of ˜π^(k)overa_2k−1-singularity (k : even).

0< i < k/2 + 1.So the blowing-ups are as follows.

−1

−1 −2

−2

−2

−2

−2

−2

−2

−2

−2

−2

−2

(k : even)

(k : odd)

σ⁽¹⁾ σ⁽²⁾ σ⁽³⁾

σ^(k)

Hence πsatisfies (♠) atP, nP =k, and we have

∆A(π^(k)).K_Y(k) = ∆A(π).KY + 2k, ∆A(π^(k))²= ∆A(π)²−4k,

∆_D(π^(k)).K_Y(k) = ∆_D(π).K_Y −η(k), ∆_D(π^(k))²= ∆_D(π)²−k,

∆A(π^(k)).∆D(π^(k)) = ∆A(π).∆D(π) + 2η(k).

Moreover, we can see that ˆψ2(∆( ˆψ3))∩(ψ1◦γ)⁻¹(P) consists of disjoint union of (−2)-curves, and ∆( ˆψ2)∩(ψ1 ◦γ)⁻¹(P) = ∅. Thus we obtain κ1(P, π) = 3k+ 2η(k), κ2(P, π) =−2η(k), δ1(P, π) = 0 andδ2(P, π) =k.

Finally, we consider the ramification locus ofϕ^(k).LetE3 be an irreducible component of (˜π^(k))^∗(E^(k)).

In the case thatk is even, X^(k) is smooth. Sinceϕ^(k)|E3 is a double cover of smooth curves, the number of its ramification points are even. Hence this

50 CHAPTER 5. PROOFS OF THEOREM 1.1.3 AND 1.1.5

ψ₁^(k) ψ₂^(k)◦ψ^(k)₃

×3

×3

×3

×3

−1

−4

−1

−4

−4

−1

−1

−2

−1

−4

−1

−4

−4

−1

−1

−2

ϕ⁰⁰ σ₃

Figure 5.2.3: The local structure ofπ^(k)˜ overa2k−1-singularity (k : odd).

number is 0 or 2 or 4. If the number is 4, then the double coverπ^(k)|ϕ^(k)(E3)

is an unbranched double cover over a smooth rational curve, which does not occur. If the number is 0, then there is another irreducible component E₃⁰ of (˜π^(k))⁻¹(E^(k)) such that ϕ^(k)|E₃⁰ is a double cover with 4 ramification points, which is a contradiction. Thereforeϕ^(k)|E3 is ramified at 2 points. So we infer that the local structure of ˜π^(k)overP is as Figure 5.2.2. Hence²(P, π) = 2k.

In the case that k is odd, there are six pointwise ramification points on Xe overE^(k).By Lemma 4.3.3, we can infer that the local structure of ˜π^(k)overP is as Figure 5.2.3. Hence²(P, π) = 2k+ 2.

Proposition 5.2.7. Let P ∈ R3 be an e7-singularity. Then π satisfies (♠) atP,ψˆ2(∆( ˆψ3))∩(ψ1◦γ)⁻¹(P)consists of disjoint union of (−2)-curves, and

∆( ˆψ2)∩(ψ1◦γ)⁻¹(P) =∅.Moreover,

κ1(P, π) = 23, κ2(P, π) =−14, ²(P, π) = 20, δ1(P, π) = 0, δ2(P, π) = 3.

Proof. The question is local on Y. We may assume that ∆ = L+T, where L and T are given by y = 0 and y²+x³ = 0, respectively. By abuse of notation, we also use the same notation to describe their strict transforms under σ⁽¹⁾◦σ⁽²⁾◦ · · · ◦σ⁽ⁱ⁾:Y⁽ⁱ⁾→Y (1≤i≤nP).

Letσ⁽¹⁾ be the blowing-up atP,andP⁽¹⁾ the singular point of (σ⁽¹⁾)^∗∆(π).

Let σ⁽²⁾ be the blowing-up at P⁽¹⁾, and P⁽²⁾ be the intersection of E⁽¹⁾, E⁽²⁾ and T. Let σ⁽³⁾ be the blowing-up at P⁽²⁾. Then ∆(π⁽²⁾) is an SNC divisor.

Note that E⁽ⁱ⁾ (i = 1,2,3) are in ∆(ψ₁⁽³⁾). Since E⁽³⁾ intersects with T, by Lemma 4.3.1,E⁽³⁾ is in ∆_A(π⁽³⁾). Similarly,E⁽¹⁾ andE⁽²⁾ are in ∆_A(π⁽³⁾).

5.2. PROOF OF THEOREM 1.1.5 51

−2

−1

−8

−1

−2

−2

−2

−1

−2

−3

−1

−3

−2

−2

−1

−1

−1

−4 −4

−2

−1

−8 ψ₁⁽⁷⁾

ψ₂⁽⁷⁾◦ψ₃⁽⁷⁾

×3

×3

×3

×3 σ₃

ϕ⁰⁰

Figure 5.2.4: The local structure of ˜π⁽⁷⁾ overe7-singularity.

LetP⁽³⁾ be the intersection ofE⁽¹⁾ andE⁽³⁾ onY⁽³⁾,andσ⁽⁴⁾ the blowing-up at P⁽³⁾. ThenE⁽⁴⁾ is not in the branch locus ofψ⁽⁴⁾₁ . Since E⁽¹⁾ ∼=P¹ and ψ1|_ψ⁻¹

1 (E⁽¹⁾) is an isomorphism, E⁽⁴⁾ is not in ∆D(π⁽⁴⁾) by Lemma 4.3.2. So E⁽⁴⁾ is not in ∆(π⁽⁴⁾).

LetP⁽⁴⁾, P⁽⁵⁾ and P⁽⁶⁾ be the intersections of T and E⁽³⁾, E⁽²⁾ and E⁽³⁾, and E⁽²⁾ and L, respectively, on Y⁽⁴⁾. Letσ⁽⁵⁾, σ⁽⁶⁾ and σ⁽⁷⁾ be the blowing-ups at P⁽⁴⁾, P⁽⁵⁾ and P⁽⁶⁾, respectively. If E⁽⁵⁾ is not in ∆_D(π⁽⁵⁾), then, by Lemma 4.3.2, E⁽⁶⁾ and E⁽⁷⁾ are not in ∆_D(π⁽⁷⁾). This is in contradiction to P ∈R3. Hence E⁽⁵⁾ is in ∆D(π⁽⁵⁾,and one can similarly check E⁽⁶⁾ and E⁽⁷⁾ are in ∆D(π⁽⁷⁾).Hence the blowing-ups are as follows:

σ⁽¹⁾ σ⁽²⁾ σ⁽³⁾ σ⁽⁴⁾

σ⁽⁵⁾ σ⁽⁶⁾ σ⁽⁷⁾

Thusπsatisfies (♠) atP,and we have

∆A(π⁽⁷⁾).K_Y(7) = ∆A(π).KY + 14, ∆A(π⁽⁷⁾)²= ∆A(π)²−28,

∆_D(π⁽⁷⁾).K_Y(7) = ∆_D(π).K_Y −3, ∆_D(π⁽⁷⁾)²= ∆_D(π)²−3,

∆A(π⁽⁷⁾).∆D(π⁽⁷⁾) = ∆A(π).∆D(π) + 6.

Hence we obtain ˆψ2(∆( ˆψ3))∩(ψ1◦γ)⁻¹(P) consists of disjoint union of (− 2)-curves, and ∆( ˆψ₂)∩(ψ₁ ◦γ)⁻¹(P) = ∅. Therefore we have κ₁(P, π) = 23, κ₂(P, π) = −14 δ₁(P, π) = 0 and δ₂(P, π) = 3. Moreover, we infer that the

52 CHAPTER 5. PROOFS OF THEOREM 1.1.3 AND 1.1.5

ψ^(k₁⁻¹⁾ ψ^(k₂⁻¹⁾◦ψ₃^(k−1) ϕ^(k−1)

π^(k−1)

−1

−1

−4

−4

−1

−1

−1

−2

−4

−1

−1

−1 −1

−4

−4

−4

−2

−2

−2

−2

−2

−2

−2

−1

×3

×3

Figure 5.2.5: The local structure of ˜π^(k−1)overdk-singularity (k : odd).

local structure of ˜π⁽⁷⁾ over P is as Figure 5.2.4 by Lemma 4.3.3. Therefore

²(P, π) = 20.

Proposition 5.2.8. LetP ∈R3 be adk-singularity (k≥4). Assume thatk is odd. Then πsatisfies (♠) atP, ψˆ₂(∆( ˆψ₃))∩(ψ₁◦γ)⁻¹(P) consists of disjoint union of(−2)-curves, and∆( ˆψ2)∩(ψ1◦γ)⁻¹(P) =∅,Moreover,

κ₁(P, π) = 2k, κ₂(P, π) =−2k+ 6, ²(P, π) = 2k−2, δ1(P, π) = 0, δ2(P, π) = 2.

Proof. The question is local on Y. We may assume that ∆ =L+T, where L and T are given by x = 0 and y²+x^k−2 = 0, respectively. Let σ⁽¹⁾ be the blowing-up at P. Then E⁽¹⁾ intersects transversally with L. By Lemma 4.3.1, we see thatE⁽¹⁾ is in ∆A(π⁽¹⁾).

First we suppose that k= 5. There are two singular points of typea1 and a3,sayP⁽¹⁾andP⁽²⁾,respectively. By Proposition 5.2.6, we can infer thatπ⁽⁴⁾ is of good type. Letσ⁽²⁾be the blowing-up atP⁽¹⁾.IfE⁽²⁾in ∆_D(π⁽²⁾),then we can see E⁽¹⁾.∆_D(π⁽⁴⁾) = 1 onY⁽⁴⁾, which is in contradiction to Lemma 4.3.2.

So E⁽²⁾ is not in the branch locus, and one can check the branch loci are as

5.2. PROOF OF THEOREM 1.1.5 53 follows:

σ⁽¹⁾ σ⁽²⁾ σ⁽³⁾ σ⁽⁴⁾

In particular, the component of the exceptional divisor intersecting with L is not in the branch locus.

Next we supposek >5. Then there are two singular points of typea₁ and dk−2, sayP⁽¹⁾ and P⁽²⁾, respectively. Letσ⁽²⁾ be the blowing-up at P⁽¹⁾. We can see thatE⁽²⁾ is not in the branch locus by induction onk.Here the branch loci are as follows:

σ⁽¹⁾ σ⁽²⁾ σ⁽³⁾ σ^(k−1)

π^(k−1)is of good type, and we have

∆^(k_A⁻¹⁾.K_Y(k−1) = ∆A(π).KY + 2(k−1), (∆^(k_A⁻¹⁾)²= ∆A(π)²−4(k−1),

∆^(k_D⁻¹⁾.K_Y(k−1) = ∆D(π).KY, (∆^(k_D⁻¹⁾)²= ∆D(π)²−2,

where ∆^(k_A⁻¹⁾:= ∆A(π^(k−1)) and ∆^(k_D⁻¹⁾:= ∆D(π^(k−1)).So we haveκ1(P, π) = 2kandκ2(P, π) =−2k+ 6.We infer that the local structure of ˜π^(k−1)overP is as Figure 5.2.5. Therefore ˆψ₂(∆( ˆψ₃))∩(ψ₁◦γ)⁻¹(P) consists of disjoint union of (−2)-curves, and ∆( ˆψ2)∩(ψ1◦γ)⁻¹(P) =∅.We obtainδ1(P, π) = 0, δ2(P, π) = 2 and²(P, π) = 2k−2.

Proposition 5.2.9. Let P ∈R₃ be the singular point of type d_k (k≥4). As-sume that k is even. Then π satisfies (♠) at P, ψˆ2(∆( ˆψ3))∩(ψ1◦γ)⁻¹(P) consists of disjoint union of(−2)-curves, and∆( ˆψ2)∩(ψ1◦γ)⁻¹(P) =∅. More-over, either

(i) κ1(P, π) = 2k+ 6, κ2(P, π) =−2k, ²(P, π) = 2k+ 4, δ1(P, π) = 0, δ2(P, π) = 2,

or

(ii) κ1(P, π) = 7k/2, κ2(P, π) =−2k, ²(P, π) = 3k, δ1(P, π) = 0, δ2(P, π) =k/2.

Proof. The question is local onY.We may assume that ∆ =L+T₁+T₂,where L, T₁ and T₂ are given by x = 0, y −x^(k−2)/2 = 0 and y +x^(k−2)/2 = 0, respectively.

Ifk= 4,then, by Lemma 4.3.2, we infer that the branch loci are as follows:

σ⁽¹⁾ σ⁽²⁾ σ⁽³⁾ σ⁽⁴⁾

54 CHAPTER 5. PROOFS OF THEOREM 1.1.3 AND 1.1.5

−2

−2

−2

−2

−2

−3

−2

−2

−1

−1

−2

−1

−1

−1

−1

−4

−4

−4

−4

−4

−4

−8

−1

−1

−2

−2 −1

−1

×3

×3

×3

×3

ψ^(k)₁ ψ^(k)₂ ◦ψ₃^(k)

σ₃

ϕ⁰⁰

Figure 5.2.6: The local structure of ˜π^(k)overd_k-singularity (k: even) in (i).

Supposek >4.Let σ⁽¹⁾ be the blowing-up atP. ThenE⁽¹⁾ is in ∆A(π⁽¹⁾), and there are two singular points of typea1 and dk−2, sayP⁽¹⁾ andP⁽²⁾. Let σ⁽²⁾ be the blowing-up atP⁽¹⁾. E⁽²⁾ is either not in ∆(π⁽²⁾) or in ∆D(π⁽²⁾).

Case (i). IfE⁽²⁾is not in ∆(π⁽²⁾),by Lemma 4.3.2 and induction onk,the branch loci are as follows:

σ⁽¹⁾ σ⁽²⁾ σ⁽³⁾ σ^(k)

Henceπsatisfies (♠) atP.In this case, we obtain

∆_A(π^(k)).K_Y(k) = ∆_A(π).K_Y + 2k, ∆_A(π^(k))²= ∆_A(π)²−4k,

∆D(π^(k)).K_Y(k) = ∆D(π).KY −2, ∆D(π^(k))²= ∆D(π)²−2,

∆A(π^(k)).∆D(π^(k)) = ∆A(π).∆D(π) + 4.

Thusκ1(P, π) = 2k+6 andκ2(P, π) =−2k.Moreover, by Lemma 4.3.3, we infer that the local structure of ˜π^(k)overP is as Figure 5.2.6. Hence ˆψ₂(∆( ˆψ₃))∩(ψ₁◦ γ)⁻¹(P) consists of disjoint union of (−2)-curves, and ∆( ˆψ2)∩(ψ1◦γ)⁻¹(P) =∅. We haveδ1(P, π) = 0, δ2(P, π) = 2 and ²(P, π) = 2k+ 4.

Case (ii). IfE⁽²⁾ is in ∆_D(π⁽²⁾), by Lemma 4.3.2 and induction onk, the

5.2. PROOF OF THEOREM 1.1.5 55

×3

×3

×3

×3

−2

−1

−8

−1

−2

−3

−2−1

−1

−8

−1

−2

−3

−1

−2

−8

−2

−1

−1

−1

−1

−2

−2

−3

−1

−8

−1

−2

−3

−2−1

−8

−3

−2

−2

−1

−2

−1

ψ₁^(k) ψ₂^(k)◦ψ^(k)₃

σ⁰₃

ϕ⁰⁰

Figure 5.2.7: The local structure of ˜π^(k) overdk-singularity (k: even) in (ii).

branch loci are as follows:

σ⁽¹⁾ σ⁽²⁾ σ⁽³⁾ σ^(k)

Hence πsatisfies (♠) atP.In this case, we obtain

∆A(π^(k)).K_Y(k) = ∆A(π).KY + 2k, ∆A(π^(k))²= ∆A(π)²−4k,

∆_D(π^(k)).K_Y(k) = ∆_D(π).K_Y −1

2k, ∆_D(π^(k))²= ∆_D(π)²−1 2k,

∆_A(π^(k)).∆_D(π^(k)) = ∆_A(π).∆_D(π) +k.

Thusκ1(P, π) = 7k/2 andκ2(P, π) =−2k. By Lemma 4.3.3, we infer that the local structure of ˜π^(k)overP is as Figure 5.2.7. Hence ˆψ₂(∆( ˆψ₃))∩(ψ₁◦γ)⁻¹(P) consists of disjoint union of (−2)-curves, and ∆( ˆψ2)∩(ψ1◦γ)⁻¹(P) = ∅. We haveδ1(P, π) = 0, δ2(P, π) =k/2 and²(P, π) = 3k.

By the above propositions, Proposition 5.2.2 is clear.

Proof of Theorem 1.1.5. The above propositions show κ₁(P, π)−²(P, π) =δ₁(P, π) +δ₂(P, π) and

κ₂(P, π) +²(P, π) = 3δ₁(P, π) + 2δ₂(P, π)

56 CHAPTER 5. PROOFS OF THEOREM 1.1.3 AND 1.1.5 for eachP ∈Sing(∆(π)).Hence the assertion is clear.

The following proposition follows from [24, Theorem 0.6] and Theorem 1.1.5.

Proposition 5.2.10. Let Y :=P², and∆ a reduced sextic curve onY with at worst simple singularities. Let π : X → Y be a generic 4-fold cover with the branch locus∆.If Gal(K/e C(Y))∼=S4,then

c²₁(X) = 8 and c2(X) = 4, or c²₁(X) = 9 and c₂(X) = 3.

Proof. Letπ:X→Y be a generic 4-fold cover whose branch locus is a reduced sextic curve with at worst simple singularities. By [24, Theorem 0.6] and its proof, δ₁ = 6, δ₂ = 4 or δ₁ = 9, δ₂ = 0. Hence the assertion follows from Theorem 1.1.5.

Chapter 6

Proof of the results in Section 1.1.3

In this chapter, we prove the results in Section 1.1.3. In Section 6.1, we construct a family of Galois closure curves for a plane quintic curve through a generic 4-fold cover π: X →Y between smooth surfaces. In Section 6.2, we determine the type of singularities of the branch locus ∆(π) and the type of singularity of singular fibres of the family. In Section 6.4, we prove the results in Section 1.1.3.

6.1 Construction of a family of Galois closure curves

We first put Γ ⊂P² a smooth curve of any degree d ≥4. We fix a system of homogeneous coordinates (T0:T1 :T2) of P²,and put (T₀^∨ :T₁^∨:T₂^∨) the one of the dual space (P²)^∨ of P² induced by (T0 : T1 : T2). Let Π : P²×P² 99K P²×(P²)^∨ be the rational map given by

(P ; Q)7→(P ; l_{P Q}),

where lP Q is the line passing through P and Q. If P = (T0 : T1 : T2) and Q = (U0 : U1 : U2) in P², then lP Q = (T₀^∨ : T₁^∨ : T₂^∨) = (T1U2 −T2U1 : T2U0−T0U2:T0U1−T1U0) in (P²)^∨.It is easy to see that Π is undetermined along the diagonal set ∆⊂P²×P².We putBl∆(P²×P²)→P²×P²the blowing-up along ∆. Then we obtain the morphism $ : Bl∆(P²×P²) → P²×(P²)^∨ induced by Π. X⊂Bl∆(P²×P²) the strict transform of Γ×Γ⊂P²×P².Then we have X ∼= Γ×Γ. We put Y := $(X), and π := $|X : X → Y, which is (d−1)-fold cover over Y.

Remark 6.1.1. (i) The surface Y ⊂ P² ×(P²)^∨ has a structure of ruled surfaces over Γ, p :Y →Γ, induced by the projection P²×(P²)^∨ →P². In particular,Y is smooth.

57

58 CHAPTER 6. PROOF OF THE RESULTS IN SECTION 1.1.3 (ii) For anyP ∈Γ,the fibrep⁻¹(P) is the dual linel_P^∨ ⊂ {P} ×(P²)^∨

corre-sponding toP ∈P².

(iii) For anyP ∈Γ, πnaturally satisfies the following commutative diagram:

{P} ×Γ p⁻¹(P)

Γ P¹

π|{P}×Γ

-∼?

∼?

πP

-(iv) We can easily see that the projectionP²×(P²)^∨→(P²)^∨ induces ad-fold coverp⁰:Y →(P²)^∨.

From now, let Γ⊂P²be a smooth quintic curve. SinceG_P ∼=S₄forP 6∈Σ, we obtain the S₄-cover ˜π : Xe → Y and the diagram in Figure 1.1.1. Let

˜

σ:S→Xe be the minimal resolution of X,e andρ:=p◦π˜◦σ˜:S →Γ.

Xe S

Y Γ

˜ ?

π

¾^σ˜

?

ρ p

-It is easy to see that ρ^∗(P) is isomorphic to the Galois closure curve Γ_P for P ∈Γ\Σ.HenceS is of general type. Furthermore, the 4-fold coverπis generic by the next lemma.

Lemma 6.1.2. The4-fold coverπis generic. In particular, Σis finite.

Proof. Let Γ^∨ be the dual curve of Γ. A singular point of Γ^∨ corresponds to either a tangent line at an inflection point or a bitangent line of Γ. Letl_P be a line onP²passing throughP ∈P². ]π⁻¹(P, l_P)≤2 if and only if (P, l_P) satisfies one of the following conditions (see Figure 6.1.1):

(B-1) I(l_P,Γ;P) = 5;

(B-2) I(lP,Γ;Q) = 4 for some Q6=P;

(C-1) (i) I(l_P,Γ;P) = 4, or

(ii) I(lP,Γ;P) = 2 and I(lP,Γ;Q) = 3 for someQ6=P; (C-2) I(lP,Γ;P) = 1 and I(lP,Γ;Q) = 3 for someQ6=P; (D-1)lP is a bitangent line with I(lP,Γ;P) = 3;

(D-2)lP is a bitangent line withlP 6=TP.

(♣)

Thus, for each case, the dual line l_P^∨ passes through a singular point of Γ^∨. Thus such points (P, l_P) are contained in p⁰⁻¹(Sing(Γ^∨)) which is finite from Remark 6.1.1 (iv). Thereforeπis generic.

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