Slope inequalities for irregular cyclic covering fibrations
SSLOPE INEQUARITIES FOR IRREGULAR CYCLIC COVERINGFIBRATIONS
HIROTO AKAIKE
Abstract.
Let f : S → B be a finite cyclic covering fibration of a fibered surface. Westudy the lower bound of slope λ f when the relative irregularity q f is positive. Introduction
Let f : S → B be a surjective morphism from a smooth projective surface S to asmooth projective curve B with connected fibers. We call it a fibration of genus g when ageneral fiber is a curve of genus g . A fibration is called relatively minimal, when any ( − C with C = − n a( − n )-curve. A fibration is called smooth when all fibers are smooth, isotrivial when all ofthe smooth fibers are isomorphic, locally trivial when it is smooth and isotrivial.Assume that f : S → B is a relatively minimal fibration of genus g ≥
2. We denote by K f = K S − f ∗ K B a relative canonical divisor. We associate three relative invariants with f : K f = K S − g − b − ,χ f := χ ( O S ) − ( g − b − ,e f := e ( S ) − g − b − , where b and e ( S ) respectively denote the genus of the base curve B and the topologicalEuler-Poincar´e characteristic of S . Then the following are well-known: • (Noether) 12 χ f = K f + e f . • (Arakelov) K f is nef. • (Ueno) χ f ≥ χ f = 0 if and only if f is locally trivial. • (Segre) e f ≥ e f = 0 if and only if f is smooth.When f is not locally trivial, we put λ f := K f χ f and call it the slope of f according to [5], in which Xiao succeeded in giving its effectivelower bound as λ f ≥ g − g . Another invariant we are interested in is the relative irregularity of f defined by q f := q ( S ) − b , where q ( S ) := dim H ( S, O S ) denotes the irregularity of S as usual. When q f ispositive, we call f an irregular fibration. Xiao showed in [5] that λ f ≥ g, h, n )introduced in [2], where Enokizono gave the lower bound of the slope for them. Note that a r X i v : . [ m a t h . AG ] M a y t is nothing more than a hyperelliptic fibration when h = 0 and n = 2. Recall that Luand Zuo obtained the lower bound of the slope for irregular double covering fibrationsin [4] and [3]. Inspired by their results, we try to generalize them to irregular primitivecyclic covering fibrations with n ≥
3. We give the lower bound of slope for those of type( g, , n ) in Theorems 3.4 and 4.7, and for those of type ( g, h, n ) with h ≥ . q f >
0. Recall that χ f and (the essential part of) K f can be expressedin terms of the so-called k -th singularity index α k defined for each non-negative integer k . The negativity referred above can be used to get some non-trivial restrictions on α which is the most difficult one to handle with among all α k ’s. Thanks to such informationtogether with an analysis of the Albanese map, we can obtain the desired slope inequalities.We also give a small contribution to the modified Xiao’s conjecture that q f ≤ (cid:100) g +12 (cid:101) holds, posed by Barja, Gonz´alez-Alonso and Naranjo in [1]. It is known to be true forfibrations of maximal Clifford index [1] and for hyperelliptic fibrations [4] among others.We show in Theorem 4.5 that q f ≤ ( g + 1 − n ) / f is a primitive cycliccovering fibration of type ( g, , n ) under some additional assumptions. For the historyaround the conjecture, see the introduction of [1].The author express his sincere gratitude to Professor Kazuhiro Konno for suggestingthis assignment, his valuable advice and support. The author also thanks Dr. MakotoEnokizono for his precious advices, allowing him to use Proposition 3.2 freely.1. Primitive cyclic covering fibrations
We recall the basis properties of primitive cyclic covering fibrations, most of which canbe found in [2].
Definition . Let f : S → B be a relatively minimal fibration of genus g ≥
2. We callit a primitive cyclic covering fibration of type ( g, h, n ), when there are (not necessarilyrelatively minimal) fibration ˜ ϕ : (cid:102) W → B of genus h ≥ n -cyclic covering˜ θ : (cid:101) S = Spec (cid:102) W (cid:32) n − (cid:77) j =0 O (cid:102) W ( − j (cid:101) d ) (cid:33) → (cid:102) W branched over a smooth curve (cid:101) R ∈ | n (cid:101) d | for some n ≥ (cid:101) d ∈ Pic( (cid:102) W ) such that f isthe relatively minimal model of ˜ f = ˜ ϕ ◦ ˜ θ .Let f : S → B be a primitive cyclic covering fibration of type ( g, h, n ). Let (cid:101) F and (cid:101) Γbe general fibers of ˜ f and ˜ ϕ , respectively. Then the restriction map ˜ θ | (cid:101) F : (cid:101) F → (cid:101) Γ is aclassical n -cyclic covering branched over (cid:101) R ∩ (cid:101) Γ. By the Hurwitz formula for ˜ θ | (cid:101) F , we get(1.1) r := (cid:101) R. (cid:101) Γ = 2 (cid:0) g − − n ( h − (cid:1) n − . From (cid:101) R ∈ | n (cid:101) d | , it follow that r is a multiple of n .Let ˜ ψ : (cid:102) W → W be the contraction morphism to a relative minimal model W → B of ˜ ϕ : (cid:102) W → B . Since ˜ ψ is a composite of blowing-ups, we can write ˜ ψ = ψ ◦ · · · ψ N ,where ψ i : W i → W i − denotes the blowing-up at x i ∈ W i − ( i = 1 , · · · , N ), W = W nd W N = (cid:102) W . We define a reduced curve R i inductively as R i − = ( ψ i ) ∗ R i starting from R N = (cid:101) R down to R = R . We also put E i = ψ − i ( x i ) and m i = mult x i R i − ( i = 1 , · · · , N ). Lemma 1.2 ([2], Lemma 1.5) . In the above situation, the following hold for any i =1 , · · · , N . (1) Either m i ∈ n Z or n Z + 1 . Furthermore, m i ∈ n Z if and only if E i is not containedin R i . (2) R i = ψ ∗ i R i − − n [ m i n ] E i , where [ t ] denotes the greatest integer not exceeding t . (3) There exists d i ∈ Pic( W i ) such that d i = ψ ∗ i d i − − [ m i n ] E i and R i ∼ n d i , d N = (cid:101) d .Remark . By [2], we can assume the following for any primitive cyclic covering fi-brations. Let ˜ σ be a generator of the covering transformation group of ˜ θ , and σ theautomorphism of S over B induced by ˜ σ . Then the natural morphism ρ : (cid:101) S → S is aminimal succession of blowing-ups that resolves all isolated fixed points of σ .We must pay a special attention when h = 0, since we have various relatively minimalmodels for ˜ ϕ : (cid:102) W → B . Using elementary transformations, one can show the following. Lemma 1.4 ([2], Lemma 3.1) . Let f : S → B be a primitive cyclic covering fibration oftype ( g, , n ) . Then there is a relatively minimal model of ˜ ϕ : (cid:102) W → B such that mult x R h ≤ r gn − for all x ∈ R h , where R h denotes the ϕ -horizontal part of R . Moreover if mult x R > r ,then mult x R ∈ n Z + 1 . When h = 0, we always assume that a relatively minimal model of ˜ ϕ : (cid:102) W → B is as inthe above lemma. Corollary 1.5.
Let the situation be the same as in Lemma 1.4. If x is a singular pointof R and m = mult x R , then n (cid:20) mn (cid:21) ≤ r . Proof.
When m ∈ n Z , the inequality clearly holds by Lemma 1 .
4. If m ∈ n Z + 1, then n [ mn ] + 1 = m . From Lemma 1 .
4, we have m ≤ r + 1. So we get n [ mn ] ≤ r . (cid:3) In closing the section, we give an easy lemma that will be usuful in the sequel.
Lemma 1.6.
Let π : C → C be a surjective morphism between smooth projective curves.Let R π and ∆ be the ramification divisor and the branch locus of π , respectively. Then, (deg( π ) − (cid:93) ∆ ≥ deg R π , where (cid:93) ∆ denotes the cardinality of ∆ as a set of points.Proof. We put ∆ = { Q , . . . , Q (cid:93) ∆ } . For any Q i ∈ ∆, we put π − ( Q i ) = { P i , . . . , P ij i } .Note that deg( π ) = r ( P i ) + · · · + r ( P ij i ) for any i = 1 , . . . , (cid:93) ∆, where r ( P ) denotes theramification index of π around P ∈ C . Then, from the property of ramification divisor,deg R π = (cid:93) ∆ (cid:88) i =1 j i (cid:88) j =1 ( r ( P ij ) − π ) (cid:93) ∆ − ( j + · · · j (cid:93) ∆ ) ≤ (deg( π ) − (cid:93) ∆ , hich is what we want. (cid:3) Singularity indices and the formulae for K f and χ f . We let f : S → B be a primitive cyclic covering fibration of type ( g, h, n ) and freely usethe notation in the previous section. We obtain a classical n -cyclic covering θ i : S i → W i branched over R i by setting S i = Spec ( n − (cid:77) j =0 O W i ( − j d i ))Since R i is reduced, S i is a normal surface. There exists a natural birational morphism S i → S i − . Set S (cid:48) = S , θ = θ , d = d and f (cid:48) = ϕ ◦ θ . Then we have a commutativediagram: (cid:101) S = S N (cid:101) θ (cid:15) (cid:15) (cid:47) (cid:47) ρ (cid:39) (cid:39) S N − θ N − (cid:15) (cid:15) (cid:47) (cid:47) · · · (cid:47) (cid:47) S = S (cid:48) θ (cid:15) (cid:15) S f (cid:5) (cid:5) (cid:102) W = W N ψ N (cid:47) (cid:47) (cid:101) ϕ (cid:43) (cid:43) W N − ψ N − (cid:47) (cid:47) · · · ψ (cid:47) (cid:47) W = W ϕ (cid:15) (cid:15) B The well-known formulae for cyclic coverings give us K f = n ( K ϕ + 2( n − d .K ˜ ϕ + ( n − ˜ d ) , (2.1) χ ˜ f = nχ ˜ ϕ + 12 n − (cid:88) j =0 j ˜ d ( j ˜ d + K ˜ ϕ ) , (2.2)(see e.g., [2]). From Lemma 1 . d = d − N (cid:88) i =1 (cid:20) m i n (cid:21) , (2.3) ˜ d .K ˜ ϕ = d .K ϕ + N (cid:88) i =1 (cid:20) m i n (cid:21) (2.4)and(2.5) K ϕ = K ϕ − N. Definition . Let Γ p and F p respectively denote fibers of ϕ : W → B and f : S → B overa point p ∈ B . For any fixed p ∈ B , we consider all singular points (including infinitelynear ones) of R on Γ p . For any positive integer k , we let α k ( F p ) be the number of singularpoints of multiplicity either kn or kn +1 among them, and call it the k -th singularity index of F p . We put α k := (cid:80) p ∈ B α k ( F p ) and call it the k -th singularity index of the fibration.We also put α := ( K ˜ ϕ + (cid:101) R ) (cid:101) R and call it the ramification index of ˜ ϕ | (cid:101) R : (cid:101) R → B .By a simple calculation, we get = (cid:88) k ≥ α k , (2.6) N (cid:88) i =1 (cid:20) m i n (cid:21) = (cid:88) k ≥ kα k , (2.7) N (cid:88) i =1 (cid:20) m i n (cid:21) = (cid:88) k ≥ k α k (2.8)and(2.9) α = ( K ϕ + R ) R − (cid:88) k ≥ nk ( nk − α k . Substituting (2.3) through (2.8) for (2.1) and (2.2), one gets K f = n ( K ϕ + 2( n − d .K ϕ + ( n − d ) − n (cid:88) k ≥ (( n − k − α k and χ ˜ f = nχ ˜ ϕ + n ( n − d .K ϕ + n ( n − n − d − n ( n − (cid:88) k ≥ ((2 n − k − k ) α k . Since K f ≥ K f , χ ˜ f = χ f and χ ˜ ϕ = χ ϕ , we obtain(2.10) K f ≥ n ( K ϕ + 2( n − d .K ϕ + ( n − d ) − n (cid:88) k ≥ (( n − k − α k and(2.11) χ f = nχ ϕ + n ( n − d .K ϕ + n ( n − n − d − n ( n − (cid:88) k ≥ ((2 n − k − k ) α k . We treat the cases h = 0 and h > Proposition 2.2.
Let f : S → B be a primitive cyclic cover fibration of type ( g, , n ) andlet α i ( i ≥ be the singularity index in Definition . . Then, ( r − K f ≥ ( r − n − rn ( n − α (2.12) + nk ≤ r (cid:88) k ≥ (cid:18) n − n nk ( r − − ( nk − − ( r − n (cid:19) α k , ( r − χ f = (2 r − n − r n ( n − α + nk ≤ r (cid:88) k ≥ (cid:18) n − n nk ( r − − ( nk − (cid:19) α k . (2.13) Proof.
Note that if α k >
0, then nk ≤ r from Corollary 1.5.We find that R ≡ − r K ϕ + M Γ for some M ∈ Z , where the symbol ≡ means thenumerical equivalence, since ϕ : W → B is a P -bundle and we have K W . Γ = − (cid:101) R. (cid:101) Γ = R. Γ = r . Hence we get R.K ϕ = − M , (2.14) = 2 rM . (2.15)Therefore we have ( K ϕ + R ) .R = 2( r − M . From this equality and (2.9), we get2( r − M = α + nk ≤ r (cid:88) k ≥ nk ( nk − α k . (2.16)On the other hand, substituting (2.14) and (2.15) for (2.10), we get K f ≥ ( r − n − rn n − M − n nk ≤ r (cid:88) k ≥ (( n − k − α k . Multiplying this by r − .
16) for it, we get (2.12). Similarly one canshow (2.13). (cid:3)
When h >
0, we have the following:
Proposition 2.3.
Let f : S → B be a primitive cyclic covering fibration of type ( g, h, n ) such that h ≥ and α i ( i ≥ the singularity index in Definition . . Put t := 2( g − − ( h − n + 1) ,T := (cid:40) g − K ϕ .R, if h = 1 , − (cid:0) ( g − − n ( h − K ϕ − ( n − h − R (cid:1) ( n − h − , if h ≥ . Then t > and T ≥ . Furthermore, tK f ≥ tx (cid:48) K ϕ ( n − h −
1) + ty (cid:48) T + tz (cid:48) α + (cid:88) k ≥ a k α k (2.17) and tχ f = ntχ ϕ + t ¯ x (cid:48) K ϕ ( n − h −
1) + t ¯ y (cid:48) T + t ¯ z (cid:48) α + (cid:88) k ≥ ¯ a k α k (2.18) hold, where x (cid:48) = ( g − n − (cid:0) ( g − n + 1) − n ( h − (cid:1) nt , ¯ x (cid:48) = ( n − n + 1)( g − − n ( h − nt ,y (cid:48) = ( n − (cid:0) − n − n (cid:1) t , ¯ y (cid:48) = ( n − n + 1)12 nt ,z (cid:48) = 2( n − ( g − nt , ¯ z (cid:48) = ( n − (cid:0) n − g − − n ( n + 1)( h − (cid:1) nt ,a k = 12 ¯ a k − nt, ¯ a k = k
12 ( n − (cid:0) g −
1) + n ( h − n − k − (cid:1) . In (2 . and (2 . , the quantity K ϕ ( n − h − is understand to be zero, when h = 1 . roof. We get t > r ≥ n ≥ g ≥
2. We shall show that T ≥
0. If h = 1,by the canonical bundle formula, we have K ϕ ≡ χ ( O W )Γ + l (cid:88) i =1 (cid:0) − k i (cid:1) Γwhere { k i | i = 1 , . . . , l } denotes the set of multiplicities of all multiple fibers of ϕ , k i ≥ K ϕ .R ≥ χ ( O W )Γ .R = 2( g − n − χ ( O W )(2.19)Since W is an elliptic surface, we have χ ( O W ) ≥ T ≥
0. If h ≥
2, weconsider the intersection matrix K ϕ K ϕ . d K ϕ . Γ K ϕ . d d d . Γ K ϕ . Γ d . Γ 0 for { K ϕ , d , Γ } . Since we have K ϕ ≥ K ϕ . d )( d . Γ)( K ϕ . Γ) − d ( K ϕ . Γ) − ( d Γ) K ϕ ≥ . (2.20)Since d . Γ = rn = 2( g − − n ( h − n ( n − , K ϕ . Γ = 2( h − , the inequality (2 .
20) is equivalent to2 (cid:0) g − − n ( h − (cid:1) K ϕ . d − n ( n − h − d ≥ n ( n − h − (cid:0) g − − n ( h − (cid:1) K ϕ . So we get 0 ≥ (cid:0) ( g − − n ( h − K ϕ − ( n − h − R (cid:1) and, hence, T ≥ n (cid:0) K ϕ + 2( n − n K ϕ .R + ( n − n ) R (cid:1) = x (cid:48) K ϕ ( n − h −
1) + y (cid:48) T + z (cid:48) ( K ϕ + R ) R and nχ ϕ + n − K ϕ .R + ( n − n − n R = nχ ϕ + ¯ x (cid:48) K ϕ ( n − h −
1) + ¯ y (cid:48) T + ¯ z (cid:48) ( K ϕ + R ) R. Hence we obtain from (2 .
10) and (2 .
11) that K f ≥ x (cid:48) K ϕ ( n − h −
1) + y (cid:48) T + z (cid:48) ( K ϕ + R ) R − n (cid:88) k ≥ (cid:0) ( n − k − (cid:1) α k (2.21)and χ f = nχ ϕ + ¯ x (cid:48) K ϕ ( n − h −
1) + ¯ y (cid:48) T + ¯ z (cid:48) ( K ϕ + R ) R − n ( n − (cid:88) k ≥ (cid:0) (2 n − k − k (cid:1) α k . (2.22) rom (2 .
9) and (2 . tK f ≥ tx (cid:48) K ϕ ( n − h −
1) + ty (cid:48) T + tz (cid:48) α + (cid:88) k ≥ (cid:0) n − ( g − k ( nk − − nt (( n − k − (cid:1) α k Since one sees a k = 2( n − ( g − k ( nk − − nt (( n − k − , we obtain (2 . . . (cid:3) Slope inequality for irregular cyclic covering fibrations.
The purpose of this section is to show the slope inequalities for irregular cyclic coveringfibrations of type ( g, h, n ), n ≥
3. We start things in a more general setting.
Definition . Let ˜ θ : ˜ S → ˜ W be a finite Galois cover (not necessarily primitive cyclic)between smooth projective varieties with Galois group G . Let α : (cid:101) S → Alb( (cid:101) S ) be theAlbanese map. For any ˜ σ ∈ G , we denote by α (˜ σ ) : Alb( (cid:101) S ) → Alb( (cid:101) S ) the morphisminduced from ˜ σ : (cid:101) S → (cid:101) S by the universality of the Albanese map. We putAlb ˜ σ ( (cid:101) S ) := Im { α (˜ σ ) − (cid:101) S ) → Alb( (cid:101) S ) } and let α ˜ σ : (cid:101) S → Alb ˜ σ ( (cid:101) S )be the morphism defined by α ˜ σ := ( α (˜ σ ) − ◦ α .The following is due to Makoto Enokizono. Proposition 3.2.
Suppose that G is a cyclic group generated by ˜ σ in the above situation.If q (cid:101) θ := q ( (cid:101) S ) − q ( (cid:102) W ) > , then the following hold. (1) dim Alb ˜ σ ( (cid:101) S ) = q (cid:101) θ . (2) If Fix( G ) := { x ∈ (cid:101) S | ˜ σ ( x ) = x } (cid:54) = ∅ , then it is contracted by α ˜ σ to the unit element ∈ Alb ˜ σ ( (cid:101) S ) . (3) If α ˜ σ ( (cid:101) S ) is a curve, then the geometric genus of α ˜ σ ( (cid:101) S ) is not less than q ˜ θ .Proof. Firstly, we show (1). By the construction of α (˜ σ ) − (cid:101) S ) → Alb( (cid:101) S ), we getthe following commutative diagram: H (Alb( (cid:101) S ) , Ω (cid:101) S ) ) ( α (˜ σ ) − ∗ (cid:47) (cid:47) α ∗ (cid:15) (cid:15) H (Alb( (cid:101) S ) , Ω (cid:101) S ) ) α ∗ (cid:15) (cid:15) H ( (cid:101) S, Ω (cid:101) S ) ˜ σ ∗ − (cid:47) (cid:47) H ( (cid:101) S, Ω (cid:101) S ) . Since α ∗ is an isomorphism, we have dim Ker( α (˜ σ ) − ∗ = dim Ker(˜ σ ∗ − G is acyclic group generated by ˜ σ , we see that Ker(˜ σ ∗ −
1) coincides with the G -invariant part H ( (cid:101) S, Ω (cid:101) S ) G of H ( (cid:101) S, Ω (cid:101) S ). On the other hand, since ˜ θ ∗ : H ( (cid:102) W , Ω (cid:102) W ) → H ( (cid:101) S, Ω (cid:101) S ) G is anisomorphism, we have dim(Ker( α (˜ σ ) − ∗ ) = q ( (cid:102) W ) and, hence,dim(Im( α (˜ σ ) − ∗ ) = q ( (cid:101) S ) − q ( (cid:102) W ) . It follows that Alb ˜ σ ( (cid:101) S ) is of dimension q (cid:101) θ . econdly, we show (2). We take a point x in Fix( G ) as the base point of the Albanesemap α : (cid:101) S → Alb( (cid:101) S ) . Let x ∈ Fix( G ). Note that we have α ˜ σ ( x ) = ( α (˜ σ ) − α ( x )) = α (˜ σ )( α ( x )) − α ( x )and that α (˜ σ )( α ( x )) − α ( x ) is the function given by ω (cid:55)→ (cid:82) xx ˜ σ ∗ ω − (cid:82) xx ω for ω ∈ H ( (cid:101) S, Ω (cid:101) S )modulo periods. Since x and x are both in Fix( G ), we find (cid:90) xx ˜ σ ∗ ω − (cid:90) xx ω = (cid:90) ˜ σ ( x )˜ σ ( x ) ω − (cid:90) xx ω = 0 . Hence we get α ˜ σ ( x ) = α (˜ σ )( α ( x )) − α ( x ) = 0. Since x can be taken arbitrarily in Fix( G ),we get (2).When α ˜ σ ( (cid:101) S ) is a curve, one can check easily that the geometric genus of α ˜ σ ( (cid:101) S ) is notless than q ˜ θ , by (1) and the universality of the Abel-Jacobi map for the normalization of α ˜ σ ( (cid:101) S ). Hence (3). (cid:3) The slope inequality in the case of h = 0 . We consider the primitive cycliccovering fibration f : S → B of type ( g, , n ) with q f >
0. Since ϕ : W → B is a ruledsurface, we have q ( W ) = b and it follows q ˜ θ = q f . We apply Proposition 3.2 to the cycliccovering ˜ θ : (cid:101) S → (cid:102) W to find that its ramification divisor Fix( G ) is contracted to a point by α ˜ σ : (cid:101) S → Alb ˜ σ ( (cid:101) S ), where ˜ σ denotes a generator of the Galois group G := Gal( (cid:101) S/ (cid:102) W ). So if α ˜ σ ( (cid:101) S ) is a surface (resp. a curve), from Mumford’s theorem (resp. Zariski’s Lemma), theintersection form is negative definite (resp. semi-definite) on Fix( G ), and we in particularget Fix( G ) < ≤ . Hence, in any way, we have (cid:101) R ≤ , (3.1)since ˜ θ ∗ (cid:101) R = n Fix( G ).Here, we remark the following. Lemma 3.3.
Let f : S → B be a primitive cyclic covering of type ( g, , n ) . If it is notlocally trivial and q f > , then r ≥ n .Proof. We assume that r < n and show that this leads us to a contradiction.Recall that r is a multiple of n . If r < n , then r ≤ n and R has to be smooth byLemmas 1.2 and 1.4. On the other hand, since q f >
0, we already know from (3.1) thatthe self-intersection number of any irreducible component C of R is non-positive.Let C be the minimal section with C = − e and Γ a fiber of ϕ : W → B . Note thatwe can choose the normalized vector bundle of rank 2 associated with W so that thereare no effective divisor numerically equivalent to C − c Γ for any positive integer c . Put C ≡ aC + b Γ with two integers a , b . We have a ≥
0. If a = 0, then we have b = 1, thatis, C is a single fiber by its irreducibility. So we may assume that a is positive.We have C = a (2 b − ae ) ≤
0. Hence 2 b ≤ ae . Furthermore, since C is irreducible and a >
0, we have ( K ϕ + C ) C ≥ C . Since K ϕ ≡ − C − e Γ, we have0 ≤ ( K ϕ + C ) C = ( a − b − ae ) ≤ , by 2 b ≤ ae and a ≥
1. Hence we get ( K ϕ + C ) C = 0 and, either a = 1 or 2 b = ae .In particular, as the first equality shows, C is smooth and ϕ | C : C → B is unramified. urthermore, we get C = 0 when a ≥ b = ae . In this case, we also have b ≤ ≤ CC = b − ae = − b . If a = 1 and 2 b < e , then b ≥ CC = b − e < − b ≤ C = C by the irreducibility of C . We remark herethat we have ( a C + b Γ)( a C + b Γ) = 0 when a i > b i = a i e for i = 1 , R are (i) R consists of several fibers(including the case R = 0), (ii) R is the minimal section with R <
0, and (iii) R consistsof several smooth curves with self-intersection numbers 0 which are unramified over B (via ϕ ). If (i) or (ii) is the case, then we have either g = 0 or r = 1, any of which isabsurd. If (iii) is the case, then f : S → B is a locally trivial fibration, which is againinadequate. (cid:3) From (cid:101) R = R − (cid:80) k ≥ n k α k , (2.15) and (3.1), we get2 rM ≤ nk ≤ r (cid:88) k ≥ n k α k . Hence from this and (2.16), we get α ≤ nk ≤ r (cid:88) k ≥ nkr ( r − − ( nk − α k . (3.2) Theorem 3.4.
Let f : S → B be a primitive cyclic covering fibration of type ( g, , n ) which is not locally trivial and q f > . Then λ f ≥ λ g,n := 8 − g + n − n − g − ( n − n − (cid:18) = 8 − r ( n − r − n ) (cid:19) . Proof.
For λ ∈ R , we put A ( λ ) := n − n (cid:18) ( r − n − r − λ (2 r − n − r (cid:19) . From Proposition 2 .
2, we get( r − K f − λ g,n χ f ) ≥ A ( λ g,n ) α + nk ≤ r (cid:88) k ≥ a k α k − nk ≤ r (cid:88) k ≥ λ g,n ¯ a k α k where a k := ( n − k ( r − − ( nk − − ( r − n, ¯ a k := n − k ( r − − ( nk − . We can check that A ( λ g,n ) ≤ r ≥ n by Lemma 3.3, a calculationshows that the inequality A ( λ g,n ) = n − n (cid:18) ( r − n − r − (cid:0) − r ( n − r − n ) (cid:1) (2 r − n − r (cid:19) ≤ r − n )( − ( n − + 2) + 2 n − n − r ≤ , nd we can check easily its validity. Therefore A ( λ g,n ) ≤
0. Hence from (3 . r − K f − λ g,n χ f ) ≥ nk ≤ r (cid:88) k ≥ (cid:18) ( n − r − r (8 − λ g,n ) nk ( r − nk ) − ( r − n (cid:19) α k . (3.3)For any integer k satisfying r n ≥ k ≥
1, we have nk ( r − nk ) ≥ n ( r − n ). Since we have( n − r − r (8 − λ g,n ) n ( r − n ) − ( r − n = 0 , the coefficient of α k in the right hand side of (3 .
3) is not negative. Therefore, we get K f − λ g,n χ f ≥ (cid:3) The slope inequality in the case of h ≥ . Before showing the slope inequalitywhen h ≥ n ≥
3, we study the upper bound of α .Recall that we decomposed ψ into a succession of blowing-ups ψ i as, ψ : (cid:102) W = W N ψ N → W N − → · · · → W ψ → W = W We define the order of blowing-up ψ (cid:48) appearing in ψ as follows. If the center of ψ (cid:48) is apoint on the branch locus of multiplicity m (cid:48) , we putord( ψ (cid:48) ) := (cid:20) m (cid:48) n (cid:21) . Moreover we introduce a partial order on these blowing-ups ψ (cid:48) and ψ (cid:48)(cid:48) appearing in ψ , ψ (cid:48) ≥ ψ (cid:48)(cid:48) def ⇐⇒ ord( ψ (cid:48) ) ≥ ord( ψ (cid:48)(cid:48) ) . Lemma 3.5.
Assume that n ≥ . Let x j ( ∈ R j ⊂ W j ) be a singular point infinitely nearto x i ∈ R i . Then the multiplicities satisfy m j ≤ m i .Proof. Though this can be found in [2], Lemma 3.7, when n = 0, we shall give a proof forthe convenience of readers.Let x i +1 be the singular point of R i +1 infinitely near to x i ∈ R i . If m i ∈ n Z , then R i +1 coincied with (cid:99) R i , the proper transform of R i by ψ i +1 , by Lemma 1 .
2. Hence m i +1 ≤ m i inthis case. If m i ∈ n Z +1, then R i +1 = (cid:99) R i + E i +1 . Hence we get m i +1 ≤ m i + 1 ∈ n Z + 2.From Lemma 1 . n ≥
3, we get m i +1 ≤ m i . (cid:3) From Lemma 3.5, we can reorder those blowing-ups appearing in ψ so that ψ i ≥ ψ j holds whenever i < j . We put, M := max { ord( ψ (cid:48) ) | ψ (cid:48) is a blowing up in ψ } . Then we can decompose ψ as ψ : (cid:102) W = (cid:99) W M ˆ ψ M → (cid:99) W M − · · · ˆ ψ → (cid:99) W = W in such a way that ord( ψ (cid:48) ) = M + 1 − i holds for any ψ (cid:48) appearing in ˆ ψ i . Lemma 3.6.
Let ψ (cid:48) be a blowing-up appearing in ˆ ψ i and (cid:101) D the proper inverse image ofthe exceptional curve of ψ (cid:48) on (cid:101) S . Then the geometric genus of (cid:101) D satisfies g ( (cid:101) D ) ≤ ( n − n ( M − i ) + n − . roof. Let m (cid:48) be the multiplicity of the singular point blown up by ψ (cid:48) , and (cid:101) E the propertransform of the exceptional curve of ψ (cid:48) on (cid:102) W .When m (cid:48) ∈ n Z + 1, since (cid:101) E is contained in (cid:101) R , (cid:101) D is a smooth rational curve.Assume that m (cid:48) ∈ n Z . From m (cid:48) = n ( M + 1 − i ), the intersection number of theexceptional curve of ψ (cid:48) and the branch locus is n ( M + 1 − i ). Hence the intersectionnumber of their proper transforms on (cid:102) W is at most n ( M + 1 − i ). On the other hand, weconsider the composite π : (cid:102) D (cid:48) → (cid:101) D ˜ θ | (cid:101) D → (cid:101) E , where (cid:102) D (cid:48) → (cid:101) D is normalization of (cid:101) D , and let B π be the branch locus π . From the Hurwitz formula for π and Lemma 1 .
6, we get2 g ( (cid:102) D (cid:48) ) − n ≤ ( n − (cid:93)B π . Since ˜ θ | ˜ D is totally ramified, (cid:93)B π ≤ (cid:93)B ˜ θ | ˜ D ≤ (cid:101) E. (cid:101) R ≤ n ( M + 1 − i ) . Therefore we get g ( (cid:101) D ) = g ( (cid:102) D (cid:48) ) ≤ ( n − n ( M − i ) + n − , which is what we want. (cid:3) Proposition 3.7.
Let f : S → B be a primitive cyclic covering fibration of type ( g, h, n ) such that q ˜ θ > , h ≥ , n ≥ and let α i ( i ≥ be the singularity index in Definition . . Then, (cid:0) g − − n ( h − (cid:1) α ≤ (cid:0) g − − n ( h − (cid:1) K ϕ ( n − h −
1) + T + (cid:88) k ≥ n ( n − n + 1) ¯ a k α k , (3.4) where ¯ a k is defined in Proposition . . If the image α ˜ σ ( (cid:101) S ) is a curve and ν ( q ˜ θ ) ≥ , where ν ( x ) := (cid:20) x − n ( n − − n − n (cid:21) , then (cid:0) g − − n ( h − (cid:1)(cid:0) α + ν ( q ˜ θ ) (cid:88) k =1 nk ( nk − α k (cid:1) ≤ (cid:0) g − − n ( h − (cid:1) K ϕ ( n − h −
1) + T + (cid:88) k ≥ ν ( q ˜ θ )+1 n ( n − n + 1) ¯ a k α k . (3.5) In (3 . and (3 . , the quantity K ϕ ( n − h − is understood to be zero when h = 1 .Proof. Firstly assume that α ˜ σ ( (cid:101) S ) is a curve of geometric genus g (cid:48) . In this case, by Propo-sition 3.2, we have g (cid:48) ≥ q ˜ θ and see that any curve of geometric genus less than g (cid:48) on (cid:101) S iscontracted by α ˜ σ . Hence, we know from Lemma 3 . ≤ i ≤ M satisfying( n − n ( M − i ) + n − ≤ g (cid:48) − , he proper transform of the exceptional curve of ˆ ψ i to (cid:101) S is contracted by α ˜ σ . Then, since q ˜ θ ≤ g (cid:48) , for any 1 ≤ i ≤ M satisfying( n − n ( M − i ) + n − ≤ q ˜ θ − , the same holds true. So we conclude that the total inverse image of (cid:98) R M − ν ( q ˜ θ ) in (cid:101) S iscontracted by α ˜ σ , where (cid:98) R M − ν ( q ˜ θ ) ⊂ (cid:99) W M − ν ( q ˜ θ ) is the image of (cid:101) R . Therefore, the totalinverse image of (cid:98) R M − ν ( q ˜ θ ) forms a negative semi-definite configuration. In particular, wehave (cid:98) R M − ν ( q ˜ θ ) ≤ . (3.6)By the construction, we have (cid:98) R M − ν ( q ˜ θ ) = R − (cid:88) k>ν ( q ˜ θ ) n k α k = ˆ x K ϕ ( n − h −
1) + ˆ yT + ˆ z ( K ϕ + R ) R − (cid:88) k>ν ( q ˜ θ ) n k α k , where ˆ x = − (cid:0) g − − n ( h − (cid:1) t , ˆ y = − t , ˆ z = (cid:0) g − − n ( h − (cid:1) t . Hence from (2 . .
6) and the above equality, we get t ˆ z (cid:0) α + ν ( q ˜ θ ) (cid:88) k =1 nk ( nk − α k (cid:1) ≤ − t ˆ x K ϕ ( n − h − − t ˆ yT + (cid:88) k>ν ( q ˜ θ ) t (cid:0) n k − nk ( nk − z (cid:1) α k . This is nothing more than (3 . α ˜ σ ( (cid:101) S ) is not a curve, we have (cid:101) R ≤ .
2. Using this insteadof (3.6), we get (3 .
4) by a similar argument as above. (cid:3)
Theorem 3.8.
Let f : S → B be a primitive cyclic covering fibration of type ( g, h, n ) which is not locally trivial and such that q ˜ θ and h are both positive, n ≥ . Put F ( g, h, l ) = ( g − − n ( g − (cid:0) ( h + 1)( n − l + 1) − (cid:1) − (cid:0) ( l + 1)( n − − (cid:1) n ( h − . (i) If F ( g, h, ≥ , then λ f ≥ λ g,h,n := 8 − n + 1) (cid:0) g − − n ( h − (cid:1) a . (ii) Assume that α ˜ σ ( (cid:101) S ) is a curve and ν ( q ˜ θ ) ≥ . If F ( g, h, ν ( q ˜ θ )) ≥ , then λ f ≥ λ g,h,n,q ˜ θ := 8 − n + 1) (cid:0) g − − n ( h − (cid:1) a ν ( q ˜ θ )+1 . Proof.
Here we restrict ourselves to the case that α ˜ σ ( (cid:101) S ) is a curve and show (ii) only,since (i) can be shown similarly. From (2.17) and (2.18), we obtain t ( K f − λ g,h,n,q ˜ θ χ f ) ≥ t ( x (cid:48) − λ g,h,n,q ˜ θ ¯ x (cid:48) ) K ϕ ( n − h −
1) + t ( y (cid:48) − λ g,h,n,q ˜ θ ¯ y (cid:48) ) T + t ( z (cid:48) − λ g,h,n,q ˜ θ ¯ z (cid:48) ) α + (cid:88) k ≥ ( a k − λ g,h,n,q ˜ θ ¯ a k ) α k − nλ g,h,n,q ˜ θ tχ ϕ (3.7) o apply (3 .
5) to the above inequality, we have to check that z (cid:48) − λ g,h,n,q ˜ θ ¯ z (cid:48) ≤ λ g,h,n,q ˜ θ ≥ − n + 1) (cid:0) g − − n ( h − (cid:1) a (3.8) = 8 − (cid:0) g − − n ( h − (cid:1) ( n − (cid:0) g −
1) + n ( h − n − (cid:1) , it is sufficient to see that8 − (cid:0) g − − n ( h − (cid:1) ( n − (cid:0) g −
1) + n ( h − n − (cid:1) ≥ n − g − n − g − − n ( n + 1)( h − , which is equivalent to( g − − n ( h − (cid:0) g − n ( n −
2) + 8 n ( n + 1)( h − n − n −
3) + 1) (cid:1) ≥ , the validity of which can be checked directly. Therefore we get z (cid:48) − λ g,h,n,q ˜ θ ¯ z (cid:48) ≤
0. Applying(3 .
5) to (3 . t ( K f − λ g,h,n,q ˜ θ χ f ) ≥ n − t (cid:18) g − − λ g,h,n,q ˜ θ (cid:0) g − − n ( h − (cid:1)(cid:19) K ϕ ( n − h − − nλ g,h,n,q ˜ θ tχ ϕ + ( n − − λ g,h,n,q ˜ θ )8 (cid:0) g − − n ( h − (cid:1) tT + nt ν ( q ˜ θ ) (cid:88) k =1 (cid:0) ( n − k ((2 n − k − λ g,h,n,q ˜ θ − n − k − (cid:1) α k + t (cid:88) k>ν ( q ˜ θ ) (cid:18) (8 − λ g,h,n,q ˜ θ )3 n n + 1) (cid:0) g − − n ( h − (cid:1) ¯ a k − n (cid:19) α k . We will show that (cid:0) ( n − k ((2 n − k − λ g,h,n,q ˜ θ − n − k − (cid:1) ≥ ≤ k ≤ ν ( q ˜ θ ) . Note that( n − k ((2 n − k − λ g,h,n,q ˜ θ − n − k − = 1 n (cid:18) nk (cid:0) ( n − nk − λ g,h,n,q ˜ θ (2 n − − n − n − − λ g,h,n,q ˜ θ ) (cid:1) − n (cid:19) . Firstly we will show that λ g,h,n,q ˜ θ ≥ n − n − . From (3 . − (cid:0) g − − n ( h − (cid:1) ( n − (cid:0) g −
1) + n ( h − n − (cid:1) ≥ n − n − . A calculation shows that it is equivalent to8 n ( n − g −
1) + (4( n − n −
3) + 8(2 n − n ( h − ≥ , which holds true clearly. So we have shown λ g,h,n,q ˜ θ ≥ n − n − and it follows that( n − k ((2 n − k − λ g,h,n,q ˜ θ − n − k − s increasing in k . Evaluating at k = 1, we get( n − k ((2 n − k − λ g,h,n,q ˜ θ − n − k − ≥ n − (cid:0) ( n − λ g,h,n,q ˜ θ − n − (cid:1) ≥ , by λ g,h,n,q ˜ θ ≥ n − n − . Since(8 − λ g,h,q ˜ θ )3 n n + 1) (cid:0) g − − n ( h − (cid:1) ¯ a k − n ≥ (8 − λ g,h,q ˜ θ )3 n n + 1) (cid:0) g − − n ( h − (cid:1) ¯ a ν ( q ˜ θ )+1 − n = 0(3.9)holds for any k ≥ ν ( q ˜ θ ) + 1, we obtain K f − λ g,h,n,q ˜ θ χ f ≥ n − (cid:18) g − − λ g,h,n,q ˜ θ (cid:0) g − − n ( h − (cid:1)(cid:19) K ϕ ( n − h − − nλ g,h,n,q ˜ θ χ ϕ + ( n − − λ g,h,n,q ˜ θ )8 (cid:0) g − − n ( h − (cid:1) T If F ( g, h, ν ( q ˜ θ )) ≥ h = 1, then by (2 .
19) we have T = 2( g − K ϕ .R ≥ g − n − χ ϕ .Hence it follows from (3 .
10) that K f − λ g, ,n,q ˜ θ χ f ≥ n + 13¯ a ν ( q ˜ θ )+1 F ( g, , ν ( q ˜ θ )) χ ϕ ≥ h = 1. If F ( g, h, ν ( q ˜ θ )) ≥ h ≥
2, then we can use Xiao’s slopeinequality K ϕ ≥ h − h χ ϕ and T ≥
0, to get K f − λ g,h,n,q ˜ θ χ f ≥ n + 13 h ¯ a ν ( q ˜ θ )+1 F ( g, h, ν ( q ˜ θ )) χ ϕ ≥ . h ≥ (cid:3) Special irregular cyclic covering fibrations of ruled surfaces.
Let f : S → B be a primitive cyclic covering fibration of type ( g, , n ) with q f > α ˜ σ : (cid:101) S → Alb ˜ σ ( (cid:101) S ) be the morphism defined as inDefinition 3.1 for the generator ˜ σ of the covering transformation group G of ˜ θ : (cid:101) S → (cid:102) W .Moreover we assume that there is a component C of Fix( G ) such that C = 0. Note thenthat α ˜ σ ( (cid:101) S ) is a curve by Proposition 3.2. Proposition 4.1.
In the above situation, there are a fibrations ˜ f (cid:48) : (cid:101) S → B (cid:48) , ϕ (cid:48) : (cid:102) W → P and a morphism θ (cid:48) : B (cid:48) → P , where B (cid:48) is s smooth curve, such that (cid:101) R is ˜ ϕ (cid:48) -vertical, q f ≤ g ( B (cid:48) ) , and they fit into the commutative diagram: (cid:101) S ˜ θ (cid:47) (cid:47) ˜ f (cid:48) (cid:15) (cid:15) (cid:102) W ˜ ϕ (cid:48) (cid:15) (cid:15) B (cid:48) θ (cid:48) (cid:47) (cid:47) P . roof. We can obtain ˜ f (cid:48) : (cid:101) S → B (cid:48) from the Stein factorization of α ˜ σ : (cid:101) S → α ˜ σ ( (cid:101) S ).Hence we have g ( B (cid:48) ) ≥ q f by Proposition 3.2, (3). We will show that the automorphism˜ σ : (cid:101) S → (cid:101) S induces an automorphism of B (cid:48) . We assume that there is a fiber F (cid:48) of ˜ f (cid:48) suchthat ˜ σ ∗ F (cid:48) has a ˜ f (cid:48) -horizontal component. Let F (cid:48) C be the fiber of f (cid:48) which contains thecurve C with C = 0. Then, from Zariski’s lemma, we see that F (cid:48) C = aC for some positiveinteger a and it follows F (cid:48) C = ˜ σ ∗ F (cid:48) C , since C is a component of Fix( G ). Hence0 < (˜ σ ∗ F (cid:48) .F (cid:48) C ) = (˜ σ ∗ F (cid:48) . ˜ σ ∗ F (cid:48) C ) = ( F (cid:48) .F (cid:48) C ) = 0 , a contradiction. Therefore ˜ σ maps fibers to fibers, and descends down to give an auto-morphism ˜ σ B (cid:48) : B (cid:48) → B (cid:48) . Furthermore we have the commutative diagram (cid:101) S ˜ θ (cid:47) (cid:47) ˜ f (cid:48) (cid:15) (cid:15) (cid:102) W ˜ ϕ (cid:48) (cid:15) (cid:15) ˜ ϕ (cid:47) (cid:47) BB (cid:48) θ (cid:48) (cid:47) (cid:47) D (cid:48) , where θ (cid:48) : B (cid:48) → D (cid:48) := B (cid:48) / (cid:104) ˜ σ B (cid:48) (cid:105) denotes the quotient map. In order to complete the proof,it suffices to see that D (cid:48) = P . This can be shown as follows. Any general fiber of ˜ ϕ is˜ ϕ (cid:48) -horizontal by (cid:101) R. (cid:101) Γ >
0. Since ˜ ϕ is ruled, we see that P dominates D (cid:48) and it follows D (cid:48) = P . (cid:3) The contraction ϕ : (cid:102) W → W is composed of several blowing-ups. We decompose it as ψ = ˇ ψ ◦ ¯ ψ as follows. Let ¯ ψ : (cid:102) W → W be the longest succession of blowing-downs suchthat we still have the morphism ¯ ϕ (cid:48) satisfying ˜ ϕ (cid:48) = ¯ ϕ (cid:48) ◦ ¯ ψ . Then we have the followingcommutative diagram. P (cid:102) W ˜ ϕ (cid:48) (cid:62) (cid:62) ¯ ψ (cid:47) (cid:47) ˜ ϕ (cid:32) (cid:32) W ¯ ϕ (cid:15) (cid:15) ˇ ψ (cid:47) (cid:47) ¯ ϕ (cid:48) (cid:79) (cid:79) W ϕ (cid:126) (cid:126) B Let R := ¯ ψ ∗ (cid:101) R be the image of (cid:101) R by ¯ ψ . Lemma 4.2.
The morphism ˇ ψ : W → W is not the identity map.Proof. We will prove this by contradiction. Suppose that ˇ ψ is the identity map. As onesees from the proof of Lemma 3.3, any irreducible curve D on W with D ≤ ϕ | D : D → B is an unramified covering when D = 0 and D is not a fiber of ϕ .Hence, any irreducible fiber of ¯ ϕ (cid:48) : W → P has to be smooth. Suppose that there isa fiber of ¯ ϕ (cid:48) whose reduced scheme is reducible, and take an irreducible component D .Then D < D coincides withthe minimal section. The unicity implies that we cannot have such reducible singularfibers. Therefore, a singular fiber of ¯ ϕ (cid:48) , if any, is a multiple fiber whose support is asmooth irreducible curve. Since R is a reduced divisor with support in fibers of ¯ ϕ (cid:48) byProposition 4.1, we see that R is smooth and ϕ | R : R → B is unramified. Then f : S → B is a locally trivial fibration, which is inadequate. (cid:3) ssume that θ (cid:48) : B (cid:48) → P is branched over ∆ ⊂ P . For any y ∈ ∆, let (cid:102) Γ (cid:48) y = (cid:80) ˜ n C C be the fiber of ˜ ϕ (cid:48) over y , and put (cid:101) R all := ˜ ϕ (cid:48)∗ ∆ , (cid:101) R r := (cid:88) y ∈ ∆ , C ⊂ (cid:102) Γ (cid:48) y , ˜ n C =1 C. Lemma 4.3.
In the above situation, (cid:101) R r (cid:22) (cid:101) R. Proof.
We put G ˜ f (cid:48) := { τ ∈ G | τ ( (cid:101) F (cid:48) ) = (cid:101) F (cid:48) for any fiber (cid:101) F (cid:48) of ˜ f (cid:48) } Since ˜ f (cid:48) ◦ τ = τ for any τ ∈ G ˜ f (cid:48) , the morphism ˜ f (cid:48) induces the morphism π : (cid:101) S/G ˜ f (cid:48) → (cid:102) W and we have the following commutative diagram. (cid:101) S (cid:33) (cid:33) ˜ θ (cid:29) (cid:29) ˜ f (cid:48) (cid:37) (cid:37) (cid:101) S/G ˜ f (cid:48) (cid:15) (cid:15) π (cid:47) (cid:47) (cid:102) W ˜ ϕ (cid:48) (cid:15) (cid:15) B (cid:48) θ (cid:48) (cid:47) (cid:47) P Note that the degree of π is equal to that of θ (cid:48) . We claim that Fix( G ˜ f (cid:48) ) = Fix( G ).This can be see as follows. It is clear that Fix( G ˜ f (cid:48) ) ⊃ Fix( G ). If there is a point x ∈ Fix( G ˜ f (cid:48) ) \ Fix( G ), then we have ˜ σ ( x ) (cid:54) = x for the generator ˜ σ of G . On the otherhand, since G ˜ f (cid:48) is a subgroup of G of order n/ deg θ (cid:48) , we have G ˜ f (cid:48) = (cid:104) ˜ σ deg θ (cid:48) (cid:105) . Hence thenumber of G -orbits of x is at most n/ deg θ (cid:48) . This contradicts that ˜ θ : (cid:101) S → (cid:102) W is totallyramified. Therefore Fix( G ˜ f (cid:48) ) = Fix( G ). Hence (cid:101) S/G ˜ f (cid:48) is smooth. Let R π be the branchlocus of π . Since ˜ θ is totally ramified, one can check easily that R π = (cid:101) R . Hence it issufficient to prove that (cid:101) R r ≤ R π . Let C be any component of (cid:101) R r . We can take analyticlocal coordinates ( U P , x ) on P , ( U (cid:102) W , y, z ) on (cid:102) W and ( U B (cid:48) , w ) on B (cid:48) such that ˜ ϕ (cid:48) ( C ) isdefined by x = 0, C is defined by y = 0, θ (cid:48)∗ x = w deg θ (cid:48) and ¯ ϕ (cid:48)∗ x = y . U B (cid:48) × P U (cid:102) W is definedby y = w deg θ (cid:48) in U B (cid:48) × U (cid:102) W . So U B (cid:48) × P U (cid:102) W → U (cid:102) W is ramified over C ∩ U (cid:102) W and U B (cid:48) × P U (cid:102) W is smooth. Hence the natural morphism (cid:101) S/G ˜ f (cid:48) → B (cid:48) × P (cid:102) W is an isomorphism around U B (cid:48) × P U (cid:102) W . Therefore we get C (cid:22) R π = (cid:101) R . (cid:3) We suppose that ˇ ψ = ˇ ψ ◦ · · · ◦ ˇ ψ u , where ˇ ψ i : ˇ W i → ˇ W i − is a blowing-up atˇ x i − ∈ ˇ W i − with exceptional curve ˇ E i ⊂ ˇ W , ˇ W = W and ˇ W u = W . Let ˇ R i be the imageof R in ˇ W i , and let ˇ x i be a singular point of ˇ R i of multiplicity ˇ m i . Lemma 4.4.
Assume that n ≥ . For ≤ i ≤ u − , we have ˇ m i ≥ q f n − + 2 . Moreover ifthere is ˇ m i such that equality sign holds, then q f n − + 2 ∈ n Z and deg θ (cid:48) = n .Proof. Let
E ⊂ W be any ( − ψ . Note that ¯ ϕ (cid:48) | : E → P is surjectiveand E .R ∈ n Z . We will show that E .R ≥ q f n − + 2. If n ≥ q f n − + 2, then the assertion isclear from Lemma 1.2, (1). Hence we may assume that n < q f n − + 2 in the following. Inparticular, we have q f ≥ n − y Lemma 4 .
3, it is enough to show that E .R r ≥ q f n − + 2 where R r is image of (cid:101) R r . Let R θ (cid:48) be the ramification divisor of θ (cid:48) : B (cid:48) → P . Since g ( B (cid:48) ) ≥ q f >
0, we have deg θ (cid:48) > R θ (cid:48) = 2 g ( B (cid:48) ) − θ (cid:48) . (4.1)By Lemma 1 . . θ (cid:48) − (cid:93) ∆ ≥ g ( B (cid:48) ) − θ (cid:48) , that is, (cid:93) ∆ ≥ θ (cid:48) − g ( B (cid:48) ) + 2 . We put R all := ¯ ψ (cid:48)∗ ∆. For any p ∈ E ∩ R all , let r p := I p ( E .R all ) be the local intersectionnumber. Since R all = ¯ ψ (cid:48)∗ ∆ consists of (cid:93) ∆ fibers of ¯ ϕ (cid:48) , one has (cid:88) E∩ R all r p = ( E .R all ) = deg( ϕ (cid:48) | E ) (cid:93) ∆ ≥ (deg ¯ ϕ (cid:48) | E ) (cid:18) θ (cid:48) − g ( B (cid:48) ) + 2 (cid:19) . (4.2)By definition, r p ≥ p ∈ ( E ∩ R all ) \ ( E ∩ R r ). On the other hand, by the Hurwitzformula for ¯ ϕ (cid:48) | E : E → P , one has2 deg ¯ ϕ (cid:48) | E − R θ (cid:48) ≥ (cid:88) E∩ R all ( r p − . Hence, 2 deg ¯ ϕ (cid:48) | E − ≥ (cid:88) E∩ R all ( r p −
1) = (cid:88) ( E∩ R all ) \ ( E∩ R r ) ( r p −
1) + (cid:88) E∩ R r ( r p − ≥ (cid:88) ( E∩ R all ) \ ( E∩ R r ) r p (cid:88) E∩ R r ( r p − (cid:88) E∩ R all r p − (cid:93) ( E ∩ R r )2 ≥ (deg ¯ ϕ (cid:48) | E ) (cid:18) θ (cid:48) − g ( B (cid:48) ) + 1 (cid:19) − (cid:93) ( E ∩ R r )2 , where the last inequality comes from (4 . g ( B (cid:48) ) ≥ q f ≥ n − ≥ deg θ (cid:48) −
1. Therefore( E .R ) ≥ ( E .R r ) ≥ (cid:93) ( E ∩ R r ) ≥ (deg ¯ ϕ (cid:48) | E ) (cid:18) θ (cid:48) − g ( B (cid:48) ) − (cid:19) + 4 ≥ θ (cid:48) − g ( B (cid:48) ) + 2 ≥ θ (cid:48) − q f + 2 . For any ˇ x i , let ˇ x i + j i be the last infinitely near singular point blown up by ˇ ψ , E i + j i theexceptional curve. Then, from the above argument, we get2deg θ (cid:48) − q f + 2 ≤ ( E i + j i .R ) = ˇ m i + j i ≤ ˇ m i . Moreover if the equality signs hold everywhere, then we getˇ m i = 2deg θ (cid:48) − q f + 2 = ( E i + j i .R ) ∈ n Z . ince deg θ (cid:48) ≤ n , we are done. (cid:3) Theorem 4.5.
Let f : S → B be a locally non-trivial primitive cyclic covering fibrationof type ( g, , n ) with q f > and n ≥ . Assume that there is a component C of Fix(˜ σ ) such that C = 0 . Then, q f ≤ g − n + 12 . Proof.
There is a singular point x of R which is blown up by ˇ ψ : W → W from Lemma4 .
2. If m := mult x R ≤ r , then2 q f n − ≤ m ≤ r gn − .
4. So we get q f ≤ ( g − n + 1) /
2. If m > r , then we have m ∈ n Z + 1 fromLemma 1 .
4. Let ˇ x be the last singular point, infinitely near to x , blown up by ˇ ψ and ˇ m its multiplicity. Then it holds that ˇ m ∈ n Z . Indeed, if ˇ m ∈ n Z + 1, the exceptional curvearizing from ˇ x is contained in branch locus. It contradicts the definition of ˇ ψ . So we getˇ m + 1 ≤ m . Therefore we get2 q f n − ≤ ˇ m + 1 ≤ m ≤ r gn − . .
4. It follows q f ≤ ( g − n + 1) / (cid:3) Therefore, the Modified Xiao’s Conjecture is true in this particular case.Now, we turn our attention to the slope. Let ˇ α k be the number of the singular pointsof R with multiplicity nk or nk + 1 appearing in ˇ ψ . Then ˇ α k ≥ . α k = 0(4.3)for any k satisfying nk + 1 ≤ q f n − + 2. We put ¯ α k := α k − ˇ α k then by (4 . α k = α k (4.4)for any k satisfying nk + 1 ≤ q f n − + 2. By the construction of ¯ ϕ (cid:48) , R is contained in fibersof ¯ ϕ (cid:48) , hence we get R ≤ . On the other hand, we have R = R − nk ≤ r (cid:88) qfn − +2 ≤ nk n k ˇ α k As R = 2 rM by (2 . rM ≤ nk ≤ r (cid:88) qfn − +2 ≤ nk n k ˇ α k (4.5) ence α + nk +1 ≤ qfn − +2 (cid:88) k ≥ nk ( nk − α k ≤ α + nk ≤ r (cid:88) k ≥ nk ( nk −
1) ¯ α k ≤ nk ≤ r (cid:88) qfn − +2 ≤ nk nkr ( r − − ( nk − α k ≤ nk ≤ r (cid:88) qfn − +2 ≤ nk nkr ( r − − ( nk − α k , where the first and the last inequalities above follow immediately from ¯ α k ≥ . .
16) and (4 . Proposition 4.6.
Under the same assumptions as in Proposition . , α + nk +1 ≤ qfn − +2 (cid:88) k ≥ nk ( nk − α k ≤ nk ≤ r (cid:88) qfn − +2 ≤ nk nkr ( r − − ( nk − α k . Using this, we will prove the following:
Theorem 4.7.
Let f : S → B be a locally non-trivial primitive cyclic covering fibrationof type ( g, , n ) such that there is a component C ⊂ Fix( G ) with C = 0 . If q f > and n ≥ , then λ f ≥ λ g,n,q f := 8 − n ( g + n − g − q f )( q f + n − (cid:18) = 8 − rn ( n − q f n − )( r − (2 + q f n − )) (cid:19) . (4.6) Proof.
We first remark that, for two real numbers x , y with x + y ≤ r , we have x ( r − x ) ≥ y ( r − y ) if and only if x ≥ y . Since we have n + (2 + 2 q f / ( n − ≤ r by the proof ofTheorem 4.5 and r ≥ n , this observation works for x = n , y = 2 + 2 q f / ( n − n ≥ q f n − , i.e., ( n − n − ≥ q f .Since n ≥ q f n − , we get n ( r − n ) ≥ (cid:0) q f n − (cid:1)(cid:0) r − (2 + 2 q f n − (cid:1) . Therefore, λ g,n ≥ λ g,n,q f , and (4 .
6) follows from Theorem 3 . n < q f n − .In this case, we have (cid:0) q f n − (cid:1)(cid:0) r − (2 + 2 q f n − (cid:1) ≥ n ( r − n )and, hence, λ g,n ≤ λ g,n,q f . Then we have A ( λ g,n,q f ) ≤
0, since the function A ( λ ) definedin the proof of Proposition 3.4 is decreasing in λ and we have already proved A ( λ g,n ) ≤ rom Proposition 2.2, we have( r − K f − λ g,n,q f χ f ) ≥ A ( λ g,n,q f ) α + nk ≤ r (cid:88) k ≥ ( a k − λ g,n,q f ¯ a k ) α k (4.7)where a k and ¯ a k are the same as in the proof of Proposition 3.4. Applying Proposition 4.6to (4.7), we get( r − K f − λ g,n,q f χ f ) ≥ nk< qfn − +2 (cid:88) k ≥ ( − A ( λ g,n,q f ) nk ( nk −
1) + a k − λ g,n,q f ¯ a k ) α k + nk ≤ r (cid:88) qfn − ≤ nk ( A ( λ g,n,q f ) nkr ( r − nk ) + a k − λ g,n,q f ¯ a k ) α k . First, we will show − A ( λ g,n,q f ) nk ( nk −
1) + a k − λ g,n,q f ¯ a k ≥ k with nk + 1 ≤ q f n − . By a simple calculation, we get − A ( λ g,n,q f ) nk ( nk − α + a k − λ g,n,q f ¯ a k = nk (cid:18) ( n − r − n ( nk − λ g,n,q f (2 n − − n − n − r − n (12 − λ g,n,q f ) (cid:19) − ( r − n. We claim that λ g,n,q f ≥ n − n − . It is equivalent to4( n + 1) q f ( g − n + 1 − q f ) + (2 n − g − n ( n − n − ≥ . From g − n + 1 ≥ q f and q f ≥ ( n − n − + 1, we easily see that it holds true. Since λ g,n,q f ≥ n − n − , the right hand side of (4.9), which is incleasing in k , is not less than n (cid:18) ( n − r − n ( n − λ g,n,q f (2 n − − n − n − r − n (12 − λ g,n,q f ) (cid:19) − ( r − n = n ( n − r − (cid:18) λ g,n,q f ( n − − n − (cid:19) − ( r − n = n ( r − n ( n − n − ≥ . Therefore we get (4.8).Secondly, we will show A ( λ g,n,q f ) nkr ( r − nk ) + a k − λ g,n,q f ¯ a k ≥ or any positive integer k satisfying r ≥ nk ≥ q f n − . By a simple calculation, we get A ( λ g,n,q f ) nkr ( r − nk ) + a k − λ g,n,q f ¯ a k = ( n − r − r nk ( r − nk )(8 − λ g,n,q f ) − ( r − n (4.11)Since we have nk ( r − nk ) ≥ (2 + 2 q f n − (cid:0) r − (2 + 2 q f n − (cid:1) for any positive integer k satisfying 2 + q f n − ≤ nk ≤ r , the right hand side of (4 .
11) is notless than( n − r − r (2 + 2 q f n − (cid:0) r − (2 + 2 q f n − (cid:1) (8 − λ g,n,q f ) − ( r − n = 0 . In sum, we have shown K f − λ g,n,q f χ f ≥ (cid:3) An example.
We construct primitive cyclic covering fibrations of type ( g, , n ) with relative minimalirregularity q f satisfying g + n − m ( q f + n −
1) for any integer m ≥
2. Hence, when m = 2, this implies that the bound of q f in Theorem 4.5 is sharp. Also, our examplesshow that the slope bound (4.6) in Theorem 4.7 is sharp.Let ϕ : W := P ( O P (cid:76) O P ( e )) → B := P be the Hirzebruch surface of degree e ≥ C a fiber of ϕ and the section with C = − e , respectively. We knowthat mC + b Γ is very ample if and only if b > me . So we take b with b > me .We take two general members D, D (cid:48) of | mC + b Γ | which intersect each other trans-versely. Let Λ be the pencil generated by D and D (cid:48) . Then Λ define the rational map ϕ Λ : W · · · → P . Let ψ be a minimal succession blowing-ups which eliminates the basepoints of Λ. We get a relatively minimal fibration ˜ ϕ (cid:48) : (cid:102) W → P by putting ˜ ϕ (cid:48) = ϕ Λ ◦ ψ . De-note by (cid:101) Γ (cid:48) a general fiber of ˜ ϕ (cid:48) and K (cid:102) W a canonical divisor of (cid:102) W . By a simple calculation,we get K (cid:102) W = 8 − x, K (cid:102) W . (cid:101) Γ (cid:48) = m − m x − m, (cid:101) Γ = 0 , (5.1)where x is the a number of blowing-ups in ψ . Note that x = ( mC + b Γ) .Let ∆ ⊂ P be a set of qn − + 2 general points, where q is an integer satisfying qn − + 2 ∈ n Z . Then there is a divisor d (cid:48) on P such that n d (cid:48) = ∆. Let (cid:101) R = ( ˜ ϕ (cid:48) ) ∗ ∆ be the fiber of˜ ϕ (cid:48) over ∆. Since ∆ is general, we can assume that (cid:101) R is both reduced and smooth.We consider a classical cyclic n -covering θ (cid:48) : B (cid:48) = Spec P (cid:18) n − (cid:77) j =0 O P ( − j d (cid:48) ) (cid:19) → P . Since ∆ is general, we can assume that the fiber product (cid:101) S := B × P (cid:102) W is smooth. Notingthat the morphism ˜ θ : (cid:101) S → (cid:102) W induced by θ (cid:48) is nothing but the natural one (cid:101) S = Spec P (cid:18) n − (cid:77) j =0 O (cid:102) W ( − j ( ϕ (cid:48) ) ∗ d (cid:48) ) (cid:19) → (cid:102) W , ne gets a commutative diagram B (cid:101) S ˜ f (cid:79) (cid:79) ˜ θ (cid:47) (cid:47) ˜ f (cid:48) (cid:15) (cid:15) (cid:102) W ˜ ϕ (cid:96) (cid:96) ˜ ϕ (cid:48) (cid:15) (cid:15) B (cid:48) θ (cid:48) (cid:47) (cid:47) P where ˜ f := ˜ ϕ ◦ ˜ θ .By the construction, we get q f = q = g ( B (cid:48) ). From the formulae K (cid:101) S = n ( K (cid:102) W + n − n (cid:101) R ) , χ ( O (cid:101) S ) = nχ ( O (cid:102) W ) + 12 n − (cid:88) j =1 n j (cid:101) R ( 1 n j (cid:101) R + K (cid:102) W ) , and (5 . K (cid:101) S = (cid:18) m − m (cid:18) qn − (cid:19) ( n − − n (cid:19) x + 8 (cid:18) n − m ( n − (cid:18) qn − (cid:19)(cid:19) , (5.2) χ ( O (cid:101) S ) = m − m (cid:18) qn − (cid:19) ( n − x + n − m ( n − (cid:18) qn − (cid:19) . (5.3)Let g be the genus of fibration f : (cid:101) S → B . Then it is easy to see that2 gn − m (cid:18) qn − (cid:19) . Hence we get K f = K (cid:101) S − g − g ( B ) −
1) = (cid:18) m − m (cid:18) qn − (cid:19) ( n − − n (cid:19) x,χ ˜ f = χ ( O (cid:101) S ) − ( g − g ( B ) −
1) = m − m (cid:18) qn − (cid:19) ( n − x. Therefore we get λ ˜ f = K f χ ˜ f = 8 − n ( g + n − g − q f )( q f + n − q = q f .We remark that ˜ f is relatively minimal. In fact the singular points of R , the image of (cid:101) R in W, are all of multiplicity q f n − + 2 ∈ n Z and can be resolved by a single blowing-up.So there is no ˜ ϕ -vertical ( − n )-curve in (cid:102) W . Therefore there is no ˜ f -vertical ( − (cid:101) S . References [1] M. A. Barja, V. G.Alonso, and J. C. Naranjo, Xiao’s Conjecture for general fibred surfaces, J. reineangew Math. (2018), 297–308.[2] M. Enokizono, Slopes of fibered surfaces with a finite cyclic automorphism, Michigan Math. J. (2017), 125–154.[3] X.Lu and K.Zuo, On the slope conjecture of Barja and Stoppino for fibred surfaces, preprint(arXiv:1504.06276v1[math.A.G]).
4] X.Lu and K.Zuo, On the slope of hyperelliptic fibrations with positive relative irregularity, Trans.Amer. Math. Soc. (2017), 909–934.[5] G.Xiao, Fibred algebraic surfaces with low slope, Math. Ann. (1978), 449–466.