Local splitting families of hyperelliptic pencils, I.

T.Arakawa and T.Ashikaga

Introduction. The aim of this paper is to construct splitting families of degenerations of hyperelliptic curves so that the original complicated singular fiber decompose into several simple singular fibers by these local deformations. Moreover we define certain invariants of germs of singular fibers and study their properties. The historical background of our study is the following:

In 1977, Moishezon [Mo] showed that any degenerate elliptic curve splits by a local deformation into several singular fibers of only one of two types. One is a rational curve with a node (type I_{1} in Kodaira’s table [Kod1]) and the other is a multiple fiber whose reduced scheme is a nonsingular elliptic curve (type _{m}I_{0}). Y. Matsumoto [Ma1] and Ue [Ue1] studied the similar problem for good torus fibrations which are topological analogue of elliptic fibrations in some sense. These results are available for studying diffeomorphism classes of global elliptic surfaces or torus fibrations ([FM,Ma2,Ue2]etc.).

In 1988, Horikawa [Ho4] showed that any degenerate genus two curve splits into several fibers of type I_{1} in his table [Ho2] (a stable curve with two elliptic components with a node) and several fibers of type 0 (the fibers “arising from rational double points” in some sense, which include many topologically different types of fibers). At the same time, he remarked that the sum of a certain invariant (Horikawa index for genus 2 in our terminology) is conserved in his splitting families.

In 1990, inspired by the discussion with Xiao Gang, Miles Reid [Re] proposed this type of problem – Morsification for fiber germs in his terminology – from the viewpoint of the study of relative canonical algebra for pencils of curves such as Mendes-Lopes’es work [Men]. Especially he pointed out the importance to seek for “atomic fibers”, i.e. the units of degenerations.

On the other hand, Nakayama [Na] studied certain two-parameter splitting degenerations of elliptic curves from the viewpoint of threefold minimal model theory.

Now in this paper, we show that any degenerate hyperelliptic curve splits into several singular fibers belong to very small classes called type 0_{0}, class I and class II defined in §1 step by step by preserving the relative hyperelliptic involution of pencils. (Theorem 3.3.) The fiber of type 0_{0} is an irreducible stable curve with one node. The fiber in class I is a stable curve with two components with two nodes. The fiber in class II is defined by a certain strong condition for the germs of singularities of the branch curve arising from it’s relative hyperelliptic involution. We only classify them for small genus case, i.e. it contains one, three and three types for genus 2, 3 and 4 respectively (Proposition 1.7).

The method of the construction of our splitting families is as follows: We first construct a one-parameter family of normal surfaces which is a double cover over a family of local projective line bundles, and second we resolve it simultaneously. A bad singular fiber corresponds to bad singularities of the branch curve of the double cover which pass through the projective line over the origin. Therefore we first construct a splitting family of branch curve singularities so that bad singularities decompose into several more simple singularities, and then we lift it to a family of local hyperelliptic pencils.

The idea of our construction of local analytic splitting family of this curve singularity essentially comes from the well-known method for the Morsification of singularity so called A’Campo–Gusein Zade theory [Ac,G], that is, a perturbation method along the way of it’s resolution process.

The method of the lifting to the family of pencils is to write down the equation itself case by case according to the situation of the family of branch curves. All our methods are completely explicit so that we can describe which types of singular fibers appear in our splitting families.

In the last section, we define two invariants for the singular fiber of a hyperelliptic pencil, i.e. the Horikawa index and the local signature. Moreover we show that the sum of these invariants are conserved in our splitting families. (Theorem 4.10.)

The Horikawa index is defined to be the fractional number which measure the contribution of the fiber to the distance from the geographical lower bound of existence of hyperellitic pencils of fixed genus. From the viewpoint of singularity theory, it measures the total badness of singularities of the branch curve apart from simple (ADE) singularities.

The local signature is defined directly from the Horikawa index and the topological Euler contribution, and the usual global signature of a compact surface with hyperelliptic pencil is the sum of these local signatures. (Proposition 4.7.) For the genus 2 case, Matsumoto [Ma4] observed this fact by using the speciality of Meyer’s signature cocycle [Mey]. (Note that the general pencil does not satisfy this localized property of the signature, we refer to [At2, Kod3, Hi2] etc.) Matsumoto also calculated the local signature of two types of singular fibers of Lefschetz fibrations, which coincides with the “atomic fibers” of genus 2 by [Ho4] and a special case of Corollary 4.11. Therefore the global signature of genus 2 fibration is written as a simple linear combination of two terms as in Corollary 4.13, which give a precise “third” solution of the the negative signature problem posed by Persson [P2]. (Note that Xiao[X1] and Ueno[U] solved this problem.) Corollary 4.13 gives us similar information for genus 3 and 4 cases.

We lastly remark that the indecomposability of the singular fibers, the conservation of invariants of the fiber for general splitting families and the decomposition rule of homotopical monodromy for splitting degenerations will be forthcoming themes.

1 Special classes of germs of singular fiber

In this section, we define very special classes of degenerations of hyperelliptic curves. We will show in §3 that any degeneration of hyperelliptic curves splits by local deformations into several fibers in these classes step by step.

1.1. Let S be a 2-dimensional complex manifold and Δ_{ϵ} = {t ∈ C ; |t| < ϵ} be a complex 1-dimensional open disk with radius ϵ. Let f : S → Δ_{ϵ} be a relatively minimal proper surjective holomorphic map. If f satisfies the conditions

(i) For any t ∈ Δ_{ϵ} \ {0}, the fiber f^{−1}(t) is a nonsingular hyperelliptic curve of genus g ≥ 2, (ii) F = f^{−1}(0) is a singular curve, we call f a degeneration of hyperelliptic curves of genus g, and F a singular fiber of f.

Now let f′ : S′ → Δ_{ϵ′} be another degeneration of hyperelliptic curves of genus g. If there is a positive real number ϵ′′ with ϵ′′ ≤ ϵ and ϵ′′ ≤ ϵ′ such that the restrictions f|_{Δ_{ϵ′′}} : S|_{Δ_{ϵ′′}} → Δ_{ϵ′′} and f′|_{Δ_{ϵ′′}} : S|_{Δ_{ϵ′′}} → Δ_{ϵ′′} are isomorphic to each other as analytic fiber spaces, then we call f and f′ are equivalent. The equivalence class of f, which we write [f, F], is called the germ of singular fiber associated with f.

The study of the germ [f, F] is translated into the study of the singularities of the branch curve of the double cover arising from the relative hyperelliptic involution in the following way: Let π : W = P^{1} × Δ_{ϵ} → Δ_{ϵ} be a trivial P^{1}-bundle and set Γ_{t} = π^{−1}(t) for t ∈ Δ_{ϵ}. Let B be a reduced divisor on W which satisfies

i) BΓ_{t} = 2g + 2 (t ∈ Δ_{ϵ}), ii) The bad points of B are at most on Γ_{0}, where the bad point P of B is the point which satisfies I_{P} (B, Γ_{π(P)}) ≥ 2 ( i.e. a singular point of B or a tangential point to Γ_{π(P)}).

Let B′ be another divisor on W′ = P^{1} × Δϵ′ which satisfies the same conditions as in the above i) and ii). We say (W, B) and (W′, B′) are equivalent if there is a positive real number ϵ′′ with ϵ′′ ≤ ϵ and ϵ′′ ≤ ϵ′ such that the restrictions (W, B)|_{Δ_{ϵ′′}} and (W′, B′)|_{Δ_{ϵ′′}} coincide with each other modulo a finite succession of elementary transformations in the sense of [Ho2, §2]. We write [W, B] the equivalence class of (W, B) and call it the germ of branch curve of a degeneration of hyperelliptic curves. It is well-known that there exists a one-to-one correspondence between the set of germs of singular fibers {[f, F]} and those of branch curves {[W, B]} (cf. [Ho2]).

We remark that there exist examples such that [f, F] and [f′, F′] are mutually different as germs of singular fiber but F and F′ have the same weighted dual graph, for instance compare 2I_{0−m} with II_{n−0}^{∗} in Namikawa-Ueno’s table [NU,p.159,172] for g = 2 case. We also refer to Matsumoto-Montesinos [MM, §8] for its topological meaning. Therefore in this paper, we always express the germ of fiber by a suitable representative of the corresponding branch curve.

We introduce some notations for a germ of branch curve (W, B). We set

B̃ = B − Γ_{0} if Γ_{0} ⊂ B, B̃ = B otherwise,

and call B̃ the horizontal part of B. Let

W = W_{0} ⟵^{τ_{1}} W_{1} ⟵^{τ_{2}} ⋯ ⟵^{τ_{r}} W_{r}  (1.1.1)

be a succession of blow-ups such that the center P_{i−1} ∈ W_{i−1} of τ_{i} (1 ≤ i ≤ r) is infinitely near to a bad point of B. Let E_{i,i} be the exceptional curve for τ_{i}. For 0 ≤ i < j ≤ r, let E_{i,j} be the proper transform of E_{i,i} by the map τ_{i,j} = τ_{i+1} ∘ ⋯ ∘ τ_{j} : W_{j} ⟶ W_{i}. We also write E_{0,j} the proper transform of Γ_{0} by τ_{0,j}. Then the reduced scheme of the total transform of Γ_{0} by τ_{0,j} is written by E_{j} = ∑_{i=0}^{j} E_{i,j}. We put B_{0} = B and let B_{i} (1 ≤ i ≤ r) be the even proper transform of B_{i−1} in the sense of [Ho1, P1], that is, B_{i} = τ_{i}^{∗}B_{i−1} − 2[m_{i}/2]E_{i,i} where m_{i} is the multiplicity of B_{i−1} at P_{i−1}.

For choosing one representative among the equivalence class of it in a normalized form in some sense, the following lemma is useful:

Lemma 1.2. ([X3,Lemma 5.1.2],[Ho3], [P2]) Let [f, F] be a germ of singular fiber of degeneration of hyperelliptic curves of genus g. Then there exists a representative (W, B) of the branch curve of [f, F] which satisfies the following condition (∗); (∗) For any bad point P of B, the multiplicity of the horizontal part mult_{P} B̃ is not greater than g + 1. Moreover such (W, B) is uniquely determined by [f, F] except for the following case (∗∗); (∗∗) There exists just two bad point P_{1}, P_{2} of B such that mult_{P_{1}} B̃ = mult_{P_{2}} B̃ = g + 1.

We call (W, B) which satisfies (∗) the germ of the normalized branch curve. Note that the branch curve is not necessarily normalized in this paper, otherwise mentioned.

1.3. Now we define very special classes of germs of singular fiber.

1.3.1. Assume B is smooth on W, meet nonsingular transversally to Γ_{0} except for one point P and is contact of order two to Γ_{0} at P. Then the associated singular fiber F is an irreducible curve of genus g − 1 with one ordinary double point. ( The genus of a component means that of its normalization.) If the normalized branch curve (W, B) has the above property, we call [f, F] a fiber germ of type 0_{0}.

1.3.2. Assume B does not contain Γ_{0} and meet nonsingular transversally to Γ_{0} except for one point P and has an ordinary singularity at P of multiplicity 2g′ + 2, where g′ is an integer with 1 ≤ g′ ≤ [(g − 1)/2]. Then F has an irreducible decomposition F′ + F′′ where F′ and F′′ are nonsingular curves of genus g′ and g − g′ − 1 respectively meeting each other transversally at two points, whose self-intersection numbers in S are (F′)^{2} = (F′′)^{2} = −2. If the normalized branch curve (W, B) has the above property, we say that [f, F] belongs to class I. In fact, this class contains [(g − 1)/2] types according the number g′.

1.3.3. Assume B contains Γ_{0} and the following conditions (a), (b), (c) hold:

(a) We produce the succession of blow-ups (1.1.1) such that any singularity of the even proper transform B_{r} is ordinary. Note that such a process is uniquely determined by B up to the order of the choice of the centers of blow-ups. Then we have: (a1) B_{r} contains E_{r}, (a2) Any singularity of B_{r} has even multiplicity, (a3) For any 0 ≤ i ≤ r, any singularity on B_{i} with even multiplicity is ordinary.

(b) If a bad point P on B is not an ordinary singularity, then at least one local branch B^{(1)} of B at P is tangential to Γ_{0}, that is, I_{P} (B^{(1)}, Γ_{0}) > mult_{P} B^{(1)}.

(c) If B has a ordinary double point at Q, then the horizontal part B̃ meets Γ_{0} at another point R such that B̃ is contact to Γ_{0} at R of order at least three (that is, after the three times blow-ups at the infinitely near point of R, at least one component of the proper transform of B̃ and Γ_{0} still meet each other.)

We say a germ of fiber [f, F] belong to class II if any representative of the branch curve (W, B) with B ⊃ Γ_{0} of [f, F] has the above properties.

1.4. Let (C, P) be an analytic germ of a plane curve singularity. If (C, P) satisfies the following condition for some m ≥ 2 and n ≥ 2, we call it an m^{n} point; C has m irreducible local branches C^{(1)}, ⋯, C^{(m)} such that C^{(i)} is nonsingular and I_{P} (C^{(i)}, C^{(j)}) = n for any 1 ≤ i < j ≤ m. (The typical equation is x^{m} + y^{mn} = 0.)

Lemma 1.5 Let P be a bad point of the branch curve B which defines a singular fiber in class II.

(i) If mult_{P} B is even, then (B, P) is an ordinary singularity. (ii) If mult_{P} B = 3, then (B, P) is a 3^{2} point. (iii) If mult_{P} B = 5, then (B, P) is a 5^{2} point.

Proof The assertion (i) is clear by the definition. We consider (ii) and (iii). Let B^{(1)}, ⋯, B^{(k)} be the set of local analytic components of B at P. Let τ_{1} : W_{1} ⟶ W be the blow-up at P and we put P_{1} = E_{0,1} ∩ E_{1,1}. Let B̃^{(i)} (1 ≤ i ≤ k) be the proper transform of B^{(i)} by τ_{i}.

Assume mult_{P} B = 3. We suppose k = 1. Since B^{(1)} is tangential to Γ_{0} by the condition (b), (B_{1}, P_{1}) is a triple or a quadruple point. But (B_{1}, P_{1}) is not an ordinary quadruple point, and so it is a triple point by the condition (a3). Then the even proper transform B_{r} in (1.3.3.a) has an ordinary triple point, which contradict the condition (a2). Suppose k = 2. If one of B^{(i)} (i = 1, 2) is tangential to Γ_{0} and the another is not, then B_{r} also has an ordinary triple point. Hence both of B^{(i)} (i = 1, 2) are tangential, and so P_{1} is a quadruple point, which must be ordinary. Therefore we have the assertion (ii).

Assume mult_{P} B = 5. We first suppose that all of B^{(i)} (1 ≤ i ≤ k) are tangential to Γ_{0}. If k = 4, then the multiplicity at P_{1} of B_{1} is 6, and so (B_{1}, P_{1}) is ordinary. Therefore (B, P) is a 5^{2} point. We will show that 1 ≤ k ≤ 3 is impossible.

Assume k = 3. We may suppose mult_{P} B^{(1)} = 2 and the other two are nonsingular. Then B̃^{(1)} is nonsingular, otherwise (B_{1}, P_{1}) is a non-ordinary 6-pl point. Therefore (B^{(1)}, P) is a simple cusp, i.e. t^{2} + x^{3} = 0. Then after the blow-up at P_{1}, B_{3} has an ordinary triple point at P_{2} = E_{1,2} ∩ E_{2,2}. This is impossible.

Assume k = 2. We first consider the case where mult_{P} B^{(1)} = 3 and B^{(2)} is nonsingular. Since P_{1} cannot be a 6-pl point, we have mult_{P_{1}} B̃^{(1)} ≤ 2. Then (B^{(1)}, P) is either t^{3}+x^{4} = 0 or t^{3} + x^{5} = 0. In each case, after 2-times blow ups, (B_{2}, P_{2}) has a triple point which is not a 3^{2} point. This is impossible. In the case where mult_{P} B^{(1)} = mult_{P} B^{(2)} = 2 , the argument is similar. k = 1 is also impossible by a similar argument.

It remains to consider that at least one of the components, say B^{(1)}, is not tangential. Then B̃^{(1)} pass through a point Q ∈ E_{1,1}\E_{0,1}. Since mult_{Q}B_{1} ≤ 4 by the condition (b), it follows from (i) and (ii) that (B_{1}, Q) is an ordinary quadruple point or a 3^{2} point or ordinary double point. In the first case, B_{1} has a triple point at P_{1} which is not a 3^{2} point. This is impossible. The second case contradicts the condition (b). In the third case, we produce the elementary transformation so that the image of E_{1,1} become a new fiber. Then the condition (c) is not satisfied. Therefore we have the assertion (iii). Q.E.D.

1.6. For low genus cases 2 ≤ g ≤ 4, we have the classification of fibers in class II. We prepare some notations for writing them. We express the information of the set of bad points of a representative of the branch curve B inside the bracket { } by writing that of B̃. {m, ⋯} means that B has an ordinary (m + 1)-pl point. {m^{n}, ⋯} means that B has an (m + 1)^{n} point. We express the weighted dual graph of the minimal singular fiber as follows: The component of the singular fiber is always nonsingular in our list, The Gothic numbers O, I, II ⋯ mean that the genus of the components are 0, 1, 2, ⋯ respectively. The coefficient means the multiplicity of the component. The lower suffix means the self-intersection number of the component, but we omit it if it is −2. The symbol “—” means “intersects transversally”.

For instance, {3, 3, 2^{2}} means that the bad points of B are two ordinary quadruple points and one 3^{2} point.(See the Figure 1.) This really appears as (g=4, iii) in Proposition 1.7. In this case, the even resolution consists of 4 times blow-ups , and the minimal singular fiber is F = 2F_{0} + F_{1} + F_{2} + 2F_{3} where g(F_{0}) = 0, g(F_{1}) = g(F_{2}) = g(F_{3}) = 1, F_{0}F_{1} = F_{0}F_{2} = F_{0}F_{3} = 1, F_{0}^{2} = F_{1}^{2} = F_{2}^{2} = −2 and F_{3}^{2} = −1. This is easily seen by the usual double covering method (cf.[P1]).

Proposition 1.7. The classification of singular fiber germs of class II for genus 2 ≤ g ≤ 4 are the followings: We have one, three and three types for g = 2, 3 and 4 respectively;

(g = 2) (i) {3, 3} I_{−1} − I_{−1}

(g = 3) (i) {3, 5} I_{−1} − II_{−1} (ii) {2^{2}, 2^{2}} 2I_{−1} − 2O − 2I_{−1} (iii) {4^{2}} 2II_{0}

Figure 1: branch curve of type {3, 3, 2^{2}}

(g = 4) (i) {3, 7} I_{−1} − III_{−1} (ii) {5, 5} II_{−1} − II_{−1} (iii) {3, 3, 2^{2}}

I | 2O − 21_{−1} | I

Proof Let (W, B) be the normalized branch curve of the germ of fiber. If B contains Γ_{0}, lemma 1.5 easily implies the list in the assertion.

Assume B does not contain Γ_{0}. We set B ∩ Γ_{0} = {P_{1}, P_{2}, ⋯, P_{k}}. We may assume that (B, P_{1}) is a triple or a 5-pl point and B intersect Γ_{0} nonsingular transversally at P_{2}, ⋯, P_{k}. Otherwise, we choose another branch curve (W′, B′) such that B′ contains the fiber by elementary transformations. Then the condition (a3) in 1.3.3 is not satisfied for (W′, B′).

Now let (W′′, B′′) be the branch curve obtained by the elementary transformation at P_{1}. Then B′′ contains the new fiber Γ′′_{0}. Let Q be the point on Γ_{0} which is the image of E_{0,1} by the contraction map. Since B does not contain Γ_{0}, the multiplicity of (B′′, Q) is even. Therefore (B′′, Q) is an ordinary singularity. Other bad points of B′′ have their multiplicity at most 6. Then we apply lemma 1.5, and obtain the assertion. Q.E.D.

2 Local analytic splitting of plane curve singularities

Let C be a normal analytic curve defined on an open set V_{0} = {(X, Y) ∈ C^{2} ; |X| < ϵ, |Y | < ϵ, } for a sufficiently small ϵ such that P = (0, 0) is the unique singular point of C. In this section, we explicitly construct certain local analytic deformations {C_{u}}_{u∈Δ_{δ}} of C such that C_{u} (u ≠ 0) has several isolated singular points. The idea is essentially same as the method in A’Campo, Gusein-Zade theory ([Ac, G]).

2.1. Let C = ∑_{j=1}^{s} C^{(j)} be the local analytic irreducible decomposition of C at P. Let V_{0} ⟵^{τ_{1}} V_{1} ⟵^{τ_{2}} ⋯ ⟵^{τ_{r}} V_{r} be the succession of blow-ups whose centers are infinitely near to P and let C̃_{r} (resp. C̃_{r}^{(j)}, 1 ≤ j ≤ s) be the proper transform of C (resp. C^{(j)}) by τ_{0,r} := τ_{1}τ_{2} ⋯ τ_{r}. Let Q be an isolated singular point of C̃_{r}.

We first assume that Q is a smooth point of the (reduced) exceptional set E_{r} = ∑_{j=1}^{r} E_{j}. Assume Q is on E_{j_{0}} (1 ≤ j_{0} ≤ r). Let U(x, y) be the canonical open coordinate neighborhood on V_{r} containing Q, that is, U ∩ E_{r} coincides with the union of the complements of one point of E_{j_{0}} and that of E_{j_{1}} (1 ≤ j_{0}, j_{1} ≤ r, j_{0} ≠ j_{1}) which are defined by y = 0 and x = 0, respectively. Let (x, y) = (α, 0) (α ≠ 0) be the coordinate of Q in U. Since the map (τ_{0,r})|_{U} : U → U_{0} is written by (x, y) ↦ (X = x^{r_{1}}y^{s_{1}}, Y = x^{r_{2}}y^{s_{2}}) for some non-negative integers r_{1}, s_{1}, r_{2} and s_{2}, U is written by

U = {(x, y) ∈ C^{2} ; |x^{r_{1}}y^{s_{1}}| < ϵ, |x^{r_{2}}y^{s_{2}}| < ϵ}.

Let C̃_{r}^{(1)}, … C̃_{r}^{(k_{0})} (k_{0} ≤ s) be the components of C̃_{r} which pass through Q (by changing the order of them if necessary). Since ϵ is sufficiently small, we may assume that the support of the divisor ∑_{j=0}^{k_{0}} C̃_{r}^{(j)} on V_{r} is contained in U. Let

g(x − α, y) = ∏_{j=1}^{k_{0}} g^{(j)}(x − α, y) = 0

be the equation of ∑_{j=0}^{k_{0}} C̃_{r}^{(j)} on U. Now let D be the divisor on V_{r} × Δ_{δ} which satisfies the following conditions:

(i) The support of D is contained in U × Δ_{δ} and the equation of D is given by

g(x − α − φ(u), y − ψ(u)) = 0  (2.1.1)

where φ(u), ψ(u) are generic holomorphic functions of u which satisfy φ(0) = ψ(0) = 0, (ii) D does not meet the divisor D′ = ∑_{j=k_{0}+1}^{s} C̃_{r}^{(j)} × Δ_{δ} on V_{r} × Δ_{δ}, (iii) For any u ∈ Δ_{δ}, let D_{u} be the restriction of D to V_{r} × {u} and we consider it as a curve on V_{r}. Then D_{u} intersects E_{r} transversally at I_{Q}(C̃_{r}, E_{j_{0}}) mutually distinct points. Moreover the point Q_{u} = {(x, y) = (α + φ(u), ψ(u))} on U is the unique singular point of D_{u}.

The existence of D is clear by taking ϵ and δ sufficiently small. Now we define a divisor M on V_{0} × Δ_{δ} by

M = (τ_{0,r})_{∗}D + ∑_{j=k_{0}+1}^{s}C̃^{(j)} × Δ_{δ}.

The natural fibration π : M → Δ_{δ} has the following properties:

(a) M_{0} = π^{−1}(0) coincides with C.

(b) For any u ∈ Δ_{δ} \ {0}, the fibers M_{u} has just two isolated singular points at P and R_{u} = τ_{0,r}(Q_{u}). Moreover the germ of singularity (M_{u}, R_{u}) coincides with (C̃_{r}, Q).

We call this situation that (C, P) splits into (C_{u}, P) and (C_{u}, R_{u}) by the perturbation at the infinitely near point Q, and we call the fibration π the splitting family of (C, P) of type A.

Example 2.2. The singularity x^{3} + y^{n} = 0 (0 < 3k < n) splits into x^{3} + y^{3k} = 0 and x^{3} + y^{n−3k} = 0 by the perturbation at the infinitely near point which appear after k times blow- ups.

2.3. Assume Q is a double point of E_{r}. We construct local deformations of (C, p) by the following two different ways:

(1) We assume Q = E_{j_{0}} ∩ E_{j_{1}} and we put α = 0 in the same situation as in 2.1. Then we can construct the fibration π : M → Δ_{δ} such that the only one condition (iii) is replaced by the following (iii)’: (iii)’ For any u ∈ Δ_{δ}, D_{u} intersects E_{r} only at nonsingular points of E_{j_{0}}+E_{j_{1}}. D_{u} intersects E_{j_{k}} (k = 0, 1) transversally at I_{Q}(C̃_{r}, E_{j_{k}}) mutually distinct points. Moreover the point Q_{u} = {(x, y) = (φ(u), ψ(u))} on U is the unique singular point of D_{u}.

We also call π a splitting family of type A.

(2) On the equation (2.1.1), we put α = 0 and ψ(u) ≡ 0. Then D_{u} (u ≠ 0) has a singularity at Q_{u} = {(x, y) = (ψ(u), 0)} on E_{j_{0}} and intersects E_{j_{1}} at I_{Q}(C̃_{r}, E_{j}) distinct points transversally. We construct π : M → Δ_{δ} in the same way. Then M_{u} (u ≠ 0) has one isolated singular point at P such that the germ (M_{u}, P) is different from (C, P), because their embedded resolution processes are distinct each other. We call π a splitting family of type B.

Example 2.4. x^{5} + y^{3} = 0 splits into (x^{4} + y^{2})(x + y) = 0 as follows: After 2 times blow-ups, the proper transform become nonsingular which meet E_{1} transversally and contacts of order 2 to E_{2}. We perturb it to the direction of generic point of E_{1}, and then blow down 2 times.

Remark 2.5. The above deformations are not necessarily well-defined in the algebraic category, because the local analytic reducibility does not induce the algebraic reducibility. (This was pointed out to the authors by Prof. Miyanishi.)

3 Splitting families of degenerations

In this section, we construct splitting families of degenerations of hyperelliptic curves. Our aim is not only to prove the existence of splitting families but also to describe explicitly the germs of singular fibers in our families.

Definition 3.1. Let [f, F] be a germ of singular fiber of a degeneration of hyperelliptic curves of genus g. Assume there exist positive real numbers ϵ and δ, a 3-dimensional complex manifold Z and a flat surjective holomorphic map h : Z → Δ_{ϵ} × Δ_{δ} such that (i) The fibration h_{0} : Z_{0} → Δ_{ϵ} × {0} arising from the restriction of h over 0 ∈ Δ_{δ} has an unique singular fiber over 0 ∈ Δ_{ϵ} such that the germ of fiber [h_{0}, h_{0}^{−1}(0)] coincide with [f, F], (ii) For any u ∈ Δ_{δ} \ {0}, the fibration h_{u} : Z_{u} → Δ_{ϵ} × {u} has l (l ≥ 2) singular fibers. The number l is independent on u. Then we call h a splitting family of [f, F].

Definition 3.2. Let G be a certain subset of the set of all germs of singular fiber of degenerations of hyperelliptic curves of genus g. We say the germ [f, F] is reduced to G via several splitting families if the following conditions are satisfied:

First [f, F] has splitting family h : Z → Δ_{ϵ} × Δ_{δ}. If any germ of singular fiber [h_{u}, h_{u}^{−1}(t)] (t ∈ Δ_{ϵ} × {u}) of any general fibration h_{u} : Z_{u} → Δ_{ϵ} × {u} (u ∈ Δ_{δ} \ {0}) belongs to G, then we stop our reducing process. Assume some of them do not belong to G. Let [h_{u}, h_{u}^{−1}(t_{0})] (t_{0} ∈ Δ_{ϵ} × {u}) be any germ of singular fiber of h_{u} which does not belong to G. Then [h_{u}, h_{u}^{−1}(t_{0})] has a splitting family h′ : Z′ → Δ_{ϵ′} × Δ_{δ′}. If any germ of singular fiber of any general fibration of h′ belongs to G, then we stop our process. If not, the germ which does not belong to G has also a splitting family.

Then this process terminate for finitely many times, that is, any germ of singular fiber of any general fibration of the terminating splitting families belong to G

The main result of this paper is the following:

Theorem 3.3. Let G be the set of the union of the germs of singular fiber of type 0_{0}, classes I and II defined in §1. Then any germ of singular fiber of a degeneration of hyperelliptic curves is reduced to G via several splitting families.

Corollary 3.4. Any degeneration of hyperelliptic curves of genus two (resp. three, resp. four) is reduced via several splitting families to germs of singular fibers of two types (resp. five types, resp. five types) whose tables are listed in 1.3.1, 1.3.2 and proposition 1.7.

For the proof of theorem 3.3, we start to prepare some lemmas. Let (W, B) be a representative of the branch curve of the germ of fiber [f, F], where W = P^{1} × Δ_{ϵ} for sufficiently small ϵ. Let P be a bad point of B. Let W = W_{0} ⟵^{τ_{1}} W_{1} ⟵^{τ_{2}} ⋯ ⟵^{τ_{r}} W_{r} be a succession of blow-ups at infinitely near points of P. We use the same notations as in 1.1.

Lemma 3.5. Assume there is a component E_{j_{1},r} (0 ≤ j_{1} ≤ r) and a point Q on E_{j_{1},r} such that (i) Q is not a double point of E_{r}, (ii) E_{j_{1},r} is not contained in B_{r} and (iii)

B_{r} has a singularity at Q or B_{r} is tangential to E_{j,r} at Q. Then [f, F] has a splitting family.

Proof Let B^{♯} be the sum of components of B which pass through P. Since ϵ is sufficiently small, B^{♯} does not pass through any other point of Γ_{0} and we have B^{♯} ∩ (B − B^{♯}) = ϕ. Now let V_{0} be a sufficiently small classical open neighborhood of P on W, and let

V_{0} × Δ_{δ} ⊃ B^{♯} \underrightarrow{π} Δ_{δ}

be the splitting family of type A of the germ (B^{♯}, P) arising from the perturbation at Q constructed in 2.1. Since ϵ and δ are sufficiently small and the functions φ(u), ψ(u) in 2.1 are general, we may assume that the analytic closure of B^{♯} in W_{0} × Δ_{δ} coincides with B^{♯} itself, which we denote the same symbol B^{♯}. Now we define a divisor B on W = W × Δ_{δ} by

B := B^{♯} + (B − B^{♯}) × Δ_{δ}.

Let ρ : B → Δ_{δ} be the natural morphism. We consider the fiber B_{u} = ρ^{−1}(u) as a divisor on W ≅ W ×{u}. Then B_{0} coincides with B and B_{u} (u ≠ 0) has the following properties:

i) B_{u}Γ_{t} = 2g + 2 (t ∈ Δ_{ϵ}), ii) B_{u} has two bad points P on Γ_{0} and R_{u} on Γ_{π(R_{u})}, where the point R_{u} is described in 2.1.

Since the problem is local for the parameter space, we may assume that there is a line bundle L on W which satisfies [B] ≅ 2L. On P^{1}-bundle P(O_{W} ⊕ O_{W} (L)), we construct a double cover μ : S ⟶ W branched along B. In order to resolve the singularities on S and to obtain a desired smooth family, we construct a relative even resolution for the family ρ of branch curves in the following way:

By the construction of ρ and Tomari’s lemma [As, Lemma 3.8], the first r-times multiplicity sequence of the even resolution of the germ (B_{u}, P) coincides with {m_{1}, …, m_{r}} for any u ∈ Δ_{δ}. Now let

W = W_{0} \underleftarrow{τ_{1}} W_{1} \underleftarrow{τ_{2}} ⋯ \underleftarrow{τ_{r}} W_{r}

be the succession of blow-ups such that the center of τ_{i} (1 ≤ i ≤ r) is {P_{i−1}} × Δ_{δ}. Let B_{i} = τ _{i}^{∗}B_{i−1} − 2[m_{i}/2]C_{i} (1 ≤ i ≤ r) be the even proper transform of B_{i−1}, where B_{0} = B and C_{i} is the exceptional set for τ_{i}. Note that B_{r} does not contain E_{j_{1},r} × Δ_{δ} by our assumption. Therefore on the coordinate open neighborhood U × Δ_{δ} = {(x, y, z)} of W × Δ_{δ} described in 2.1, the equation of B_{r} is written by g(x − α − φ(u), y − ψ(u)) = 0. Now let Q be the curve on W_{r} × Δ_{δ} defined by the image of the holomorphic map

\begin{matrix}Δ_{δ} & ⟶ & U × Δ_{δ} ⊂ W_{r} × Δ_{δ} \\ u & ⟼ & (x, y, u) = (φ(u), ψ(u), u). \end{matrix}

Then B_{r} is equisingular along the locus Q in the sence that the restricted germs of singularity ((B_{r})_{|π_{r}^{−1}(u)}, Q_{|π_{r}^{−1}(u)}) have the same even resolution process for any u ∈ Δ_{δ}, where π_{r} : B_{r} ⟶ Δ_{δ} is the natural map. Therefore we have the similar process for Q. Lastly the singularities of (B − B^{♯}) × Δ_{δ} are of course simultaneously resolved and we complete the relative even resolution for the family ρ. Namely, by the succession of blow-ups τ^{♭} : W^{♭} → W, the even proper transform B^{♭} of B become nonsingular and the restriction of τ^{♭} over any u ∈ Δ_{δ} coincides with the even resolution of B_{u}. Since B^{♭} is an even divisor on W^{♭}, we can construct the nonsingular double covering π : M^{♭} → W^{♭} which branches along B^{♭} and obtain natural morphisms M^{♭} ⟶^{ρ_{1}} Δ_{ϵ} × Δ_{δ} →^{ρ_{2}} Δ_{δ}.

Now assume a certain fiber of the fibration M_{|(ρ_{1}ρ_{2})^{−1}(u_{0})}^{♭} ⟶ Δ_{ϵ} × {u_{0}} contains a (−1)-curve for some u_{0} ∈ Δ_{δ}. Then by Kodaira’s stability theorem [Kod2], every fiber of M_{|(ρ_{1}ρ_{2})^{−1}(u)}^{♭} ⟶ Δ_{ϵ} × {u} contains a (−1)-curve for each u ∈ Δ_{δ}, and those (−1)-curves are simultaneously contracted by Fujiki-Nakano [FN]. Repeating these processes, we reach a family of relatively minimal fibrations M → Δ_{ϵ} × Δ_{δ} → Δ_{δ}. By our construction, this family has the following properties:

(i) f_{0} : M_{0} → Δ_{ϵ} × {0} has a unique singular fiber over 0 ∈ Δ_{ϵ} whose fiber germ coincides with [f, F]. (ii) For any u ∈ Δ_{δ} \ {0}, f_{u} : M_{u} → Δ_{ϵ} × {u} has just two singular fibers f_{u}^{−1}(0) and f_{u}^{−1}(π(R_{u})). Hence we have the assertion. Q.E.D.

Example 3.6. Assume B ⊅ Γ_{0}, BΓ_{0} = 6 and B has a triple point x^{3} + t^{6n} = 0 at P ∈ Γ_{0} with I_{P} (B, Γ_{0}) = 3 and meet transversally to Γ_{0} at other three points. By 2k-times blowing-ups (1 ≤ k ≤ n − 1), the infinitely near point Q of P become x^{3} + t^{′6(n−k)} = 0 and B_{2k} does not contain E_{2k,2k}. Then we use the method in Lemma 3.5. B_{u} has two bad points x^{3} + t^{6k} = 0 and x^{3} + t^{6(n−k)} = 0 at mutually distinct two points as in Example 2.2. ( In the Figure 2, the solid line is a component of the branch locus and the dotted line is not so.) The associated family of genus two fibrations f : M → Δ_{δ} has the following properties: f_{0} has an unique singular fiber

I_{−1} − \overbrace{O − ⋯ − O}^{n} −I_{−1}

and f_{u} (u ≠ 0) has two singular fibers

I_{−1} − \overbrace{O − ⋯ − O}^{k} −I_{−1} , I_{−1} − \overbrace{O − ⋯ − O}^{n−k} −I_{−1}.

Lemma 3.7. Under the same conditions i), ii) as in Lemma 3.5, we further assume Q = E_{j_{0},r} ∩ E_{j_{1},r} for some 0 ≤ j_{0} < j_{1}. Then [f, F] has a splitting family.

Proof If E_{j_{0},r} ⊄ B, then we can construct a splitting family as same as in Lemma 3.5 by using the splitting family of type A for (B, P) defined in 2.3 (1).

Assume E_{j_{0},r} ⊂ B. Then we use the splitting family of type B for (B, P) defined in 2.3 (2) arising from the perturbation of the branch curve at Q into the generic point of E_{j_{0},r}, Figure 2: Construction of the branch curve in Example 3.6

13 and construct a family ρ : B → Δ_{δ} of branch curves as in Lemma 3.5. ρ has a relative even resolution and the associated family of relatively minimal hyperelliptic fibrations f : M → Δ_{ϵ} × Δ_{δ} → Δ_{δ} has the following properties:

(i) f_{0} : M_{0} → Δ_{ϵ} × {0} has a unique singular fiber over 0 ∈ Δ_{ϵ} whose fiber germ coincides with [f, F], (ii) For any u ∈ Δ_{δ} \ {0}, the fibration f_{u} : M_{u} → Δ_{ϵ} × {u} has k + 1 singular fibers, where k = I_{Q}(B_{r} − E_{j_{0},r}, E_{j_{1},r}). One of them is f_{u}^{−1}(0) and the others are fibers of type 0_{0}.

Indeed, we denote by (B_{u})_{i} (1 ≤ i ≤ r, u ≠ 0) the even proper transform by τ_{1,i} of B_{u} = ρ^{−1}(u) on W ≅ W × {u}. Since E_{j_{1},r} is contracted to Q′ = τ_{j_{1},r}(Q) by the map τ_{j_{1},r} : W_{r} → W_{j_{1}−1}, the image by τ_{j_{1},r} of the components of (B_{u})_{r} which intersect to E_{j_{1},r} pass through Q′. Since the family ρ arises from the perturbation of type B at Q = E_{j_{0},r} ∩ E_{j_{1},r}, (B_{u})_{j_{1}−1} has at least k nonsingular local components C^{(1)}, …, C^{(k)} at Q′ which intersect transversally each other at this point. Moreover C^{(j)} (1 ≤ j ≤ k) also pass through the point Q′_{u} = τ_{j_{1},r}(Q_{u}) where Q_{u} is defined in 2.3 (2). Now let U′ be a small classical open set of W_{j_{1}−1} containing Q′ and Q′_{u}, and we consider the restricted fibration π_{j−1}|_{U} : W_{j−1}|_{U} → Δ_{ϵ}. Then it is easy to see that each C^{(j)} (1 ≤ j ≤ k) contacts of order 2 at a point of certain mutually distinct fibers π|_{U}^{−1}(t_{j}) (t_{j} ≠ 0). Therefore B_{u} also contacts of order 2 at just k points of fibers Γ_{t_{j}} (j = 1, …, k) of π_{0} : W → Δ_{δ}, and intersects transversally to the fiber of π_{0} at other points. Hence the fibration f has the above properties. Q.E.D.

Example 3.8. Assume B ⊃ Γ_{0}, B̃Γ_{0} = 8, B̃ has a triple point x^{6}+t^{3} = 0 at P with I_{P} (B̃, Γ_{0}) = 6 and intersects transversally to Γ_{0} at other two points. Let τ_{1} : W_{1} → W be the blow-up at P. Then B_{1} ⊅ E_{1,1} , B_{1} ⊃ E_{0,1} and B̃_{1} = B_{1} − E_{0,1} has an ordinary triple point at Q = E_{0,1}∩E_{1,1}. By the method in Lemma 3.7, B_{u} (u ≠ 0) has two ordinary triple points on Γ_{0} and three tangential points on mutually distinct three fibers. (see Figure 3.) The associated family of genus three fibrations f : M → Δ_{δ} has the following properties: f_{0} has an unique singular fiber

O − 2O − 2I − O
|  |
O  O

and f_{u} (u ≠ 0) has four singular fibers. One of them is

O |

O − 2O − I

| I

and the other three are of type 0_{0}.

Figure 3: Construction of the branch curve in Example 3.8

Lemma 3.9. Assume B ⊅ Γ_{0} and B has at least two bad points on Γ_{0}. Then [f, F] has a splitting family.

Proof Let P_{1}, …, P_{s} be bad points of B. Let h_{i} : Δ_{ϵ} → W (i = 1, 2) be sections of π : W → Δ_{ϵ} such that the image h_{i}(Δ_{ϵ}) does not pass through any of P_{1}, …, P_{s}. Then the pair (h_{1} : h_{2}) defines a homogeneous fiber coordinate of the P^{1}-bundle π. Let (x, t) be the coordinate of the open set U := W −h_{2}(Δ_{ϵ}) ≅ Δ_{ϵ} ×C, where x = h_{1}/h_{2}. We may assume that the support of the divisor B on W is contained in U. We set P_{1} = {(x, t) = (α, 0)} (α ∈ C). Then the equation of B in U is written by f_{1}(x − α, t)f_{2}(x − α, t) = 0 where f_{1}, f_{2} are holomorphic functions of (x, t) which satisfy f_{1}(α, 0) = 0 and f_{2}(α, 0) ≠ 0. For a sufficiently small δ, we define a divisor B′ on W = W × Δ_{δ} by

f_{1}(x − α, t − cu)f_{2}(x, t) = 0,

where c is a complex number whose absolute value is sufficiently small. Then the analytic closure B of B′ in W coincides with B′ itself. Let φ : Δ_{δ} ⟶ W be the map defined by u ↦ (x, t, u) = (α, cu, u). Then the family of branch curves B → Δ_{δ} on W has a relative even resolution by the blow-ups whose centers are infinitely near to the locus φ(Δ_{δ}) and {P_{i}} × Δ_{δ} (2 ≤ i ≤ s). Then we obtain a family of hyperelliptic fibrations M → Δ_{ϵ} × Δ_{δ} such that M_{u} → Δ_{ϵ} ×{u} (u ≠ 0) has two singular fibers. One of them has one bad point on its branch curve, and the other has (s − 1) bad points. Q.E.D.

Lemma 3.10. We assume (i) B ⊃ Γ_{0}, (ii) the even proper transform B_{1} of B by τ_{1} : W_{1} → W contains E_{1} = E_{0,1} + E_{1,1}, (iii) B̃_{1} = B_{1} − E_{1} does not pass through P_{1} = E_{0,1} ∩ E_{1,1} and (iv) B̃_{1} intersects both E_{0,1} and E_{1,1}. Then [f, F] has a splitting family.

Proof We take the coordinate (t_{1}, x) on the open set U ≅ Δ_{ϵ} × C of W_{1} such that the support of the divisor B̃_{1} on W_{1} is contained in U, the equation of E_{0,1} and E_{1,1} on U are t_{1} = 0 and x = 0 respectively and the map πτ_{1} : W_{1}|_{U} → Δ_{ϵ} is given by (t_{1}, x) ↦ t = t_{1}x. Let (t_{1}, x) = (α_{1}, 0), ⋯ (α_{s}, 0), (0, β_{1}), ⋯, (0, β_{s′}) be the coordinates on U of the intersection points of B̃_{1} to E_{1} (α_{i} ≠ 0, β_{j} ≠ 0). Then the equation of B_{1} on U is written by

t_{1}x ∏_{i=1}^{s} f_{i}(t_{1} − α_{1}, x) ∏_{j=1}^{s′} f_{s+j}(t_{1}, x − β_{j}) = 0.

Now we define a divisor B′ on the open coordinate neighborhood U × Δ_{δ} = {(t_{1}, x, u)} by

(t_{1}x − cu^{2}) ∏_{i=1}^{s} f_{i}(t_{1} − α_{1}, x − \frac{cu^{2}}{α_{1}} ) ∏_{j=1}^{s′} f_{s+j}(t_{1} − \frac{cu^{2}}{β_{1}} , x − β_{j}) = 0

where c is a complex number whose absolute value is sufficiently small. Let B_{1} be the closure of B^{1} in W_{1} × Δ_{δ}. Let φ_{i} (1 ≤ i ≤ s), ψ_{j} (1 ≤ j ≤ s′), be the map from Δ_{δ} to U × Δ_{δ} ⊂ W_{1} × Δ_{δ} defined by

φ_{i} : u ⟼ (α_{i}, \frac{cu}{α_{i}}, u), ψ_{j} : u ⟼ (\frac{cu}{β_{j}}, β_{j}, u).

Then by the succession of blow-ups at the infinitely near locus of φ_{i}(Δ_{δ}) and ψ_{i}(Δ_{δ}), we have a relative even resolution of the family of branch curves W_{1} ×Δ_{δ} ⊃ B ⟶ Δ_{δ} except the point P_{1}. Let M̃ → W ×Δ_{δ} be the double cover branched along B̃. Then M̃ is smooth except for one point P̃_{1}, which is a 3-dimensional A_{1}-singularity. P̃_{1} is resolved without affecting the general fiber of M̃ ⟶ Δ_{δ} by the method of Atiyah [At1] (see also [B]). Let M → Δ_{ϵ} × Δ_{δ} → Δ_{δ} be the family of relatively minimal hyperelliptic fibrations obtained from the relative (–1) contractions of the resolved threefold of M̃. Then M_{u} → Δ_{ϵ} × {u} has two singular fibers f_{u}^{−1}(0) and f_{u}^{−1}(u) for any u ≠ 0. Q.E.D.

Lemma 3.11. Assume (i) B ⊃ Γ_{0}, (ii) B̃ = B −Γ_{0} intersects nonsingular transversally to Γ_{0} at a point P and (iii) B̃ has no intersection point where B̃ contact of order more than two to Γ_{0}. Then [f, F] has a splitting family.

Proof We set B̃ ∩ Γ_{0} = {P, P_{1}, …, P_{s}}. Let U ≅ C × Δ_{ϵ} = {(x, t)} be an open coordinate on W such that B̃ ∩ Γ_{0} is contained in U, the map is given by (x, t) ↦ t and the local branch of B̃ passing through P is given by x = 0. Then the equation of B on U is written by txf_{1}(x − α_{1}, t) ⋯ f_{s}(x − α_{s}, t) = 0 where f_{i}(x − α_{i}, t) is the equation of the equation of B̃ passing through P_{i} = {(x, t) = (α_{i}, 0)} (1 ≤ i ≤ s). Now we define a divisor B′ on U × Δ_{δ} by the equation

(tx − u^{2}) ∏_{i=1}^{s} f_{i} (x − α_{i}, t − \frac{u^{2}}{α_{i}} + \frac{u^{2}}{α_{i}^{2}} (x_{i} − α_{i})) = 0.

Then B′ pass through the point P̃_{i} = (α_{i}, \frac{u^{2}}{α_{i}} , u) (1 ≤ i ≤ s). Since the tangent line of the curve tx − u^{2} = 0 at P̃_{i} coincides with t − \frac{u^{2}}{α_{i}} + \frac{u^{2}}{α_{i}^{2}} (x_{i} − α_{i}) = 0 (where u is fixed) and this curve contacts of order 2 to this tangent line, the germs of singularities (B, P_{i}) and (B_{u}, P̃_{i}) have the same even resolution process by the assumption iii).

Let B be the closure of B′ on W × Δ_{δ}. Then B coincides with B′ ∪ {∞} where ∞ is a certain point on Γ_{0}. The family W × Δ_{δ} ⊃ B → Δ_{δ} has a relative even resolution except for P, and the A_{1}-singularity over P on its double cover is resolved by [At]. The induced splitting family of hyperelliptic fibrations has s singular fibers on the general fibration. Q.E.D.

Example 3.12. In the situation of Lemma 3.11, we assume that B̃ is given by x(x^{2g+1} − 1) = 0. Then the divisor B′ is written by (tx − u^{2})(x^{2g+1} − 1) = 0. Note that if u ≠ 0, then B′_{u} has 2g + 1 ordinary double points at (x, t) = (ζ^{ν}, u^{2}ζ^{−ν}) (0 ≤ ν ≤ 2g) where ζ is a primitive (2g + 1)-th root of unity. (See Figure 4.) The associated family of genus g fibrations f : M → Δ_{δ} has the following properties:

f_{0} has a unique singular fiber of the form F = 2E_{0} + E_{1} + ⋯ + E_{2g+2} where E_{j}’s are nonsingular rational curves with E_{0}^{2} = −g − 1, E_{j}^{2} = −2, E_{0}E_{j} = 1 (1 ≤ j ≤ 2g + 2) and E_{i}E_{j} = 0 (1 ≤ i < j ≤ 2g + 2).

f_{u} (u ≠ 0) has 2g + 1 singular fibers of the following forms: F′ = E_{1} + E_{2} where E_{1} is a nonsingular curve of genus g − 1 and E_{2} is a nonsingular rational curve, E_{1}^{2} = E_{2}^{2} = −2 and E_{1}E_{2} = 2.

Lemma 3.13. Assume (i) B ⊅ Γ_{0} and (ii) B has an ordinary double point at a point P ∈ Γ_{0}. Then [f, F] has a splitting family.

Proof Let C^{(i)} (i = 1, 2) be the components of B passing through P. Then the equation of B on an open set U of W is written by f_{1}(x, t)f_{2}(x, t)g(x, t) = 0 where f_{i}(x, t) (i = 1, 2) is the equation of C^{(i)}. Let B′ be the divisor on U × Δ_{δ} defined by

(f_{1}(x, t)f_{2}(x, t) − u^{2}) g(x, t) = 0.

Then the closure of B′ in W × Δ_{δ} similarly induce a splitting family f : M → Δ_{ϵ} × Δ_{δ} as in Lemma 3.11. The singular fibers of f_{u} (u ≠ 0) are as follows:

i) if B has another bad points except P, then f_{u} has three singular fibers. Moreover two of them are of type 0_{0}. ii) If P is the unique bad points of B, then f has two singular fibers of type 0_{0}. Q.E.D.

Example 3.14 Let ([f, F]) be the germ given in example 3.12. By combining example 3.12 and lemma 3.13, ([f, F]) splits step by step into 4g + 2 fibers of type 0_{0}. (Compare it with [Ma4, p.140],[Ma5].)

Lemma 3.15. We assume B_{r} has an ordinary singularity Q of odd multiplicity k ≥ 3. Then [f, F] has a splitting family.

Proof We only consider the case where Q is contained in a unique E_{j_{1},r} (1 ≤ j_{1} ≤ r) and E_{j_{1}} ⊂ B_{r} (The other cases are similar and we omit them). By the canonical coordinate U(x, y) as in 2.1, B_{r} is written by yf_{1}(x − α, y) ⋯ f_{k−1}(x − α, y)g(x, y) = 0, where f_{i}(x − α, y) = 0 (1 ≤ i ≤ k − 1) is the equation of each branch curves of B_{r} − E_{j_{1},r} passing through P = (α, 0) and E_{j_{1},r} is defined by y = 0. Then we define a divisor B′ on U × Δ_{δ} by

yf_{1} (x − α − φ(u), y − ψ(u)) ∏_{i=2}^{k−1} f_{i}(x − α, y)g(x, y) = 0

for generic holomorphic functions φ(u), ψ(u) with φ(0) = ψ(0) = 0. Then B′ ⊂ U × {u} has k −1 ordinary double points. The analytic closure B on W_{r} ×Δ_{δ} of B′ has the natural properties similar as in 2.1.

Note that the locus (x, y, u) = (α, 0, u) on U×Δ_{δ} has the same multiplicity sequence for the even resolution for any u ∈ Δ_{δ}. Therefore the family of the branch curves W_{r} × Δ_{δ} ⊂ B → Δ_{δ} has the relative even resolution except for the m − 1 ordinary double points, which can be also resolved by [At]. Therefore we have a splitting family f : M → Δ_{δ} such that f_{u} (u ≠ 0) has k singular fibers and k − 1 of them are of the following type: the branch curve does not contain the fiber and the bad point is an ordinary double point. The fiber F′ is written by F′ = A + B where A is a nonsingular curve of genus g − 1 and B is a nonsingular rational curve, and A^{2} = B^{2} = −2, AB = 2. Q.E.D.

Proof of Theorem 3.3

Let (W, B) be a representative of the branch curve of [f, F] which has no splitting family.

Step 1 Assume B ⊃ Γ_{0}. Then we will show that the germ [f, F] belongs to class II. Lemma 3.15 implies the condition (a2) in 1.3.3 and lemmas 3.5 and 3.7 imply (a3) and (a1). Lemma 3.11 implies (c).

We consider (b). Assume B has a bad point P of odd multiplicity such that the local branch of B at P does not contain any tangential component to Γ_{0}. By the blow-up τ_{1} at P, we get B_{1}, which contains E_{0,1} + E_{1,1} and does not pass through E_{0,1} ∩ E_{1,1}. Hence we conclude that B has no other bad point, otherwise [f, F] has a splitting family by Lemma 3.10. Let τ^{(1)} : W_{1} → W_{0}^{(1)} be the contraction of E_{0,0} and we set B^{(0)} = τ_{∗}^{(1)}B_{1}. The pair (W_{0}^{(1)}, B^{(1)}) is also a representative of the branch curve of [f, F] and B^{(1)} contains the new fiber Γ_{0} of W_{0}^{(1)}. If B^{(1)} has a bad point P^{(1)} of odd multiplicity without tangential component, P^{(1)} is the unique bad point of B^{(1)}. Then we produce the elementary transformation (W_{0}^{(2)}, B^{(2)}) of (W_{0}^{(1)}, B^{(1)}) at P^{(1)} · · ·. After finitely many times, this situation terminates. Otherwise, {P, P^{(1)}, P^{(2)}, …} coincides with the set of infinitely near points of P and their multiplicities drop for finitely many steps. On the other hand, P^{(i)} is the unique bad point of B^{(i)} without tangential component such that B^{(i)} contains Γ_{0}. This implies (B^{(i)} − Γ_{0})Γ_{0} < 2g + 2 for some i, which is absurd. Therefore after finite times of those elementary transformations, any bad point of the obtained branch curve B^{(r)} either has even multiplicity or has no tangential component. In the former case, B^{(r)} has ordinary singularity at that point by (a2). Hence (b) holds.

Step 2 Assume B ⊅ Γ_{0}. Then the bad point P of B is unique by Lemma 3.9. If m = multi_{P} B ≥ 1 is odd, then the problem is reduced to Step 1 by the elementary transformation at P.

Assume m is even. Then P is an ordinary singularity by lemmas 3.5 and 3.7.

If m = 2, then [f, F] splits into two singular fibers of type 0_{0} by Lemma 3.13. If m ≥ 4, then [f, F] belongs to class I. Hence we proved that any germ [f, F] which does not belong to classes 0_{0}, I or II has a splitting family.

Step3 It remains to prove the finiteness of our reducing process to class G. Let f : M → Δ_{ϵ} × Δ_{δ} → Δ_{δ} be any splitting family constructed in this section for a given (f, F), and let (f_{u}, F′) be any singular fiber germ of the fibration f_{u} (∀u ≠ 0). Then there are some representatives of branch curves (B, W) and (B′, W′) of (f, F) and (f_{u}, F′) respectively so that, for any bad point P′ of B′, there exists a bad point P of B such that (B′, P′) is a (perturbed) deformation of (B, P) in our sense. Then one of the following conditions is satisfied:

(i) The germ of the singularity (B′, P′) coincides with (B, P), (ii) mult_{P′}B′ < mult_{P} B, (iii) mult_{P′}B′ = mult_{P} B and the minimal number of times of blow-ups for (B′, P′) such that the reduced scheme of the total transform of the horizontal component B̃′ of B′ is strictly less than that of (B, P).

Therefore the assertion follows. Q.E.D.

4 Horikawa index and signature

In this section, we define the Horikawa index and the local signature of a germ of singular fiber of a degeneration of hyperelliptic curves. By using these notions, we again present the result of theorem 3.3 and apply it to the study of compact complex surfaces with hyperelliptic pencils, especially to their signature problem.

4.1. Let f : S ⟶ Δ_{ϵ} be a degeneration of hyperelliptic curves of genus g with an unique singular fiber F. Let (W, B) be a representative of the branch curve of the germ [f, F]. Let τ_{r,0} : W_{r} ⟶ W_{0} = W be the succession of blow-ups such that the even proper transform B_{r} is nonsingular. Assume the number r of times of blow-ups is taken to be minimal which enjoy the above property. Let m_{1}, ⋯, m_{r} be the multiplicity sequence of τ_{r,0} as in 1.1. Let S′ ⟶ W be the normal double cover branched along B on the total space of the line bundle of the square root of B. Let {Q_{i}}_{1≤i≤ℓ_{0}} be the set of isolated singular points on S′. Let ρ : S̃ ⟶ S′ be the canonical resolution of S′, i.e. S̃ is the double cover of W_{r} branched along B_{r} on the total space of the line bundle of the square root of B_{r}. The natural fibration S̃ ⟶ Δ_{ϵ} is not relatively minimal in general. Let α_{(W,B)} be the number of times of contraction maps of (−1) curves from S̃ to the original relative minimal model S. Let p_{g}(S′, Q_{i}) and K_{(S′,Q_{i})}^{2} be the geometric genus and the square of the canonical cycle for the resolution ρ of the germ of singularity (S′, Q_{i}) (1 ≤ i ≤ ℓ_{0}) respectively. Note that the formula in [Ho1, Lemma 6] implies that

∑_{i=1}^{ℓ} p_{g}(S′, Q_{i}) = \frac{1}{2} ∑_{i=1}^{r}[\frac{m_{i}}{2} ]([\frac{m_{i}}{2} ] − 1), ∑_{i=1}^{ℓ} K_{(S′,Q_{i})}^{2} = −2 ∑_{i=1}^{r}([\frac{m_{i}}{2} ] − 1)^{2}.

Definition-Lemma 4.2. We put

H([f, F]) := ∑_{i=1}^{ℓ}\{\frac{4(g − 1)}{g}p_{g}(S′, Q_{i})+K_{(S′,Q_{i})}^{2}\}+α_{(W,B)} = \frac{2}{g} ∑_{i=1}^{r}([\frac{m_{i}}{2} ]−1)(g−[\frac{m_{i}}{2} ])+α_{(W,B)}.

This is well-defined as an invariant for [f, F], i.e. is determined independently of the choice of a representative (W, B) of the branch curve of [f, F]. We call it Horikawa index (or H-index for short) of [f, F].

Since this notion is essential in our discussion, we give two proofs for the welldefinedness. One comes from a global compactification argument, and the other comes from a local computational argument.

The first proof of lemma 4.2. Let P_{1}^{(0)}, ⋯, P_{ℓ_{0}}^{(0)} be all the bad points of B. By Artin’s theorem [Ar], we may assume that the germ of singularity (B, P_{i}^{(0)}) (1 ≤ i ≤ ℓ_{0}) is algebraic. We fix an inclusion W = P^{1} × Δ_{ϵ} ↪ P^{1} × P^{1} := Σ_{0}. We denote by π̄ : Σ_{0} ⟶ P^{1} the extension of the projection π : W ⟶ Δ_{ϵ} and identify the original fiber Γ_{0} with π̄^{−1}(0). Then it is easy to see that there exists a reduced divisor B̄^{(0)} on Σ_{0} such that

(i) B̄^{(0)} is linearly equivalent to (2g + 2)C_{0} + 2k_{0}Γ_{0} for a sufficiently large integer k_{0} where C_{0} is a fiber for another fibration of Σ_{0}, (ii) The germs of bad points of B̄^{(0)} on Γ_{0} coincide with (B, P_{1}^{(0)}), ⋯, (B, P_{ℓ_{0}}^{(0)}). Let (W^{(1)}, B^{(1)}) be a representative of the branch curve of [f, F] obtained from (W, B) by one-time elementary transformation. This transformation is globalized from (Σ_{0}, B̄^{(0)}) to (Σ_{1}, B̄^{(1)}) where B̄^{(1)} is a divisor on Hirzebruch surface Σ_{1} of degree 1. Let S^{(i)} be the normal double cover of Σ_{i} branched along B̄^{(i)} on the total space of the line bundle of the square root of B̄^{(i)} for i = 0, 1. By usual double covering method (cf.[P1]), we easily have

ω_{S^{(i)}}^{2} = \frac{4(g − 1)}{g}(χ(O_{S^{(i)}}) − g − 1)

where ω_{S^{(i)}} is the dualizing sheaf.

Now let Q_{1}^{(i)}, ⋯, Q_{ℓ_{i}}^{(i)} be the singular points on the fiber over the origin of the fibration S^{(i)} ⟶ P^{1}. Let S̃^{(i)} ⟶ S^{(i)} be the canonical resolution with respect to these singularities. Let S̃^{(i)} ⟶ S^{∗(i)} be the succession of contractions of (−1) curves such that the fiber over the origin of the fibration S^{∗(i)} ⟶ P^{1} does not contain any (−1) curve. Let α^{(i)} be the number of times of this (−1) contractions. Then we have

χ(O_{S^{∗(i)}}) − χ(O_{S^{(i)}}) = − ∑_{j=1}^{ℓ_{i}} p_{g}(S^{(i)}, Q_{j}^{(i)}), ω_{S^{∗(i)}}^{2} − ω_{S^{(i)}}^{2} = ∑_{j=1}^{ℓ_{i}} K_{(S^{(i)},Q_{j}^{(i)})}^{2} + α^{(i)}

for i = 0, 1. On the other hand, the fibrations S^{∗(i)} ⟶ P^{1} for i = 0, 1 are isomorphic to each other over P^{1}\{0}, and after resolving the singularities on these locuses and contracting (−1) curves, we obtain the same surface by the uniqueness of the relative minimal model. Therefore we have

χ(O_{S^{∗(0)}}) = χ(O_{S^{∗(1)}}), ω_{S^{∗(0)}}^{2} = ω_{S^{∗(1)}}^{2}.

From these, we have

\begin{aligned}& ∑_{j=1}^{ℓ_{0}}\{\frac{4(g−1)}{g}p_{g}(S^{(0)}, Q_{j}^{(0)}) + K_{(S^{(0)},Q_{i}^{(0)})}^{2}\} + α^{(0)} \\ = & ∑_{j=1}^{ℓ_{1}}\{\frac{4(g−1)}{g}p_{g}(S^{(1)}, Q_{j}^{(1)}) + K_{(S^{(1)},Q_{i}^{(1)})}^{2}\} + α^{(1)}. \end{aligned}

Hence the assertion follows. Q.E.D.

The second proof of lemma 4.2. Let (W′, B′) be a resulting pair after one-time elementary transformation of (W, B). Let W′_{r′} ⟶ W′ be the minimal even resolution of B′, and let m′_{1}, ⋯, m′_{r′} be their multiplicity sequence. We put

H = \frac{2}{g} ∑_{i=1}^{r}([\frac{m_{i}}{2} ] − 1)(g − [\frac{m_{i}}{2} ]) + α, H′ = \frac{2}{g} ∑_{i=1}^{r′}([\frac{m′_{i}}{2} ] − 1)(g − [\frac{m′_{i}}{2} ]) + α′

where α = α_{(W,B)} and α′ = α_{(W′,B′)}. It suffices to prove H = H′. Note that 0 ≤ m_{1} ≤ 2g + 3.

(i) Suppose m_{1} = 0, that is, B does not pass through P_{1}. We may assume that m′_{1} = 2g+2, m′_{2} = m_{1}, m′_{3} = m_{2} …. Since there are two more (−1) curves over the proper transform of the fiber on the double cover of W_{r′}, we have α′ = α + 2. Hence H′ = H + \frac{2}{g}([\frac{m′_{1}}{2} ] − 1)(g − [\frac{m′_{1}}{2} ]) + 2 = H. (ii) Suppose m_{1} = 1 and B does not contain Γ_{0}. We may assume m′_{1} = 2g + 2, m′_{2} = 2, m′_{3} = m_{1}, m′_{4} = m_{2} … and we have α′ = α + 2. Hence we similarly have H′ = H. (iii) Suppose m_{1} = 1 and B contains Γ_{0}. We may assume m′_{1} = 2g + 3, m′_{2} = 2, m′_{3} = m_{1}, m′_{4} = m_{2} … and we have α′ = α + 2. Hence H′ = H. (iv) Suppose 2 ≤ m_{1} ≤ 2g+1. In this case, we may assume the i-th even proper transform of B and B′ coincides with each other for i ≥ 1. Especially we have m′_{2} = m_{2}, m′_{3} = m_{3} … and α′ = α. Moreover we have the following equality:

m′_{1} = \{ \begin{matrix}2g + 2 − m_{1} & if Γ_{0} ⊄ B and m_{1} ≡ 0 (2) \\ 2g + 3 − m_{1} & if Γ_{0} ⊄ B and m_{1} ≡ 1 (2) \\ 2g + 3 − m_{1} & if Γ_{0} ⊂ B and m_{1} ≡ 0 (2) \\ 2g + 4 − m_{1} & if Γ_{0} ⊂ B and m_{1} ≡ 1 (2) \end{matrix}

Indeed, we first assume Γ_{0} ⊄ B and m_{1} is even. Let W ⟵^{τ_{1}} W_{1} ⟶^{τ′_{1}} W′ be the elementary transformation. The 1-st even proper transform B_{1} coincides with the proper transform of B by τ_{1},and also coincides with the proper transform of B′ by τ′_{1}. The proper transform E_{0,1} of Γ_{0} by τ_{1} coincides with the exceptional curve for τ′_{1} and vice versa for E_{1,1}. Therefore we have m′_{1} = B_{1}E_{0,1} = (τ_{1}^{∗}B − m_{1}E_{1,1})E_{0,1} = BΓ_{0} − m_{1} = 2g + 2 − m_{1}. The other cases are similar. In any case, we have ([\frac{m′_{1}}{2} ] − 1)(g − [\frac{m′_{1}}{2} ]) = ([\frac{m_{1}}{2} ] − 1)(g − [\frac{m_{1}}{2} ]), which induce H′ = H.

(v) Suppose m_{1} = 2g + 2 or 2g + 3. Then (W′, B′) → (W, B) is also an elementary transformation whose center has multiplicity less than two in B′. Hence the assertion follows from (i), (ii) and (iii). Q.E.D.

Remark 4.3. The notion of H-index was introduced for the first time by Horikawa [Ho2] for genus 2 case. He classified germs of singular fibers of genus 2 into six types 0, I_{k}, II_{k}, III_{k}, IV_{k} and V, and their H-indices are 0, 2k−1, 2k, 2k−1, 2k and 1 respectively. (see also Reid [Re]).

H-index is related to the invariants of compact surfaces by the following theorem, whose proof is essentially seen in Horikawa [Ho3, Theo.2.1] and Persson [P2, Prop.2.12] (see also Xiao[X2]):

Theorem 4.4. (Horikawa-Persson)

(i) For any germ [f, F] of a degeneration of hyperelliptic curves, we have H([f, F]) ≥ 0. (ii) Let φ : V ⟶ C be a global hyperelliptic pencil of genus g, that is, h is a surjective proper holomorphic map from a nonsingular compact surface to a nonsingular compact curve such that the general fiber is a hyperelliptic curve of genus g. Let F_{1}, ⋯, F_{k} be all its singular fibers. Then we have

K_{V/C}^{2} = \frac{4(g − 1)}{g}χ_{φ} + ∑_{i=1}^{k} H([φ, F_{i}]),

where K_{V/C} is the relative canonical sheaf and χ_{φ} = deg φ_{∗}K_{V/C}.

Example 4.5. The calculation of H := H([f, F]) for the germs defined in §1 is the following:

(1) If [f, F] is of type 0_{0}, then H = 0.

(2) If [f, F] belong to class I with 1 ≤ g′ ≤ [(g − 1)/2], then H = 2g′(g − g′ − 1)/g.

(3) If [f, F] belong to class II with 2 ≤ g ≤ 4, then H of the germs corresponding to proposition 1.7 are as follows: (g = 2) (i) H = 1, (g = 3) (i) H = 5/3, (ii) H = 10/3, (iii) H = 8/3, (g = 4) (i) H = 2, (ii) H = 3, (iii) H = 4.

Next we study the signature of compact surfaces with hyperelliptic pencils from the viewpoint of H-index.

Definition 4.6. We put

σ([f, F]) := \frac{1}{2g + 1}\{gH([f, F]) − (g + 1)E([f, F])\}

and call it the local signature of the germ [f.F], where E([f, F]) := χ_{top}(F) − 2g + 2 is the Euler contribution of F (cf.[P3]).

Proposition 4.7. Let φ : V ⟶ C be a global hyperelliptic pencil of genus g, and let F_{1}, ⋯, F_{k} be all its singular fibers. Then we have

σ(V) = ∑_{i=1}^{k} σ([φ, F_{i}]),

where σ(V) is the global signature of V, i.e. the signature of the intersection form of the 2−homology of V.

Proof We use the relative formulation of Xiao [X2,X3]. Set e_{φ} = ∑_{i=1}^{k} E(F_{i}). Noether’s formula and Hirzebruch’s signature theorem ([Hi1]) says that

12χ_{φ} = K_{V/C}^{2} + e_{φ}, 3σ(V) = K_{V/C}^{2} − 2e_{φ}.

We set H(V) = ∑_{i=1}^{k} H([φ, F_{i}]). It follows from theorem 4.4 (ii) and the above formulas that

H(V) = \frac{2g + 1}{3g}K_{V/C}^{2} − \frac{g − 1}{3g}e_{φ} = \frac{2g + 1}{g}σ(V) + \frac{g + 1}{g}e_{φ}.

Therefore the assertion follows. Q.E.D.

Example 4.8. The calculation of σ := σ([f, F]) for the germs defined in §1 is the following:

(1) If [f, F] is of type 0_{0}, then σ = −(g + 1)/(2g + 1).

(2) If [f, F] belong to class I with 1 ≤ g′ ≤ [(g − 1)/2], then σ = 2(gg′ − g^{′2} − g − g′ − 1)/(2g + 1).

(3) If [f, F] belong to class II with 2 ≤ g ≤ 4, then σ of the germs corresponding to proposition 1.7 are as follows: (g = 2) (i) σ = −1/5, (g = 3) (i) σ = 1/7, (ii) σ = −6/7, (iii) σ = 0, (g = 4) (i) σ = 1/3, (ii) σ = 7/9, (iii) σ = −1.

Definition 4.9. Let X([f, F]) be a certain invariant of the germ of singular fiber [f, F]. Let h : Z ⟶ Δ_{ϵ} × Δ_{δ} be a splitting family of [f, F] in the sense of definition 3.1. We say the family h conserves the invariant X if the following condition is satisfied: Let F_{1,u}, ⋯, F_{ℓ,u} be all singular fibers of the fibration h_{u} : Z_{u} ⟶ Δ_{ϵ} × {u} for any u ∈ Δ_{δ}. Then we have

X([f, F]) = X([h_{u}, F_{1,u}]) + ⋯ + X([h_{u}, F_{ℓ,u}]).

The following is a more precise version of theorem 3.3.

Theorem 4.10. Let G be the union of the sets of germs of singular fibers of type 0_{0}, in class I and in class II defined in 1.3. Then for any germ [f, F] of a degeneration of hyperelliptic curves, there exists a reduction of [f, F] to G via splitting families which conserves the H-index and the local signature, that is, these invariants are conserved in any splitting family for this reduction.

Proof Step 1 Let h : Z ⟶ Δ_{ϵ} × Δ_{δ} be any splitting family constructed in §3. By our construction, h is induced by the double cover over a family of local P^{1}- bundle W = W × Δ_{δ} whose branch locus is a family of divisors W ⊃ B ⟶^{ρ} Δ_{δ}. Let = {m_{1,u}, ⋯, m_{r_{u},u}} be the multiplicity sequence of the minimal even resolution of (W_{u}, B_{u}) for u ∈ Δ_{δ}. We note that, bad points on several fibers of ρ_{u} contribute to this set for u ≠ 0, while bad points are on an unique fiber for u = 0. Now we define the subsequence T_{u} of the sequence {[m_{1,u}/2], ⋯, [m_{r_{u},u}/2]} which consists of the elements of numbers greater than 1, and we consider it as a set, i.e. we neglect its order. (Note that the multiplicity with [m_{u,i}/2] ≤ 1 does not contribute the H-index of the fiber.) Then it follows from our construction that the splitting family h conserves this set, i.e. we have

T_{0} = T_{u} for any u ∈ Δ_{δ}. Moreover the number of times of blow-downs of (−1) curves from the canonical resolution to its relative minimal model is independent of u by [Kod2]. Therefore h conserves the H-index.

Step 2 Next we show the conservation of the Euler contribution, which induce the conservation of the local signature by the definition and Step 1.

We first consider the splitting family of lemma 3.5. We use the notations in this lemma. By the stability of (−1) curves, we may identify M^{♭} with M. Let Ẽ be the set of components of F which dominates E_{j,r} by the generically two-to-one map h_{0} : M_{0} ⟶ W_{r}. Note that Ẽ is irreducible or consists of two irreducible components. Let A_{1}, ⋯, A_{s}, A′_{1}, ⋯, A′_{s′}, D be all the connected components of F −Ẽ, where A_{i} (1 ≤ i ≤ s) intersect Ẽ at one point, A′_{j} (1 ≤ j ≤ s) intersect Ẽ at two points and h_{0}(D) = Q. Then we have

χ_{top}(F) = χ_{top}(Ẽ) + ∑_{i=1}^{s} χ_{top}(A_{i}) + ∑_{j=1}^{s′} χ_{top}(A′_{j}) − s − 2s′ + χ_{top}(D) − ϵ(D),

where ϵ(D) (1 or 2) is the number of the points of the intersection of D and Ẽ.

Now we consider singular fibers F′ = f_{u}^{−1}(0), F′′ = f_{u}^{−1}(π(R_{u})) of h_{u} : M_{u} ⟶ W_{r}×{u} for u ≠ 0. Let Ẽ′ be the set of components of F′ which dominates E_{j,r} × {u} by h_{u}. Then F′ − Ẽ′ has connected components which are isomorphic to A_{1}, ⋯, A_{s}, A′_{1}, ⋯, A′_{s′}, and we have χ_{top}(F′) = χ_{top}(Ẽ′) + ∑_{i=1}^{s} χ_{top}(A_{i}) + ∑_{j=1}^{s′} χ_{top}(A′_{j}) − s − 2s′. Moreover our method of the construction implies

χ_{top}(Ẽ′) = χ_{top}(Ẽ) − 2[\frac{k}{2}],

where k = I_{Q}(B_{r}, E_{j,r}). On the other hand, F′′ consists of the irreducible component Γ̃ which dominates the fiber P^{1} by h_{u} and the connected component which is isomorphic to D. We have χ_{top}(F′′) = χ_{top}(Γ̃) + χ_{top}(D) − ϵ(D). Moreover our construction implies

χ_{top}(Γ̃) = 2 − 2g + 2[\frac{k}{2}].

Therefore we have E(F) = E(F′) + E(F′′). For other splitting families constructed in §3, we have the assertion by the similar argument. We omit it.

Hence the claim of the theorem follows from theorem 3.3. Q.E.D.

Corollary 4.11. Assume the germ [f, F] has vanishing H-index. Then [f, F] is reduced to germs of type 0_{0} via splitting families.

Proof Note that the germ has vanishing H-index if and only if it has a representative of branch curve which has at most simple singularities, in other words whose normal double cover has at most rational double points. Especially a germ in class II has positive H-index by the definition. Therefore example 4.5 says that a germ which belongs to G has vanishing H-index if and only if it is of type 0_{0}. Thus the assertion follows from theorem 4.10. Q.E.D.

Remark 4.12 The surface of general type with a pencil whose slope has minimal value automatically has a hyperelliptic pencil such that all the singular fibers have vanishing H-indices ([X2, Kon]). Therefore, considering the above corollary, one can ask whether such a surface has a reduction to surfaces with at most 0_{0} fibers by global splitting families or not. This type of problem is called “global Morsification problem”.

Corollary 4.13. Let φ : V ⟶ C be a global hyperelliptic pencil of genus 2 ≤ g ≤ 4. Then the global signature σ(V) is written as follows:

(g = 2) σ(V) = −(3/5)k_{0_{0}} − (1/5)k_{(II,i)},

(g = 3) σ(V) = −(4/7)k_{0_{0}} − (6/7)k_{(I,1)} + (1/7)k_{(II,i)} − (6/7)k_{(II,ii)},

(g = 4) σ(V) = −(5/9)k_{0_{0}} − (2/3)k_{(I,1)} + (1/3)k_{(II,i)} + (7/9)k_{(II,ii)} − k_{(II,iii)},

where k_{0_{0}}, ⋯, k_{(II,iii)} are the sum of numbers of germs of the corresponding type which are obtained by applying every singular fibers of φ to our reducing process via local splitting families. For instance, k_{(I,1)} is the number of germs in class I with g′ = 1, and k_{(II,i)} is the number of germs of type (i) in class II written in proposition 1.7.

Proof The assertion is clear from proposition 4.7, theorem 4.10, proposition 1.7 and example 4.8. Q.E.D.

Remark 4.14. For g = 2, The above formula due to Matsumoto, Horikawa and corollary 4.11 as we explained in Introduction. For g ≥ 3, corollary 4.13 and example 4.8 tell us that there are many singular fibers with positive local signature. In some sense, this fact support the existence of surfaces with hyperelliptic pencils whose global signatures are positive, which was discovered by Xiao and Chen [C].

Acknowledgement The authors thanks to Professors Eiji Horikawa, Kazuhiro Konno, Miles Reid and Yukio Matsumoto for variable discussions. Thanks are also due to Professors Masayoshi Miyanishi, Xiao Gang, Sampei Usui and Dr. Shigeru Takamura for useful advices. The paper is written during the second-named author’s stay at Max-Planck-Institut für Mathematik, which he thanks for its hospitality.

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Tatsuya Arakawa Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560, Japan. E-mail:arakawa@mathsun01.math.sci.osaka-u.ac.jp

Tadashi Ashikaga Max-Planck-Institut für Mathematik, Gottfried-Claren-Straβe, 53173, Bonn, Germanny and Faculty of Engineering, Tôhoku-Gakuin University, Tagajo, Miyagi, 985, Japan. E-mail:tashikaga@tjcc.tohoku-gakuin.ac.jp