数理解析研究所講究録 1233 巻 2001 年 18-50

18

Towards the Classification ofAtoms of Degenerations, I
Splitting Criteria via Configurations ofSingular Fibers

1 ぐン {?}イ

Shigeru Takamura

Research Institute for Mathematical Sciences, Kyoto University
Oiwakecho, Kitashirakawa, Sakyo-ku
Kyoto 606-8502 JAPANe-mail:takamura@kurims.kyoto-u.ac.jp

VERSION: July 31, 2001

Abstract

Motivated by the classification problem of atomic degenerations, in ourseries of papers, we make asystematic study for splitting deformations of degenerations of complex curves. We provide various new methods to constructsplitting deformations, and deduce many splitting criteria of degenerations,be applied to the classification of atomic degenerations. Roughly,
which
our criteria are separated into two tyPes; in the first type the criteria areexpressed in terms of the configuration of asingular fiber, and in the second
type, in terms of sub-divisors of asingular fiber. In both types, our constructions are ‘visible, in that we can view how the singular fiber is deformed. In
the present paper, we demonstrate splitting criteria of the first type.
$\mathrm{w}\mathrm{i}\mathrm{U}$;Secondary
thematical Subject Classification: Primary
iwords: Degeneration of complex curves, Complex surface, Singular fiber, Riemann surf
formation of complex structures, Splittings of singular fibers, Atomic degeneration, Monodr$14\mathrm{D}05,14\mathrm{J}15$$14\mathrm{H}15,32\mathrm{S}30$

19
Introduction
This paper constitutes one part of our series of papers on degenerations. By afrom asmooth complex
degeneration, we mean aproper surjective map :to the unit disk such that the fiber over the origin is singular and any
surfaceother fiber is asmooth curve of genus $g(g\geq 1)$ . Adeformation of adegeneration iscalled asplitting deformation, provided that it induces asplitting of its singular fiber.
We notice that it may occur that adegeneration admits no splitting deformation
at all, in which case the degeneration is called atomic. Our main problem is toclassify atomic degenerations of arbitrary genera (see [Re]). The classification has
been known only for the very low genus cases; for the genus 1case, by Moishezon[Mo], and for the genus 2case, by Horikawa [Ho] (see also \S 6.3), where they usedthe double covering method for constructing splitting deformations.
Recent progress for the genus 3case was made by Ashikaga and Arakawa [AA],who obtained results on the classification of atomic degenerations of hyperelliptic
curves of genus 3. Their method is also based on the double covering method. Unfortunately, this method fails to work for degenerations of non-hyperelliptic curves.
Some new idea is needed for constructing splitting deformations of degenerations of
non-hyperelliptic curves even for the genus 3case (note that for the genus 1and 2). In our series
cases, all curves are hyperelliptic, but this is not the case for genus
deforsplitting
constructingformethods
of papers we develop completely different
mations, and apply them to the classification of atomic degenerations for the genus, Ta]. The aim of this paper is to study the relation between
3,4 and 5cases [the configurations of singular fibers and the existence of splitting deformations. We
first show that two types of degenerations are atomic.
$\pi$$Marrow\triangle$$\triangle$$l\vee I$$\geq 3$$\mathrm{T}\mathrm{a},\mathrm{I}\mathrm{I}\mathrm{I}$be a degeneration of curves such that the singular
Theorem 2.0.2 Let :fiber $X$ is either (I) a reduced curve with one node, or (II) a multiple of a smoothis atomic.
curve of multiplicity at least 2. Then :$\pi$$Marrow\triangle$$\pi$$Marrow\triangle$

We remark that the proof of Theorem 2.0.2 carrries over to arbitrary dimensions
beto show that adegeneration of type (II) is atomic, i.e. letting :degeneration of compact complex manifolds of arbitrary dimension, if the singular
is atomic.
fiber $X$ is amultiple of asmooth complex manifold, then :We demondeformations.
Next, we shall state results on existence of splitting
strate several splitting criteria via the configuration of the singular fiber. Roughly,
these criteria are classified into two types; the first one is in terms of some singularities on the singular fiber and the second one is in terms of the existence of irreducible
components of multiplicity 1satisfying certain properties (see the list of splitting
criteria in the bottom of this introduction). Most of our criteria also give the explicit
description of splittings of singular fibers. We note that the commutativity of some
topological monodromies follows from one of these criteria (see Proposition 6.1.2).
From our criteria, we will see that many degenerations withgular fibers always admit splitting deformations. Together with Theorem 2.0.2 it is
$\pi$

$\pi$

$Marrow\triangle$$\mathrm{a}$$Marrow\triangle$$\mathrm{n}\mathrm{o}\mathrm{n}- \mathrm{s}\mathrm{t}\mathrm{a}\mathrm{r}- \mathrm{s}\mathrm{h}\mathrm{a}\mathrm{p}\mathrm{e}\mathrm{d}^{1}\sin-$lSee \S 4.

20interesting to know whether the following is true or not.
Conjecture 6.3.1 A degeneration is atomic if and only if its singularfiber is either
a reduced curve with one node, or a multiple of a smooth curve.

, [Ta] for results on this conjecture. (Actually, this conjecture seems
See
too optimistic for higher genus cases. Amore reasonable conjecture is given byreplacing ‘atomic’ by ‘absolutely atomic’, where adegeneration:isabsolutely atomic provided that all degenerations with the same topological type as
are atomic.) In order to classify atomic degenerations, the results of thispaper enable us to use the induction with respect to genus (see \S 6.3 for details);
let
be aset of degenerations : Alof curves of genus such that$[\mathrm{T}\mathrm{a},\mathrm{I}\mathrm{I}\mathrm{I}]$$\pi$$Marrow\triangle$$\pi:Marrow\triangle$$g$$\Lambda_{g}$$\pi$$arrow\triangle$$g$(1) the singular fiber $X$ has amultiple node2 (here we exclude the case where $X$
is areduced curve with only one node), or(2) $X$ contains an irreducible componentof multiplicity 1satisfying the following:ifis connected, then either genus(00), oris aprojective line intersecting other irreducible components at at least two
points.
$\ominus_{0}$$\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}^{3}$$X\backslash \ominus_{0}$$\geq 1$$\ominus_{0}$As aconsequence of our splitting criteria, we obtain the following.

Theorem 6.3.2 $Suppose^{4}$ that Conjecture 6.3.1 is valid for genus $\leq g-1$ .
:is a degeneration in , then is not atomic.
$\pi$$Marrow\triangle$$\Lambda_{g}$

If

$\pi$

Hence, if the assumption of this theorem is fulfilled, to determine atomic degenerations of curves of genus , it suffices to check the splittability of degenerations$\pi:Marrow\triangle$
such that$g$(A) $X=\pi^{-1}(0)$ is star-shape, or(B) $X$ is not star-shaped and (B.I) $X$ has no multiple node and (B.2) if $X$ has anirreducible componentof multiplicity 1, thenis aprojective line, andintersects other irreducible components of $X$ only at one point.
$\ominus_{0}$$\ominus_{0}$, we develop another method for constructing splitting deformations, whichIn
uses ‘barkable’ sub-divisors in singular fibers. This method is quite powerful andworks for degenerations satisfying (A) or (B).
$[\mathrm{T}\mathrm{a},\mathrm{I}\mathrm{I}\mathrm{I}]$

List of splitting criteria via configurations of singular fibers(In most cases, we assume that adegeneration is normally minimal (see \S 1). This
assumption is not restrictive at all. See \S 1. We notice that in some cases, twodifferent criteria are applicable to one degeneration.)
multiple node is either an intersection point of two irreducible components of the same
multiplicity, or aself-intersection point of all irreducible component.
is not connected, we pose no condition.
assumption is valid for $g=2$ and 3.
$2\mathrm{A}$$3\mathrm{I}\mathrm{f}X\backslash \ominus_{0}$$4\mathrm{T}\mathrm{h}\mathrm{i}\mathrm{s}$21
be nomally minimal such that the singular
Criterion 5.1.2 Let :$X$ has a multiple nodeof multiplicity at least 2. Then there exists a splitting
fiber
splits $X$ into
is a reduced
and , where:
which
deformation of$X$
by replacing the multiple node by
is obtained fromcurve with one node andmultiple annulus.
$Marrow\triangle$$\pi$$Marrow\triangle_{:}$$\pi$$X_{1}$$X_{2}$$X_{1}$$X_{2}$$a$is normally minimal such that the singular fiber
Criterion 5.1.3 Let :$X$ contains a multiple node (of multiplicity). Thenis atomic if and
$X$
node.
with
one
only if is a reduced curve$Marrow\triangle$$\pi$$\pi:Marrow\triangle$$\geq 1$be relatively minimal. Suppose that the singular
Criterion 5.2.2 Let : M
germ
a
that
p
such
of p in X is either
fiber X has a point(1) a multiple of a plane curve $singularity^{5}$ of multiplicity at least 2, or$arrow\triangle$$\pi$(2) a plane curve singularity such that if it is a node, then

Then

$\pi$

:

$Marrow\triangle$$X\backslash p$

is not smooth.

admits a splitting deformation.

be normally minimal. Suppose that the singular
Criterion 6.1.1 Let :$X$ contains an irreducible component
isof multiplicity 1such thatfiber
, $(l\geq 2)$ all connected compO(topologically) disconnected. Denote by , ,admits a splitting deformation which splits $X$. Then :nents of$(i=1, 2, \ldots, l)$ is obtained from $X$ by ‘smoothing’, , where, ,intois the omission of .
, . Here, ,, ,$Marrow\triangle$$\pi$$X\backslash \Theta_{0}$$\mathrm{O}_{0}$$\mathrm{Y}_{1}$$\mathrm{Y}\iota$$\mathrm{Y}_{2}$$\ldots$$X\backslash \Theta_{0}$$X_{1}$$X_{2}$$\ldots$$\ldots$$Marrow\triangle$$X_{i}$$X_{l}$$\check{\mathrm{Y}}_{k}$$\check{\mathrm{Y}}_{i}$$\mathrm{Y}_{i}$$\mathrm{Y}_{l}$$\mathrm{Y}_{2}$$\mathrm{Y}_{1}$$\pi$$\ldots$

be normally minimal such that the singular fiber
Criterion 6.2.1 Let :becontains an irreducible componentof multiplicity 1. Let :$\pi$$Marrow\triangle$$X$$\mathrm{O}_{0}$$\pi_{1}$$M_{1}arrow\triangle$

in M. Suppose thatthe restriction of to a tubular neighborhoodof$\mathrm{Y}^{+}:=M_{1}\cap
X$ intowhich splits
admits a splitting deformation:
admits a splitting deforrnation 1which splits $X$. Then :,,,along the boundary.
by gluing
is obtained from, , where,,into$M_{1}$$\pi$$l\vee I_{1}arrow\triangle$$\pi_{1}$$\mathrm{Y}_{1}^{+}$$\pi$$\mathrm{Y}_{l}^{+}$$\mathrm{Y}_{2}^{+}$$\ldots$$X_{1}$$X_{2}$$\ldots$$X\backslash \Theta_{0}$$\Psi_{1}$$X_{l}$$Marrow\triangle$$X_{i}$$\mathrm{Y}_{i}^{+}$$\mathrm{O}_{0}^{-}-$

Acknowledgment. Iwould like to express my deep gratitude to Professor Tadashi
Ashikaga for valuable discussions and warm encouragement. It is also my great
pleasure to thank Professor Fumio Sakai for valuable advice and suggestions after
he read the early draft of this paper. Ialso would like to thank Professors Toru
Gocho and Mizuho Ishizaka for fruitful discussions. Ialso would like to thank the
Max-Planck-Institut fiir Mathematik at Bonn, and the Research Institute for Mathematical Sciences at Kyoto University for their hospitality and financial support.

1Preparation
In this paper, $\triangle:=\{s\in \mathbb{C} : |s|<1\}$ stands for the unit disk. Let :be aproper surjective hplomorphic map from asmooth complex surface $M$ to$\pi$$5\mathrm{I}\mathrm{n}$this paper aplane curve singularity always means areduced one$Marrow\triangle$$\triangle$,

22, $(s\neq 0)$ is asmooth complex curve ofis singular, andsuch thatis adegeneration of complex curves ofgenus $g(g\geq 1)$ . We say that :$X:=\pi^{-1}(0)$. Two degenerations
genus with the singular fiber:and:are called topologically equivalent if there are orientation preserving
homeomorphisms
and, which make the following diagram
commutative:$\pi^{-1}(s)$$\pi^{-1}(0)$$Marrow\triangle$

$\pi$

$g$

$If_{1}arrow\triangle$$\pi_{1}$$M_{2}arrow\triangle$$\pi_{2}$$h:\trianglearrow\triangle$$H:\mathbb{J}f_{1}arrow M_{2}$$\mathrm{J}f_{1}arrow M_{2}H$$\pi_{1\downarrow}$$\downarrow\pi_{2}$$h$$\trianglearrow\triangle$.

Next, we introduce basic terminology concerned with deformations of degenera, where is sufficiently small. Suppose thattions. We set
is asmooth complex 3-manifold, and:is aproper surjective$M_{t}:=\Psi^{-1}(\triangle\cross\{t\})$holomorphic map. We set
and
:
.
$M$
, the composite map
Since
is smooth and
: IX
is a
submersion, and so
is smooth. We say that :i adeformation
of :
if :
coincides with :. For consistency, we
mainly use the notation
instead of.
We introduce aspecial class of deformations of adegeneration. Suppose that:is relatively minimal, i.e. its singular fiber contains no (-1)-curve(exceptional curve of the first kind). Adeformation:
is said to be
asplitting defor notion of :, provided that for $t\neq 0$ ,:has
at least two singular fibers. In this case, if,, $X_{l}(l\geq 2)$ axe singular fibers:, then we say that $X$ splits intoof
, , , . We note that a
splitting of the singular fiber induces afactorization of the topological monodromyof :. Letting :be the topological monodromy around in :,we have.
Next, we define the notion of splitting deformations for adegeneration : $Marrow$
which is not relatively minimal. We first introduce some notation. Let us take asequence of blow down maps$\triangle^{\mathrm{t}}:=\{t\in \mathbb{C} :|t|<\delta\}$$\delta$$\Psi$$\mathcal{M}$$\mathcal{M}$$arrow\triangle\cross\triangle^{\mathrm{t}}$$\pi_{t}:=\Psi|_{M_{t}}$$\dim\triangle^{\mathrm{t}}=1$$Marrow\triangle$$\Psi$$arrow\triangle\cross\triangle^{\mathrm{t}}$$\mathrm{s}$$M_{0}arrow\triangle\cross\{0\}$$\pi_{0}$$\mathcal{M}$$\cross\{t\}$$arrow\triangle^{\mathrm{t}}$$\mathrm{p}\mathrm{r}_{2}\mathrm{o}\Psi$$M_{t}$$\pi$$\Lambda f_{t}arrow \mathrm{I}\mathrm{S}$$Marrow\triangle$$\pi$$\triangle\cross\{t\}$$\triangle_{t}$$Marrow \mathrm{I}\mathrm{S}$$\pi$$\Psi$$\pi$$\pi_{t}$$X_{1},X_{2}$$M_{t}arrow\triangle_{t}$$\pi_{t}$$Marrow\triangle\cross\triangle\dagger$$Marrow\triangle$$M_{t}arrow\triangle_{t}$$\ldots$$X_{1}$$X_{2}$$\ldots$$X_{l}$$\gamma$$\pi$$Marrow\triangle$$X_{i}$$\gamma$$\pi_{t}$$M_{t}arrow\triangle_{t}$$\gamma=\gamma_{1}\gamma_{2}\cdots\gamma_{l}$$\pi$$\triangle$$Marrow M_{1}arrow M_{2}arrow f_{1}f_{2}f_{3}...arrow M_{r}f_{r}$and degenerations(1)$f_{i}$$\pi_{i}$:$M_{}arrow\triangle$. $(i=1,2, \ldots,r)$ where: $M_{i-1}arrow M_{i}$ is ablow down of a(-1)-curve inis naturally induced from:, and$M_{-1}.\cdot$$M_{}arrow\triangle$(2)$\pi_{r}$,:$\pi:-1$$M_{f}arrow\triangle$

and the map:$\pi_{i}$$M_{i-1}arrow\triangle$

is arelatively minimal.

Given adeformation I:of :, we shall construct a
deformation:of the relatively minimal degeneration :.
First, recall that by Kodaira’s stability theorem [K02], any (-1)-curve in acomplex
surface is preserved under an arbitrary deformation of the surface. Thus, there
exists afamily of (-1)-curves in. We blow down them simultaneously to obtai$\mathcal{M}$$\Psi_{f}$$arrow\triangle\cross\triangle\dagger$$\pi$$Marrow\triangle$$\mathcal{M}_{r}arrow\triangle\cross\triangle^{\mathrm{t}}$$\pi_{r}$$\Lambda f_{r}arrow\triangle$$\mathcal{M}$$\mathrm{n}$23::. Again, by Kodaira’s stability, thereofadeformation, which we blow down simultaneously to obtain
exists afamily of (-1)-curves in::. We repeat this process and finally
adeformationof::. Namely, given adeformationofobtain adeformation
\
t
r
i
a
n
g
l
e
\
d
a
g
e
r
\rightarrow \triangle, we obtain adeformation
:
of :
:
\
t
r
i
a
n
g
l
e
\
d
a
g
e
r
:
. We say that I:ofis asplitting defor mation of, provided that::is asplitting deformation of the. We say that adegeneration is atomic:
relatively minimal degeneration
if it admits no splitting deformation at all.
In this paper, instead of relatively minimal degenerations, we mainly use normally minimal degenerations, because they reflect the topological type (or topological monodromies) of degenerations. See \S 4. Recall that :is normallyminimal if $X$ satisfies the following conditions:$\Psi_{1}$$\mathcal{M}_{1}arrow\triangle$$M_{1}arrow\triangle$$\pi_{1}$$\mathrm{J}/I_{2}$$\Psi_{2}$$\mathcal{M}_{2}arrow\triangle$$\mathcal{M}_{r}arrow\triangle$$\Psi_{r}$$\Psi$$\mathcal{M}$$\pi$$\mathcal{M}$$Marrow\triangle$$arrow\triangle$$\mathcal{M}_{r}arrow\triangle$$\Psi_{r}$$\pi_{r}$$\cross$$\cross\triangle^{\mathrm{t}}$$NI_{r}arrow\triangle$$\pi$(1) the reduced part

$X_{\mathrm{r}\mathrm{e}\mathrm{d}}:= \sum_{i}\ominus_{i}$is a(-1)-curve, then(2) ifleast three points.
$\Theta_{i}$$\Theta_{i}$$\mathcal{M}_{r}arrow\triangle\cross\triangle\dagger$$\Psi_{r}$$M_{r}arrow\triangle$$\pi_{r}$$M_{r}arrow\triangle$$\pi_{r}$$Marrow\triangle$$\pi$$\cross$$M_{2}arrow\triangle$$\pi_{2}$$Marrow\triangle$

is normal crossing, andintersects other irreducible components at at

In this case, we also say that the singular fiber $X$ is normally minimal. The following
lemma is useful.

be a normally minimal degeneration of complex:
Lemma 1.0.1 Let
is a deformation of :curves of genus . Suppose that :$(t\neq0)$:
has at least two normally minimal singular fibers.
such that.
is a splitting deformation of :Then :$Marrow\triangle$$\pi$$\Psi$$g$$\pi_{t}$$\Psi$$\mathcal{M}$$\mathcal{M}$$arrow\triangle\cross\triangle^{\mathrm{t}}$$\pi$$Marrow\triangle$$M_{t}arrow\triangle$$arrow\triangle\cross\triangle^{\mathrm{t}}$$\pi$$Marrow\triangle$

Proof We first show the statement for the case $g\geq 2$ . Let

be the:, and letbe the deforrelatively minimal model of :andare normallymation of , which is determined from V. Suppose that$(i=1,
2)$ in
.
of
Then
the
is
image
:
fibersofminimal singulararoundis nontrivial (seealso singular, because the topological monodromy of] . If $g=1$ , this argument is valid except that none,
[MM2], and also [ES,amultipleofasmooth elliptic curve, in which case, the topologiis
andofcal monodromy is trivial. However, amultiple of asmooth elliptic curve is clearly
relatively minimal (it contains no projective line at all), so we completes the proof.
$Marrow\triangle$$\pi$$\Psi_{r}$$\pi_{r}$:$M_{r}arrow\triangle$$\mathcal{M}_{r}arrow\triangle\cross\triangle\dagger$$\mathrm{Y}_{2}$$\mathrm{Y}_{1}$$\pi_{r}$$\mathrm{J}/I_{r,t}arrow\triangle_{t}$$\pi_{r,t}$$\mathrm{Y}_{i}$$\pi_{r}$$\mathrm{I}\mathrm{m}$$\mathrm{Y}_{1}$$\mathrm{S}\mathrm{T}$$\Lambda/I_{r,t}$$\mathrm{Y}_{i}$$)$$\mathrm{Y}_{2}$$\square$

2Atomic degenerations
In this section, we exhibit two types of atomic degenerations.
be a degeneration of curves such that the singular
Theorem 2.0.2 Let :fiber $X$ is either (I) a reduced curve with one node, or (II) a multiple of a smoothis atomic
curve of multiplicity at least 2. Then :$\pi$$Marrow\triangle$$\pi$$Marrow\triangle$

24
We notice that in the type (I), X has one or two irreducible components, in the later
case, two smooth irreducible components intersecting at one point transversally. The
type (II) means that X is of the form mO, where rn 2, and 0 is asmooth curve.
$\ovalbox{\tt\small REJECT}$

Remark 2.0.3 We remark that the proof of Theorem 2.0.2 carrries over to arbitrary
dimensions to show that adegeneration of type (II) is atomic, i.e. letting :be adegeneration of compact complex manifolds of arbitrary dimension, if thesingular fiber $X$ is amultiple of asmooth complex manifold, then :is$\Lambda f$$\pi$$arrow$$\triangle$$Marrow\triangle$$\pi$

atomic,We first demonstrate that if $X$ is areduced curve with one node, then :is atomic. We prove this by contradiction. Assume that
:
i a
$X$
$X_{l}(l\geq
2)$,
deformationof,,splittingwhich splits
into. We noticethat adeformation of anode is either equisingular, or smoothing. Hence X. is anequisingular deformation of $X$ , and so it is also areduced curve with one node.
, we have $\chi(M)=\chi(M_{t})$ , where $\chi(M)$ stands for the
Since $M$ is diffeomorphic totopological Euler characteristic of $M$ . From this equation, we deduce the following
relation of Euler characteristics (see [BPV]
):
$\pi$

$\Psi$$\mathcal{M}$$Marrow\triangle$$arrow\triangle\cross\triangle^{\mathrm{t}}$$\mathrm{s}$$X_{1}$$\pi$$X_{2}$$\ldots$$\cdot$$M_{t}$$\mathrm{p}97$(2.0.1)$\mathrm{X}(\mathrm{X})-(2-2g)=.\cdot\sum_{=1}^{l}[\chi(X\dot{.})-(2-2g)]$

Since $X$ and$X_{1},X_{2}$,$\ldots$,$X_{l}$.

are reduced curves with one node, we have$\chi(X)=\chi(X_{1})=\cdots=\chi(X_{l})=2-\underline{9}g+1$.

Then (2.0.1) implies that $1=l$ , which gives the contradiction.
Note: We can also show the above statement purely analytically by the computation$(l\geq 2)$ , then the nodeof Ext1 (cf. [Pal]). In fact, if $X$ splits into
, ,( -singularity)of $X$ splits into nodes. However, an -singularity does not admit
any splitting. This gives acontradiction.
$X_{1},X_{2}$$l$$A_{1}$$X\iota$$\ldots$$A_{1}$

3The proof of Theorem 2.0.2 for the type (II)Next, we shall demonstrate that if $X$ is amultiple of asmooth curve, then : $Marrow$is atomic. The proof is quite intricate and long, so we separate the statement into
several claims to clarify the main step of the proof; for adeformation :, we first construct an unramified covering :of :, and thenshow that the Stein factorization offactors through asmooth family over a
disk.
$\pi$

$\triangle$$\pi_{t}$$\pi$$Marrow\triangle$$p_{t}$$M_{t}arrow\triangle_{t}$$M_{t}arrow M_{t}$$\pi_{t}\mathrm{o}p_{t}$3.1Preparation

First, we construct an unramified cyclic $m$ covering of
consider aline bundle
on $M$ . Notice that
$L=\mathcal{O}_{M}(\ominus)$$\Lambda f$. For this purpose, we$L^{\otimes m}\cong O_{M}$, because $?n$ is25
the principal divisor defined by the holomorphic function . We set $F_{s}:=\pi^{-1}(s)$(so $F_{0}=m\ominus$ ). Then has the following property: (1) For $s\neq 0$ , the restriction
is atrivial bundle on , and (2) the restrictionis aline bundle on.
that, and letNext, we take an open covering
be local trivializa. We take anon-vanishingof , with coordinates
tions
holomorphic section
define} of$M$$MarrowM$in . The map :given byisasmooth hypersurface
an unramified cyclic $m$ -covering. From the property of the line bundle , (1) for$s\neq 0$ ,has $m$ connected components such that each connected component
is connected, andanis diffeomorphic to , and (2)unramified cyclic m-covering.
is atomic, we shall prove that for an arbitrary
In order to show that :
\
t
r
i
a
n
g
l
e
\
d
a
g
e
r
\rightarrow\triangle:
has aunique singular fiber, andof ,deformation :to
0. For this purpose, we firstdiffeomorphicis
it is of the form mOt, where;notice thatis diffeomorphic toconstruct an unramified cyclic covering of$(x,t)\mapsto(f(x),t)$
is an unramified cyclic,, and the map, where we
cyclic $m$ covering-covering. Thus we have an
.
by
(This
possible,
from
that
on
is
inducedongive the complex structure) By construction, setting $NI_{t}:=\rho^{-1}(M_{t})$ , the restriction
because iscyclic $m$ -covering. Applying the Steinis also anof to:, we obtain acommutative diagram:
factorization to the map
$\pi$

$L$

$F_{s}$$L|_{F_{s}}$$\ominus \mathrm{s}\mathrm{u}\mathrm{c}\mathrm{h}$$L|_{\ominus}$$(L|_{\ominus})^{\otimes m}\cong \mathcal{O}_{\ominus}$$M= \bigcup_{\alpha}U_{\alpha}$$L$$U_{\alpha}\cross \mathbb{C}$$U_{\alpha}\cross \mathbb{C}$$(z_{\alpha}, \zeta_{\alpha})\in U_{\alpha}\cross \mathbb{C}$$L^{\otimes(-m)}\cong \mathcal{O}_{M}.\underline{\mathrm{E}}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}^{6}\tau_{\alpha}(z_{\alpha})\zeta_{\alpha}^{m}+1=0$$\tau=\underline{\{}\tau_{\alpha}$

$L$

$f$

$f(z_{\alpha}, \zeta_{\alpha})=z_{\alpha}$$L$$f^{-1}(F_{s})$$\ominus\sim:=f^{-1}(\Theta)$$F_{s}$$\pi$$\Psi$$\mathcal{M}$$f|\ominus\sim:\Theta\simarrow\ominus \mathrm{i}\mathrm{s}$$Marrow\triangle$$\pi$$\cross$$\pi_{t}$$M_{t}arrow\triangle_{t}$$\ominus_{t}$$\mathcal{M}$$\mathcal{M}$$M\cross\triangle^{\mathrm{t}}arrow M\cross\triangle^{\mathrm{t}}$$M\cross\triangle^{\mathrm{t}}$$\rho:\overline{\mathcal{M}}arrow \mathcal{M}$$m$$\underline{\mathrm{u}\mathrm{n}\mathrm{r}}\mathrm{a}\mathrm{m}\mathrm{i}\mathrm{f}\mathrm{i}\mathrm{e}\mathrm{d}$$\mathcal{M}$$\mathcal{M}$$\rho$$\rho$$\mathrm{u}\mathrm{n}\mathrm{r}\mathrm{a}\mathrm{m}\mathrm{i}\mathrm{f}\mathrm{f}\underline{\mathrm{i}\mathrm{e}\mathrm{d}}.$$\overline{M_{t}}arrow M_{t}$$p_{t}$$\rho$$M_{t}$$\underline{\mathrm{u}}\mathrm{n}\mathrm{r}\mathrm{a}\mathrm{m}\mathrm{i}\mathrm{f}\mathrm{i}\mathrm{e}\mathrm{d}$$\pi_{t}\mathrm{o}p_{t}$$M_{t}arrow\triangle$(3.1.1)$\overline{M_{t}}arrow M_{t}p_{t}$$\pi_{t}$$\tilde{\pi}_{\underline{t\downarrow}\downarrow}\triangle_{t}arrow\triangle\overline{p}_{t}$

$t$

,

:
is an -covering, and (2)is asmooth7 curve, and:is aproper surjective map such that all fibers are (topologically)
connected. We notice that since is acyclic covering, from the commutativity of
the above diagram, it is easy to check that is also acyclic covering.
$\triangle_{t}\simarrow\triangle$$\triangle_{t}-$$m$$\overline{p}_{t}$$\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\underline{\mathrm{e}}(1)$$M_{t}arrow\triangle_{t}\sim$$\overline{\pi}_{t}$$p_{t}$$\overline{p}_{t}$3.2

The proof of Theorem 2.0.2 for the type (II)After the above preparation, we prove Theorem 2.0.2 for the type (II). The key
ingredients of the proof are the following two claims, which together imply that the. In:
Stein factorization (3.1.1) is nothing but the stable reduction ofis sufficiently small.
what follows, we always assume that$\pi_{t}$$\mathrm{J}/I_{t}arrow\triangle_{t}$$|t|$Claim A$\tilde{\acute{\mathit{1}}\mathrm{r}}_{t}$:$\overline{\mathrm{J}/I_{t}}arrow\triangle_{t}\sim$is asmooth family, i.e. all fibers of$\overline{\pi}_{t}$equations are compatible with the transition functions of .
is normal,
Stein Factorization Theorem implies that sincewell known, anormal curve is smooth.
$L$$6\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{s}\mathrm{e}$$\overline{M_{t}}$$7\mathrm{T}\mathrm{h}\mathrm{e}$are smooth.

$\tilde{\Delta}_{t}$is also normal. As is

26
Claim

is an open disk.

$\mathrm{B}\triangle_{t}\sim$

has only
Assuming Claims Aand for amoment, we will verify that :. First, we note the following.
one singular fiber, and it is of the form$\mathrm{B}$$\pi_{t}$$\Lambda f_{t}arrow\triangle_{t}$$m\ominus_{t}$is a cyclic $m$ covering whereareLemma 3.2.1 Suppose thatdisks. Then the covering transformation group fixes exactly one point inopen
possibly after coordinate change.
, and is given by the map$\triangle\sim and$

$p:\triangle\simarrow\triangle$$\triangle$$unit^{8}$$z\mapsto z^{m}$$\triangle\sim$$p$

Proof. Let 7:

be agenerator of the covering transformation group. Then
, which is isomorphic to the fractional linear transformation
is an element of, the transformationof the unit disk (Poincare’ disk). Fromgroup
is an elliptic element. Thus it fixes exactly one point in , and is of the form
is given by.
possibly after coordinate change. Thus$\triangle\simarrow\triangle\sim$$\gamma$$\mathrm{A}\mathrm{u}\mathrm{t}(\triangle)\sim$$\gamma^{m}=1$$\mathrm{P}\mathrm{S}\mathrm{L}_{2}(\mathbb{R})$$\triangle\sim$$\gamma$$\gamma$$z\mapsto e^{2\pi\cdot/m}.z$$p:\triangle\simarrow\triangle$$z\mapsto z^{m}$$\square$:Now we complete the proof of the theorem. By Claim ,isbe agenerator of the covering transformation group ofasmooth family. Let, the transformation. By the construction of the Stein factorization ofdetermines agenerator of the covering transformation group ofsuchthat the following diagram commutes.
$\mathrm{A}$$\overline{M_{t}}arrow\triangle_{t}-$$\tilde{\pi}_{t}$$\tilde{\gamma}_{t}$$M_{t}arrow M_{t}$$\pi_{t}\mathrm{o}p_{t}$$\triangle_{t}\simarrow\triangle_{t}$$\tilde{\gamma}_{t}$$\gamma_{t}$(3.2.1)$\tilde{\pi}_{t\downarrow}\overline{M_{t}}arrow\overline{M_{t}}\tilde{\gamma}$$\{$ $\tilde{\pi}_{l}$$\triangle_{t^{arrow\triangle}t}\sim\gamma c\sim$.

:Namely, the pairaction on, andgenerates an equivariant::
by this action. Recall thatis the quotient ofisis also adisk. Applying Lemma 3.2.1 to the cyclic
adisk, while by Claim ,$m$ covering, we see that fixes exactly one point, say on . From thecommutativity of the diagram (3.2.1), we have$\mathbb{Z}_{m}$$(\tilde{\gamma}_{t},\gamma_{t})$$\pi_{t}$$M_{t}arrow\triangle_{t}$$\tilde{\pi}_{t}$$\mathrm{B}$$\tilde{\pi}_{t}$$\overline{M_{t}}arrow\triangle_{t}\sim$$M_{t}arrow\triangle_{t}\sim$$\triangle_{t}$$\triangle_{t}\sim$$\triangle_{t}\simarrow\triangle_{t}$$\triangle_{t}\sim$$\tilde{x}_{t}$$\gamma_{t}$Lemma 3.2.2 The action on
stabilizes precisely one fiber
except this fiber this action cyclically permutes the $m$ fibers in each orbit.
$\overline{M_{t}}$$\ominus_{t}:=\sim\tilde{\pi}_{t}^{-1}(\tilde{x}_{t})$$\tilde{\gamma}_{t}$and$\ominus_{t}\subset\overline{M_{t}}arrow\ominus_{t}\subset\overline{M_{t}}\sim\tilde{\gamma}\sim\tilde{\pi}_{\mathrm{t}}\downarrow$$\{$ $\tilde{\pi}_{l}$$\tilde{x}_{t}\in\triangle_{t}arrow\tilde{x}_{t}\in\triangle_{t}\sim\gamma\iota\sim$:Asequivariant$\pi_{t}$$8\mathrm{T}\mathrm{h}\mathrm{i}\mathrm{s}$.

is the quotient of the smooth family-action, it follows from Lemma 3.2.2 that :$\Lambda f_{t}arrow\triangle_{t}$$\mathbb{Z}_{m}$$\tilde{\pi}_{t}$$\pi_{t}$:by thehas aunique$\overline{M_{t}}arrow\triangle_{t}\sim$$M_{t}arrow\triangle_{t}$is not restrictive at all; any open disk is biholomorphic to the unit one

27. This fiber is a multiple of a smooth curve,
singular fiber over the point-action onisis unramified cyclic, so in particular, thebecause
,
where
is
the
image
unramified cyclic action. Namely, the singular fiber is
$m$
of
the
order
-actiontheequalsmultiplicity(themap
quotient
under theofdiffeomorphic to 0. In fact, the restriction of toon ). Finally, we claim thatof discriminant of . (Note thatis asmooth family over the reduced partis adisk. See Remark 3.3.3 below.) By Ehresmann’s Theorem, any fiber. Thus, assuming Claims Aand , we proved Theorem
is diffeomorphic to2.0.2 and so it remains to demonstrate these claims.
$x_{t}:=\overline{p}_{t}(\tilde{x}_{t})$$\tilde{\Theta}_{t}$$\overline{l\mathcal{V}I_{t}}arrow M_{t}$$\mathbb{Z}_{m}$$m\ominus_{t}$$\ominus_{t}$$\tilde{\gamma}_{t}$$\overline{\ominus}_{t}$$\overline{\ominus}_{t}$$\Psi$$\ominus_{t}$$\Psi$$\mathrm{D}_{\mathrm{r}\mathrm{e}\mathrm{d}}$$\bigcup_{t}\Theta_{t}$$\ominus_{t}$$\mathrm{D}_{\mathrm{r}\mathrm{e}\mathrm{d}}$$\ominus_{0}=0$$\mathrm{B}$Proof of Claim A

3.3

is asmooth family, i.e. any fiber of
We will show thatcrucial step in the proof of the theorem.

is smooth. This is a$\tilde{\acute{J}\mathrm{T}}t$$\tilde{\pi}_{t}$Step 1. Preparation, and set $x_{i}:=\pi_{t}(X_{i})$ . We
be the singular fibers of :, ,Letneed to introduce notation associated to the basic diagram:$X_{1},X_{2}$$X_{\mathrm{d}}$$M_{t}arrow\triangle_{t}$$\pi_{t}$$\ldots$(3.3.1)$\overline{M}$$arrow Mp_{l}$
$t$

$t$

$\pi_{t}$$\overline{\pi}_{\underline{t\downarrow}}\triangle$$tarrow\triangle\overline{p}_{\mathrm{t}}\downarrow$$t$

We set:(so
have, and let be the ramification:). Since the covering degree of$\overline{p_{t}}(1x_{i}):=\{\overline{x}_{i}^{(1)},\overline{x}_{i}^{(2)}, \cdots,\tilde{x}_{i}^{(N.)}.\}$$z\mapsto z^{r:}$$\overline{p}_{t}$around

(3.3.2)$\tilde{x}_{i}^{(j)}$of$m$is , we$\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{x}^{9}$$r_{i}$$\triangle_{t}\simarrow\triangle_{t}$$\overline{p}_{t}$$\#(\overline{p}_{t}^{-1}(x_{i}))=r_{i}N_{i}$$m=r_{i}\cdot$$\tilde{x}_{i}^{(j)}$.

i not amultiple
is apositive integer and, whereWe writeofdivisor, i.e.
} $=1$ . (Note that does not depend on ,{is asingular fiberis acyclic covering.) Next, recalling that
because :is not a, where is apositive integer and, we write:of
multiple divisor. Notice that$\tilde{\mathrm{Y}}_{i}^{(j)}$$\tilde{X}_{i}^{(j)}=\tilde{a}.\cdot\tilde{\mathrm{Y}}_{i}^{(j)}$$\mathrm{s}$$\tilde{a}_{i}$$\tilde{\mathrm{Y}}_{i}^{(j)}$$j$$\overline{a}_{i}$$\mathrm{c}\mathrm{o}\mathrm{e}\mathrm{f}\mathrm{f}\mathrm{i}\mathrm{c}\mathrm{i}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{s}$$\mathrm{g}\mathrm{c}\mathrm{d}$$\overline{p}_{t}$$X_{i}$$\triangle_{t}arrow\triangle_{t}$$\mathrm{Y}_{i}$$X_{i}=a_{i}\mathrm{Y}.\cdot$$M_{t}arrow\triangle_{t}$$\pi_{t}$(3.3.3)$a_{i}$$(\overline{p}_{tt}0^{=_{\iota}}’)^{-1}(x_{i})=r_{i}\overline{a}_{i}\tilde{\mathrm{Y}}_{i}^{(j)}$,of. As is unramified, theis the ramification index of atover the point is amultiple fiber of multiplicity . Thus from:the commutativity of the diagram (3.3.1), together with (3.3.3), we havewhere$\mathrm{f}\mathrm{f}\mathrm{i}\mathrm{b}\mathrm{e}\mathrm{r}^{10}$$\overline{x}_{i}^{\langle j)}$$p_{t}$$\overline{p}_{t}$$r_{i}$$\overline{M_{t}}arrow\triangle_{t}$$a_{i}$$x_{i}$$\pi_{t}\mathrm{o}p_{t}$(3.3.4)$a_{i}=r_{i}\overline{a}_{i}$.

We notice$9r_{i}$is acyclic covering.
:
does not depend on , because
connected components
areconnected;
there
is notfiber$10\mathrm{T}\mathrm{h}\mathrm{e}$$j$$(\pi_{t}0\overline{p}_{t})^{-1}(x:)$$\tilde{\Delta}_{t}\neg\Delta_{t}$$\overline{p}_{t}$$N_{j}$

28
Lemma 3.3.1

Proof. Indeed,$\mathrm{m}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT} \mathrm{i}_{\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}}N_{i^{Q}i}$.

$m\tilde{a}\dot{.}=r:N_{i}\tilde{a}.\cdot=a_{i}N.\cdot$, where the first and second equalities follows

from (3.3.2) and (3.3.4) respectively.

$\square$Next, we note that if there is asingular fiber of , then it is afiber over some
. In fact, if is asingular fiber of , then the image
is asingular fiber of. Therefore, to prove Claim , it is enough to demonstrate that for any
, the
fiber
is smooth.
$\tilde{\pi}_{t}$$\tilde{X}$$\tilde{x}_{i}^{(j)}$$p_{t}(\tilde{X})$$\tilde{\pi}_{t}$$\tilde{x}_{i}^{(j)}$$\mathrm{A}$$\pi_{t}$$\tilde{X}^{(j)}\dot{.}=\tilde{\pi}_{t}^{-1}(x_{i}^{j)})\dashv$

Step 2. All$\tilde{X}^{(j)}.\cdot$

are smooth

Now we shall show that all
are smooth. Although the proof is involved, thepartessentialof the idea is to relate the singular fibers ofand the singularfiber of. Namely, using the diagramll$\tilde{X}^{(j)}.\cdot$$\pi_{t}\mathrm{o}p_{t}$$\pi_{0}\mathrm{o}p_{0}$$\overline{\mathcal{M}}arrow \mathcal{M}\rho$$arrow\triangle\cross\triangle^{\mathrm{t}}\Psi$,

we relates the singular fibers of the following two diagrams (diagram) by taking the limit $tarrow 0$ :$\overline{M_{t}}p_{C}arrow M_{t}arrow\triangle_{t}\pi_{C}$and’$|‘ \mathrm{e}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{d}\mathrm{d}\mathrm{e}\mathrm{d}$$\overline{M}_{0}parrow M_{0}0\pi_{t}arrow\triangle 0$in the above.

Step 2.1 We consider the discriminantof ;it is acomplex subspace(plane curve) of
through $(0, 0)$ , and defined by the locus where the rank of
is not maximal. Roughly, is {: $\Psi^{-1}(s,t)$ is singular}, but
possibly non-reduced. For our discussion, we rather use the reduced partof D.
By the Weierstrass Preparation Theorem, the reduced plane curveis definedby aWeierstrass polynomial$\mathrm{D}\subset\triangle\cross\triangle^{\mathrm{t}}$$\Psi$$\triangle\cross\triangle^{\mathrm{t}}$$d\Psi$$(s,t)\in\triangle\cross\triangle^{\mathrm{t}}$$\mathrm{D}$$\mathrm{D}_{\mathrm{r}\mathrm{e}\mathrm{d}}$$\mathrm{D}_{\mathrm{r}\mathrm{e}\mathrm{d}}$,(3.3.5)$s^{\mathrm{n}}+c_{\mathrm{n}-1}(t)s^{\mathrm{n}-1}+c_{\mathrm{n}-2}(t)s^{\mathrm{n}-2}+\cdots+c_{0}(t)=0$

where
is aholomorphic function with
. By the definition of the reduced part, this equation contains no multiple root, in other words, the discriminantof the above Weierstrass polynomial does not vanish identically (but possibly
vanishes for some ). Now we claim that
, where is the number of the singular:fibers in. Indeed, when $t=0$ , (3.3.5) is, which clearly has
amultiple root, so $\mathrm{A}(0)=0$ . Since zeroes of the holomorphic function
areisolated,does not vanish for sufficiently small $t(t\neq 0)$ . Consequently, (3.3.5)
has distinct roots, and so has precisely singular fibers, implying that.
This verifies the claim, and we have$\mathrm{q}.(0)=0$$c.\cdot(t)$$\Delta(t)$$\mathrm{n}=\mathrm{d}$$t$$\pi_{t}$$\mathrm{d}$$M_{t}arrow\triangle_{t}$$s^{\mathrm{n}}=0$$\mathrm{A}(2)$$\Delta(t)$$\mathrm{n}$(3.3.6)$\mathrm{n}=\mathrm{d}$$\mathrm{n}$$\pi_{t}$$\mathrm{D}_{\mathrm{r}\mathrm{e}\mathrm{d}}=\{s^{\mathrm{d}}+\mathrm{c}\mathrm{d}-\mathrm{i}(\mathrm{i})s^{\mathrm{d}-1}+c_{\mathrm{d}-2}(t)s^{\mathrm{d}-2}+\cdots+c_{0}(t)=0\}$.

llWe do not use the Stein factorization of the map, but it is worth while pointing outthat it factors through anormal surface , which possibly has asingularity. In contrast, the Steinfactorization for the map with aone-dimensional base factors through asmooth curve$\Psi$$S$$\circ\rho$29
Next, we define aramified -fold$\mathrm{d}$$\phi$:$\mathrm{D}_{\mathrm{r}\mathrm{e}\mathrm{d}}arrow\triangle^{\mathrm{t}}$distinct points
amultiple pointby $(s, t)\mapsto t$ . Then

for $t\neq 0$
for $t=0$ .

$\mathrm{d}$$\phi^{-1}(t)=\{$$s^{\mathrm{d}}=0$, we consider theand.
3-manifold
For the remainder
complexinthehypersurface
instead ofof the proof, to emphasize the parameter , we use ‘precise’ notation
etc. Notice thatT\^o relate the singular fibers of

Step 2.2$\pi_{t}\mathrm{o}p_{t}$$\pi_{0}\underline{\mathrm{o}p}_{0}$$’\tilde{H}:=(\Psi 0\rho)^{-1}(\mathrm{D}_{\mathrm{r}\mathrm{e}\mathrm{d}})$$\mathcal{M}$$\tilde{X}_{i,t}^{(j)}$$t$$\tilde{X}_{i}^{(j)}$(3.3.7)$H$ $\cap\overline{M_{t}}=\{$

the disjoint union of all$\tilde{X}_{t}^{(j)}.\cdot$,$\mathrm{d}m\ominus\sim$

for $t\neq 0$
for $t=0$ ,is unramified
as follows. Since
where we can see, hence the fiber ofover(locally biholomorphic), we
.
, so
is
the multiple pointis smooth for all
By the first equation of (3.3.7), our goal is to show that$t\neq 0$ . To demonstrate this, fixing an arbitrary point, we take alocally defines O.
around in , such that
local
. For later discussion,be adefining equation of around inLet$f(z_{1},z_{2},t)$. By the first equation of (3.3.7),
instead of
we use the notation, so we can write(disjoint union) and$\gamma\{\cap\overline{M}_{0}=\mathrm{d}m\ominus\sim$$\pi_{0}^{-1}(0)=m\ominus \mathrm{a}\mathrm{n}\mathrm{d}$$p_{0}$$\mathrm{h}\mathrm{a}\mathrm{v}\underline{\mathrm{e}}(\pi_{0}\mathrm{o}p_{0})^{-1}\underline{(}0)=m\underline{\ominus\sim}$$s^{\mathrm{d}}=0$$\pi_{0}\mathrm{o}p_{0}$$??\cap M_{0}=\mathrm{d}m\Theta$$\mathrm{d}m\ominus$$H\cap\overline{NI_{t}}$$y\in\tilde{\Theta}(=p_{0}^{-1}(\ominus))$$\overline{M}$$\mathrm{c}\mathrm{o}\mathrm{o}\mathrm{r}\mathrm{d}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{t}\mathrm{e}^{12}(z_{1}, z_{2},t)$$z_{1}=0\underline{t},=0$$y$$\mathrm{H}-$$\mathrm{f}(\mathrm{z}\mathrm{i}, z_{2}, t)=0$$y$$\mathcal{M}$$f_{t}(z_{1}, z_{2})$$\overline{X}_{i,t}^{(j)}=a_{i}\overline{\mathrm{Y}}_{i,t}^{(j)}$$\tilde{\mathcal{H}}\cap M_{t}=\Pi_{i=1}^{\mathrm{d}}(\Pi_{j=1}^{N_{*}}.\tilde{X}_{i,t}^{(j)})$(3.3.8)andhence$f_{t}= \prod_{i=1}^{\mathrm{d}}f_{j}^{a},i$$g_{i,t}^{(j)}=0$

defineswhere,$f_{t}.

\cdot,=\prod_{j=1}^{N}.g_{i,t}^{(j)}$,locally. By the second equation of (3.3.7),
in (3.3.8), we have$\overline{\mathrm{Y}}_{i,t}^{(j)}$$\mathrm{s}\mathrm{e}\mathrm{t}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{g}^{13}t=0$(3.3.9)$z_{1}^{\mathrm{d}m}=f_{0}= \prod_{i=1}^{\mathrm{d}}f_{i,\dot{0}}^{a}$$f_{0}(z_{1}, z_{2})=z_{1}^{\mathrm{d}m}$,,

is apositive integer,, whereand so we may express
is anon-vanishing holomorphic function. By the comparison of the degrees
andof in (3.3.9), we have$g_{i,0}^{(j)}(z_{1}, z_{2})i.u_{i}^{(j)}(z_{1}, z_{2}) \oint_{=Z_{1}}\mathrm{j})$$d_{i}^{(j)}$$u_{i}^{(j)}$$z_{1}$(3.3.10)$\mathrm{d}m=\sum_{i=1}^{\mathrm{d}}a_{i}(d_{i}^{(1)}+d_{i}^{(2)}+\cdots+d_{i}^{(N.)}.)$.

Now we show the key lemma.
is asubmersion. Since is:the definition of deformations,also asubmersion. By the Implicit Function Theorem, we:unramified,It acoordinate of .
may ‘lift’
such that 1is homeomorphically mapped to apath
along apath intake the limit$(s,t)\mapsto
t$ . For example, in $s^{2}-t^{3}=(s-t^{3/2})(s+t^{3/2})$ ,
\
t
r
i
a
n
g
l
e
\
d
a
g
e
r
,
covering
under the ramifiedin,
we must choose apath on which each factor,$t-0$
takingso
two factors are multi-valued on
is single-valued$\mathrm{p}\mathrm{r}_{2}\circ\Psi$$12\mathrm{B}\mathrm{y}$$0\rho$$\mathrm{p}\mathrm{r}_{2}\mathrm{o}$$\mathcal{M}$$\mathcal{M}-\Delta \mathrm{x}$$\Delta^{\mathrm{t}}\wedge$$\Delta^{\mathrm{t}}$$\rho$$arrow\Delta^{\uparrow}\underline{\mathrm{i}\mathrm{s}}$$t\in\Delta^{\uparrow}$$\mathcal{M}$$\mathrm{o}$$13\mathrm{Y}\mathrm{V}\mathrm{e}$$l$$tarrow \mathrm{O}$$\mathrm{D}_{\mathrm{r}\mathrm{e}\mathrm{d}}$$\mathrm{D}_{\mathrm{r}\mathrm{e}\mathrm{d}}arrow\Delta^{\uparrow}$$l$$\mathrm{D}_{\mathrm{r}\mathrm{e}\mathrm{d}}$30Lemma 3.3.2$d_{\ovalbox{\tt\small REJECT}}^{(1)}\ovalbox{\tt\small REJECT}$$d\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}^{2)}\ovalbox{\tt\small REJECT}$$d\mathrm{r}_{\ovalbox{\tt\small REJECT}}^{(\mathrm{A}^{r_{\ovalbox{\tt\small REJECT}}})}\ovalbox{\tt\small REJECT}$\cdots$\ovalbox{\tt\small REJECT}_{\ovalbox{\tt\small REJECT}}\ovalbox{\tt\small REJECT}$$\ovalbox{\tt\small REJECT}$

for

1

i

$\ovalbox{\tt\small REJECT} 1\ovalbox{\tt\small REJECT} 2\ovalbox{\tt\small REJECT}$\ldots$\ovalbox{\tt\small REJECT}$d.

Proof. First, we noteby (3.3.10)$\mathrm{d}\uparrow n=\sum_{i=1}^{\mathrm{d}}a_{i}(d_{i}^{(1)}+d_{i}^{(2)}+\cdots+d_{i}^{(N.)})$(3.3.11)by$\geq\sum_{i=1}^{\mathrm{d}}a_{i}N_{i}$$d_{i}^{(1)}$,$d_{i}^{(2)}$,$\ldots$,$d_{i}^{(N:)}\geq 1$

by Lemma 3.3.1.

$= \sum_{i=1}^{\mathrm{d}}\tilde{a}.\cdot m$, which implies that
Thus we have
, andthis inequality is an equality. In particular, (3.3.11) is also an equality, and so. This complete the proof.
$\mathrm{d}m\geq\sum_{i=1}^{\mathrm{d}}\tilde{a}_{i}m$$\tilde{a}_{1}=\tilde{a}_{2}=\cdots=\tilde{a}_{\mathrm{d}}=1$$d^{(1)}\dot{.}=d^{(2)}.\cdot=\cdots=d^{(N_{})}.\cdot=1$$\square$Now it is immediate to complete the proof of Claim A. From
, we have. On the other hand, fromsmooth,becauseitis locally,is. Thus for sufficiently small
defined by,is smooth, and so,is smooth. This completes the proof of Claim A.
$\tilde{a}_{i}=1$$\tilde{X}_{i,t}^{(j)}=\mathrm{Y}_{i,t}^{(j)}L$$z_{1}\cdot$$\tilde{d_{i}}=1,\tilde{\mathrm{Y}}_{0}^{(j)}\dot{.}$$u^{(j)}.\cdot(z_{1}, z_{2})=0$$t,\tilde{\mathrm{Y}}_{t}^{(j)}.\cdot$$\tilde{X}_{i,t}^{(j)}=\tilde{\mathrm{Y}}_{t}^{(j)}.\cdot$

Remark 3.3.3 If, i.e.
:(see (3.3.6)) is adisk in$\mathrm{d}=1$$\mathrm{D}_{\mathrm{r}\mathrm{e}\mathrm{d}}=\{s+c_{0}(t)=0\}$$\pi_{t}$Proof of Claim

3.4

We shall show Claimhas only one singular fiber, then$M_{t}arrow\triangle_{t}$$\triangle\cross\triangle^{\mathrm{t}}$.

$\mathrm{B}$which asserts thatis adisk.
below is based on,
atopological argument, and by shrinking
and , we regard them withclosed manifolds with boundary. We first take:and:which make the following diagram commute:$\triangle_{t}\sim$$\mathrm{B}$$\mathrm{T}\mathrm{h}\mathrm{e}_{\sim}\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{o}\mathrm{f}$$M_{t},\overline{M_{t}}$$\triangle_{t}$$\triangle_{t}$$\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}\mathrm{e}\mathrm{o}\mathrm{m}\mathrm{o}\mathrm{r}\mathrm{p}\mathrm{h}\mathrm{i}\mathrm{s}\mathrm{m}\mathrm{s}^{14}\phi_{t}$$\overline{\phi}_{t}$$M_{0}arrow If_{t}$$\partial\triangle 0arrow\partial\triangle_{t}$$\pi 0\downarrow\downarrow\partial M_{0^{arrow\partial M_{t}}}^{\phi_{t}}\partial\triangle_{0^{arrow\partial\triangle}}^{\overline{\phi}_{t}}$$\pi_{t}$$t$.

(Namely, the restriction
to the boundary
is fiber-preserving.) Recall$M_{t}arrow
M_{t}$
that we constructed:: $M_{0}arrow M_{0}$ via the diffeomorphism
from: $M_{0}arrow M_{t}$ . Hence there is anatural diffeomorphism:, which(that is,is alifting of), and the restriction oftois$-^{\partial M_{0}}$$\underline{\mathrm{o}\mathrm{f}}\overline{\phi}_{t}$$p_{t}$$p_{0}$$\phi_{t}$$\Phi_{t}$$\Phi_{t}\mathrm{o}p_{t}=p_{0}\mathrm{o}\phi_{t}$$\phi_{t}$$14\mathrm{F}\mathrm{o}\mathrm{r}$the existence of$\phi_{t}$, see Lemma 3.5.1 in \S 3.5 Supplement below$\overline{M}_{0}arrow\overline{\mathrm{J}f_{t}}$$\partial\overline{\mathrm{J}f}_{0}$$\Phi_{t}$31fiber-preserving, i.e. the following diagram commutes$\partial\overline{\mathrm{J}/I}_{0}\partial\overline{M_{t}}\underline{\Phi_{t}}$$\{$ $\tilde{\pi}\iota$$\tilde{\pi}_{0}\{$$\partial\triangle\sim$$0^{arrow\partial\triangle}\overline{\Phi}_{t}\sim$
$t$

where
and letover$\overline{\Phi}_{t}$$\iota_{0}$,, where,is adiffeomorphism. Now we fix afiber
is afiber ofbe the natural embedding. Then:be the natural embedding.
, and let :$y_{0}\in\partial\triangle 0\sim$$C_{0}:=\tilde{\pi}_{0}^{-1}(y_{0})$$C_{t}:=\Phi_{t}(C_{0})$$C_{0}rightarrow M_{0}$$\tilde{\pi}_{t}$$C_{t}arrow\overline{M_{t}}$$y_{t}:=\overline{\Phi}_{t}(y_{0})\in\partial\triangle_{t}\sim$$\iota_{t}$$c_{\underline{1}_{\underline{\overline{\Phi}_{t}}}\downarrow}y_{0}\in_{\partial\triangle_{t}y_{t}\in\partial\triangle}0\subset\partial\overline{NI}_{0^{arrow}}^{\Phi_{t}}C_{t}\subset\partial\overline{M_{t}}\tilde{\pi}_{0}$$\overline{\pi}\iota$-t$\cdot$isis adisk. Note that
After this preparation, we can demonstrate that).
toisomorphic(whichisboundary
aconnected15
areal compact surface withis homotopically equivalent to the bouquet
is , thenThus if the genus ofof $2g$ circles, and so$\triangle_{t}-$$\triangle_{t}-$$S^{1}$$\triangle_{t}-$$\triangle_{t}\sim$$g$$S^{1}\vee S^{1}\vee\cdots\vee S^{1}$

the free group of rank $2g$ .

$\pi_{1}(\triangle_{t})-$. For this, we first take the homotopy exact
Hence it suffices to show that.
:sequence associated to the differentiable fiber$\pi_{1}(\triangle_{t})\sim=1$$\mathrm{b}\mathrm{u}\mathrm{n}\mathrm{d}1\mathrm{e}^{16}\overline{\pi}_{0}$(3.4.1)$\overline{M}_{0}arrow\triangle 0-$$\pi_{2}(\triangle_{0})\simarrow\pi_{1}(C_{0})arrow\pi_{1}(\overline{M}_{0})\underline{\iota}_{0*}arrow\pi_{1}(\triangle_{0})-arrow 1$

is adifferentiable fiber bundle, so we:
Next, noting that from Claim ,may take the homotopy exact sequence associated to it.
$\overline{\mathrm{J}/I_{t}}arrow\triangle_{t}\sim$$\mathrm{A}$(3.4.2)$\overline{\pi}_{t}$$\pi_{2}(\triangle_{t})-arrow\pi_{1}(C_{t})arrow\pi_{1}(\overline{M_{t}})\iota_{t*}arrow\pi_{1}(\triangle_{t})\simarrow 1$

The following commutative diagram relates (3.4.1) and (3.4.2):(3.4.3)$\pi_{2}(\triangle_{0})\simarrow\pi_{1}(C_{0})\sim\underline{\iota_{0*}}\pi_{1}(\overline{M}_{0})arrow\pi_{1}(\triangle_{0})\simarrow 1$

$\{$

$\{$

$\pi_{2}(\triangle_{t})\simarrow\pi_{1}($ $C_{t})arrow\pi_{1}(\iota_{t*}$ $\overline{M_{t}})arrow\pi_{1}(\triangle_{t})\simarrow 1$,

is adisk, we havewhere the vertical arrows are induced by . Since
is an isomorphism. Two vertical arrows are also isomorphisms,, and sobecause they are induced by the diffeomorphism . From the commutativity of theis an isomorphism. Then the exactness of (3.4.1)
diagram (3.4.3), we see thatis adisk.
and soimplies that$\pi_{1}(\triangle 0)=-$$\triangle 0\sim$$\Phi_{t}$$\pi_{2}(\triangle_{0})-=1$$\iota_{0*}$$\Phi_{t}$$\iota_{t*}$$\pi_{1}(\triangle_{t})=1\sim$$\triangle_{t}\sim$is connected, and so
is connected.
, the boundary
the construction of
categorydifferentiableinbundlethefamilyisafiberEhresmann’s Theorem, asmooth$\overline{M_{t}}$$15\mathrm{B}\mathrm{y}$$16\mathrm{B}\mathrm{y}$$\partial\overline{M_{t}}$$\partial\tilde{\Delta}_{t}$32Supplement: Construction of diffeon orphisms

3.5\triangle \dag er i adeformation of\rightarrow \triangle:
Suppose that:
. Note that the:
restriction
is afiber. The following lemma may beknown to the geometers, but for the convenience of the reader, we include the proof.
(Hereafter, for consistency, we denote:
instead of :)$\Psi$$\mathcal{M}$$\cross$$\pi$$\mathrm{s}$$\partial M_{t}arrow\partial\triangle_{t}$$\pi_{t}|_{\partial M_{t}}$$Marrow\triangle$$\mathrm{b}\mathrm{u}\mathrm{n}\mathrm{d}1\mathrm{e}^{17}$$\pi_{0}$$M_{0}arrow\triangle 0$$Marrow\triangle$$\pi$Lemma 3.5.1 There exists a diffeomorphism:
such that the restriction
preserves fibers, that is, there exists a diffeomorphism:
which makes the following diagram commute:$\Lambda f_{0}arrow M_{t}$$\phi_{t}$$\phi_{t}|_{\partial M_{0}}$$\partial\triangle 0arrow\triangle_{t}$$\overline{\phi}_{t}$$\pi 0\downarrow_{\overline{\phi}_{t}}\downarrow\partial M_{0^{arrow\partial M_{t}}}^{\phi_{t}}\partial\triangle_{0}arrow\partial\triangle$$\pi_{t}$$t$.

Warning: Although the restriction of
to the boundarymaps
and , this is not case for itself.
$\phi_{t}$$\pi_{0}$$\partial M_{0}$commutes with$\phi_{t}$$\pi_{t}$Proof. For simplicity, we assume thatis the unit disk. We choose ,that $0<r_{2}<r_{1}<1$ , and define an open covering, where$\triangle$$r_{1}$$r_{2}\in \mathbb{R}$so$\triangle\cross\triangle^{\mathrm{t}}=U_{\mathrm{i}\mathrm{n}}\cup U_{\mathrm{o}\mathrm{u}\mathrm{t}}$$U_{\mathrm{i}\mathrm{n}}:=\{(s,t)\in\triangle\cross\triangle^{\mathrm{t}} :,|s|<r_{1}\}$$U_{\mathrm{o}\mathrm{u}\mathrm{t}}:=\{(s, t)\in\triangle\cross\triangle\dagger :|s|>r_{2}\}$.

We then take an open covering, whereand. Taking sufficiently close to 1, we assume that
contains
no singular fiber, i.e. the restriction
is afiber bundle. In particular,
is asubmersion. Hence there exists avector fieldonsuch that$\mathcal{M}$$\mathcal{M}_{\mathrm{o}\mathrm{u}\mathrm{t}}:=\Psi^{-1}(U_{\mathrm{o}\mathrm{u}\mathrm{t}})$$=\mathcal{M}_{\mathrm{i}\mathrm{n}}\cup \mathcal{M}_{\mathrm{o}\mathrm{u}\mathrm{t}}$$\mathcal{M}_{\mathrm{i}\mathrm{n}}:=\Psi^{-1}(U_{\mathrm{i}\mathrm{n}})$$r_{1}$$\mathcal{M}_{\mathrm{o}\mathrm{u}\mathrm{t}}$$\Psi_{\mathrm{o}\mathrm{u}\mathrm{t}}:=\Psi|_{\mathcal{M}_{\mathrm{o}\mathrm{u}\mathrm{t}}}$$\Psi_{\mathrm{o}\mathrm{u}\mathrm{t}}$$\mathcal{M}_{\mathrm{o}\mathrm{u}\mathrm{t}}$$v_{\mathrm{o}\mathrm{u}\mathrm{t}}$(3.5.1)$d \Psi_{\mathrm{o}\mathrm{u}\mathrm{t}}(v_{\mathrm{o}\mathrm{u}\mathrm{t}})=\frac{\partial}{\partial t}$

Similarly, we setmap:
and so a$\mathrm{p}\mathrm{r}_{2}\mathrm{o}\Psi_{\mathrm{i}\mathrm{n}}$$\Psi_{\mathrm{o}\mathrm{u}\mathrm{t}}:=\Psi|_{\mathcal{M}_{\mathrm{o}\mathrm{u}t}}$$\mathcal{M}_{\mathrm{o}\mathrm{u}\mathrm{t}}arrow\triangle^{\mathrm{t}}$$\mathrm{s}\mathrm{u}\mathrm{b}\mathrm{m}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}^{18}$.

. By the definition of deformations, the composite

i afiber bundle with smooth complex surfaces as fibers,onsuch that$\mathrm{s}$. Thus there exists avector field(3.5.2)$d( \mathrm{p}\mathrm{r}_{2}\mathrm{o}\Psi_{\mathrm{i}\mathrm{n}})(v_{\mathrm{i}\mathrm{n}})=\frac{\partial}{\partial t}$$\mathcal{M}_{\mathrm{i}\mathrm{n}}$$v_{\mathrm{i}\mathrm{n}}$.

Notice that in (3.5.1),
is avector field on, while in (3.5.2), it is avector
field on. We shall ‘patch’ two vector fields
andby apartition of unity,and define avector field on;we first define open subsets(resp.
as follows. Take ,satisfying $0<r_{1}’<r_{2}<r_{1}<r_{2}’<1$ , andset$\triangle\cross\triangle^{\mathrm{t}}$$\partial\overline{t}\partial$$\triangle^{\mathrm{t}}$$v_{\mathrm{i}\mathrm{n}}$$U_{\mathrm{o}\mathrm{u}\mathrm{t}}’\subset U_{\mathrm{o}\mathrm{u}\mathrm{t}}^{1})$$U_{\mathrm{i}\mathrm{n}}’\subset U_{\mathrm{i}\mathrm{n}}$$r_{1}’$$U_{\mathrm{i}\mathrm{n}}’:=\{(s,t)\in\triangle\cross\triangle\dagger :$17\mathrm{I}\mathrm{n}$$v_{\mathrm{o}\mathrm{u}\mathrm{t}}$$\mathcal{M}$$v$$r_{2}’\in \mathbb{R}$|s|<r_{1}’\}$,$U_{\mathrm{o}\mathrm{u}\mathrm{t}}’:=\{(s,t\rangle\in\triangle\cross\triangle^{\mathrm{t}} :this subsection, by afiber bundle we always mean adifferentiable one.
:asingular fiber, and so it is not afiber bundle.

$18\Psi_{\mathrm{i}\mathrm{n}}$${\rm Min}$$arrow\Delta \mathrm{x}\Delta^{\uparrow}\mathrm{h}\mathrm{s}$.

|s|>r_{2}’\}$

33. Now we put
and.
. Using apartition of unity, we can construct avector field

Notice that
Then
on Asuch that$\mathcal{M}_{\mathrm{i}\mathrm{n}}’:=\Psi^{-1}(U_{\mathrm{i}\mathrm{n}}’)$$U_{\mathrm{i}\mathrm{n}}’\cap U_{\mathrm{o}\mathrm{u}\mathrm{t}}’=\emptyset$$\mathcal{M}_{\mathrm{o}\mathrm{u}\mathrm{t}}’:=\Psi^{-1}(U_{\mathrm{o}\mathrm{u}\mathrm{t}}’)$$\mathcal{M}_{\mathrm{i}\mathrm{n}}’\cap \mathcal{M}_{\mathrm{o}\mathrm{u}\mathrm{t}}’=\emptyset$$v$$v_{\mathrm{i}\mathrm{n}}$$v=\{$$v_{\mathrm{o}\mathrm{u}\mathrm{t}}$onon$\mathrm{J}1_{\mathrm{i}\mathrm{n}}’$$\mathcal{M}_{\mathrm{o}\mathrm{u}\mathrm{t}}’$Finally, we integrate the vector field on
to obtain aone-parameter family of$M_{0}arrow
M_{t}$:
diffeomorphisms
with the desired property.
$v$$\mathcal{M}$$\phi_{t}$$\square$

4Topological monodromies and singular fibers
Before we proceed to state splitting criteria, we briefly review the relation between
topological monodromies and configurations of singular fibers (see [MM2] andfor details). First, we recall the topological monodromy of adegeneration : $Marrow$. For this purpose, it is convenient to consider $M$ andas manifolds withboundary, so
is the closed unit disk. We write, and. Using apartition of unity, we construct avector field on
set. Then the integration of yields aone-parametersuch that:(see Figure 1). In particular,family of diffeomorphisms
is aselfhomeomorphism of . Setting $h:=h_{2\pi}$ , we refer to as the topological monodromy.
of :$[\mathrm{T}\mathrm{a},\mathrm{I}\mathrm{I}]$$\pi$$\triangle$$\triangle$$\partial\triangle=\{e^{\mathrm{i}\theta} :$\triangle$

0\leq\theta\leq 2\pi\}$$C_{\theta}:=\pi^{-1}(e^{\mathrm{i}\theta})$$\partial M$$v$$d\pi(v)=\partial/\partial\theta$$h_{\theta}$$v$$C_{0}arrow C_{\theta}$$C_{0}$$\pi$$h_{2\pi}$$h$$Marrow\triangle$

Figure 1:
Topological monodromies are very special homeomorphisms; they are either pe,). Recall that ahome0riodic or pseud0-periodic (see [MM2], and also [ES,is isotopic
morphism of acurve is (1) periodic if for some positive integer $m$ ,, on , the reto the identity, and (2) pseudO-periodic if for some loops , ,is periodic. (In [MM2], periodic homeomorphism
striction on$\mathrm{I}\mathrm{m}$$h$$\mathrm{S}\mathrm{T}]$$h^{m}$$C$$l_{\mathrm{I}}$$l_{n}$$l_{2}$$\ldots$$h$$C\backslash \{l_{1}, l_{2}, \ldots, l_{n}\}$$C$$\mathrm{s}$34
are considered to be special cases of pseud0-periodic homeomorphisms by taking. However for our discussion it is convenient to distinguish periodic homeomorphisms with pseud0-periodic ones.) According to whether the top0logical monodromy is periodic or pseud0-periodic, the singular fiber is star-shaped
or non-star shaped. In some sense, anon-star-shaped singular fiber is obtained by‘bonding’ star-shaped ones (see [MM2] and [Ta, ).
$\{l_{1}, l_{2}, \ldots, l_{n}\}=\emptyset$$\mathrm{I}\mathrm{I}]$

Remark 4.0.2 Based on atopological argument, Matsumoto and Montesinos [MM2]showed that the configuration of the singular fiber of adegeneration is completely, we gave an algebr0-geometricdetermined by its topological monodromy. In
proof for their results, and clarified the relation between topological monodromies
and quotient singularities.
$[\mathrm{T}\mathrm{a},\mathrm{I}\mathrm{I}]$Now the followings are the simplest examples for periodic and pseud0-periodic homeomorphisms respectively:
Example 4.0.3 (Periodic) is an unramified periodic homeomorphism, that is,
the quotient map
is aunramified cyclic covering.
$h$$Carrow C/\langle h\rangle$

Example 4.0.4 (PseudO-periodic) is aright Dehn twist along one loop 1on, so the restriction of tois isotopic to the identity.
$h$

$C$

$h$

$C\backslash l$

Adegeneration with the topological monodromy in Example 4.0.3 has asingular, where $m$ is the order of , and 0is asmooth curve which is the quotient offiber
by the action of . On the other hand, the singular fiber of adegeneration with the
topological monodromy in Example 4.0.3 is areduced curve with one node (this node
is obtained by ‘pinching’ 1on ). By Theorem 2.0.2, both of these degenerations
are atomic. Namely, all degenerations with the simplest topological monodromies
are atomic. To the contrary, if the topological monodromy is ‘complicated’, what
can we say about splittability? In this case, the singular fiber is also complicated,so the reader may imagine that they are not atomic (complicated objects should not
be atoms!). In the later half of this paper, we will show that this intuition is true.
$h$$m\ominus$

$C$

$h$

$C$

5Splitting criteria via configurations, I
In this and subsequent sections, we will give splitting criteria of degenerations interms of configurations of their singular fibers. As aconsequence of these criteria, wewill see that many degenerations with non-star-shaped singular fibers always admit
splitting deformations. We point out that these criteria are powerful for determining
atomic degenerations by induction with respect to genus (see \S 6.3 for details).
In the discussion below we often use the realization of $M$ as agraph of ;for a, the graph of is defined bydegeneration :
$g$

$\pi$

$\pi$

$Marrow\triangle$$\pi$Graph(\pi ) $=\{(x,s)\in M\cross\triangle : \pi(x)-s=0\}$ .

Of course, Graph(\pi ) is asmooth hypersurface in, and $M$ is canonically$x\inM\mapsto(x,\pi(x))\inM\cross\triangle$
isomorphic to Graph(Tr) by. Under this isomorphism$M\cross\triangle$

35
IS corresponds to the projection (r, E Graph(zr) $+$ sEb. In
the map 7
the discussion below, we identify Graph(Tr) with M via the canonical isomorphism,
and we write M instead of Graph(Tr).
$l^{\ovalbox{\tt\small REJECT}}lf\ovalbox{\tt\small REJECT}$$\ovalbox{\tt\small REJECT} \mathrm{s})$$\ovalbox{\tt\small REJECT}$

5.1

Criterion in terms of nodes

In this subsection, we shall provide splitting criteria in terms of some singularity on
the singular fiber. We start with adefinition. Consider asingularity$V_{m}:=\{(x, y)\in \mathbb{C}^{2} :x^{m}y^{m}=0\}$,is amultiple node of multiplicity $m$ .
where $m$ is apositive integer. We say that$m\geq2$
Byabuse of terminology, we also say,
is non-reduced.
Note that whenis amultiple node.
that the origin of$:=\{(x, y, s,t)\in \mathbb{C}^{4} : (xy+t)^{m}-s=0\}$ in
We consider ahypersurface
by $(x, y, s, t)\mapsto(s, t)$ . Clearly,:, and define aholomorphic map. Next, we shall$\Psi^{-1}(0,0)=V_{m}$ , and so Iis atw0-parameter deformation ofcompute the discriminant of V. Since$V_{m}$$V_{m}$$V_{m}$$\mathcal{M}$$\Psi$$\mathbb{C}^{4}$$\mathcal{M}$$arrow \mathbb{C}^{2}$$V_{m}$$\frac{\partial\Psi}{\partial x}=mx(xy+t)^{m-1}$,$\frac{\partial\Psi}{\partial y}=my(xy+t)^{m-1}$,

if and only if either (1) $x=y=0$ or (2) $xy+t=0$ .
we have
We note that $t^{m}-s=0$ for (1), and $s=0$ for (2).
$\partial\Psi/\partial x=\partial\Psi/\partial y=0$

Lemma 5.1.1 The discriminant of
To be explicit, for $t\neq 0$ ,(1)$\Psi^{-1}(t^{m}, t)$$\Psi$

consists of curves$s$$=t^{m}$and $s=0$ inis a disjoint union of $m-1$ annuli and a node,(2) $\Psi^{-1}(0,t)$ is a multiple of an annulus of multiplicity $m$ .

}

m-1

–

$+$

$s=0$

$s=t^{m}$Figure 2:

Proof. The fiber$\Psi^{-1}(t^{m},t)(t\neq 0)$

is defined by.

$xy[(xy)^{m-1}+{}_{m}\mathrm{C}_{1}(xy)^{m-2}t+\cdots+{}_{m}\mathrm{C}_{i}(xy)^{m-i-1}t^{i}+\cdot\cdot ‘ +{}_{m}\mathrm{C}_{1}t^{m-1}]=0$$\mathbb{C}^{2}$.

36, where $\alpha:\in \mathbb{C}(i=1,2, \ldots, m-1)$
This equation factorizes as$X^{m-1}+{}_{m}\mathrm{C}_{1}X^{m-2}+\cdots+{}_{m}\mathrm{C}_{i}X^{m-i-1}+\cdots+{}_{m}\mathrm{C}_{1}=0$
are the solutions of. Hence$\Psi^{-1}(t^{m}, t)(t\neq 0)$ is adisjoint union of anode $xy=0$ and $m-1$ annuli $xy+\alpha_{i}=0$$(i=1,2, \ldots, m-1)$ . On the other hand, $\Psi^{-1}(0,t)=\{(xy+t)^{m}=0\}$ is amultiple
annulus of multiplicity .
$xy \prod_{=1}^{m-1}\dot{.}(xy+\alpha.\cdot t)=0$$m$$\square$Now we can show the following.

Criterion 5.1.2 Let :$X$ has a multiple node

be normally minimal such that the singular fiberof multiplicity at least 2. Then there exists a splitting, which splits $X$ into
and , whereis a reduced
deformation of :curve with one node andis obtained from $X$ by replacing the multiple node bya multiple annulus (see Figure 4for example).
$Marrow\triangle$

$\pi$

$p$

$\pi$

$Marrow\triangle$$X_{2}$$X_{1}$$X_{1}$$X_{2}$$p$

Proof. Take an open covering $M=M_{0}\cup M_{1}$ , such that (1)around (hence
is the multiple node), and (2)multiple node (see Figure 3). We take local coordinates$M_{0}\cap X$$p$$M_{0}$$M_{1}\cap X$

is an open ballis ‘outside’ thearound ,$(z\rho, \zeta_{\beta})\in M_{0}$

$p$

$M$

Figure 3:then we have
. ThenAs$\pi(z\rho, \zeta\rho)=z_{\beta}^{m}\zeta_{\beta}^{m}$. Next, we take local coordinates, where, we have$\mathrm{J}\mathrm{r}(z_{\alpha}, \zeta_{\alpha})=\zeta_{\alpha}^{m}f_{\alpha}(z_{\alpha}, (;_{\alpha})$$p$$\pi(z_{\alpha}, \zeta_{\alpha})=\pi(z_{\beta}, \zeta_{\beta})$$f_{\alpha}$near
is anon-vanishing holomorphic function.

$\zeta_{\alpha}^{m}f_{\alpha}(z_{\alpha}, \zeta_{\alpha})=z_{\beta}^{m}\zeta_{\beta}^{m}$$(z_{\alpha}, \zeta_{\alpha})\in M_{1}$.

Note that the holomorphic function
on the right has an $m$ -th root
, whichis asingle-valued function. Thusalso has asingle valued -th root function
such that. Rewriting
by , the gluing map of
andis of the form
$z_{\beta}\zeta_{\beta}$$z_{\beta}^{m}\zeta_{\beta}^{m}$$m$$\zeta_{\alpha}^{m}f_{\alpha}$$\zeta_{\alpha}f_{\alpha}^{1/m}$$\zeta_{\alpha}f_{\alpha}^{1/m}=z_{\beta}\zeta_{\beta}$$\zeta_{\alpha}f_{\alpha}^{1/m}$$M_{0}$$(_{\alpha}$$M_{1}$$z_{\alpha}=\phi_{\alpha\beta}(z\rho,\zeta\rho)$where$\phi_{\alpha\beta}$

is holomorphic.

,$\zeta_{\alpha}=z_{\beta}\zeta_{\beta}$

around$p$,

37
Now we consider asmooth hypersurface$\mathcal{M}_{0}$in$\{(z_{\beta}, \zeta_{\beta}, s,t)\in M_{0}\cross\triangle\cross\triangle^{\mathrm{t}} :We also define asmooth hypersurface

in$\mathcal{M}_{1}$(z\rho\zeta_{\beta}+t)^{m}-s=0\}$$\Psi_{i}$$t\neq 0$:$\mathcal{M}_{i}arrow \mathrm{I}\mathrm{S}$.

by$M_{1}\cross\triangle\cross\triangle^{\mathrm{t}}$$\{(x,s,t)\in M_{1}\cross\triangle\cross\triangle\dagger :Letgiven by$M_{0}\cross\triangle\cross\triangle\dagger$\pi(x)-s=0\}$.

be the natural projection. From Lemma 5.1.1, for$\cross\triangle^{\mathrm{t}}(i=0,1)$,(5.1.1)$\Psi_{0}^{-1}(s,t)=\{$

disjoint union of $m-1$ annuli and a node,
amultiple annulus of multiplicity $m$ ,$s=t^{m}$,$s=0$ .

On the other hand, we have
$X\cap M_{1}$(5.1.2)$\Psi_{1}^{-1}(s, t)=\{$

Now we glue$\mathcal{M}_{0}$

with

,

smooth,$s=0$ ,

otherwise.

by$\mathcal{M}_{1}$$(_{\alpha}=z_{\beta}\zeta_{\beta}+t$$z_{\alpha}=\phi_{\alpha\beta}(z_{\beta}, (_{\beta}),$Note that this map transforms the defining equation of. Letting :Then we obtain acomplex 3-manifold
projection, we consider two fibers:$\Psi$$\mathcal{M}$$X_{1}=\Psi^{-1}(t^{m}, t)$,.

$\mathcal{M}_{0}$$\mathcal{M}$$X_{2}=\Psi^{-1^{\mathit{1}}}(0, t)$nearto that of.
be the natural$\mathcal{M}_{1}$$p$$arrow\triangle\cross\triangle^{\mathrm{t}}$.

) From (5.1.1) and (5.1.2),
is areduced(
andare fibers of :is obtained from $X$ by replacing the multiple node
curve with one node, andby amultiple annulus, and no other singular fibers. As both of
and
are
\
t
r
i
a
n
g
l
e
\
d
a
g
e
r
\rightarrow\triangle
is asplitting
normally minimal, it follows from Lemma 1.0.1 that :.
anddeformation, which splits $X$ into
$X_{1}$$X_{2}$$\pi_{t}$$M_{t}arrow\triangle_{t}.$$X_{1}$$X_{2}$$X_{1}$$\Psi$$\mathcal{M}$$X_{2}$$\cross$$X_{2}$$X_{1}$$\square$\triangle \dag er also works for the case where is a\rightarrow \triangle
The above construction of I :is not necessarily asplitting
multiple node of multiplicity 1. But :is smooth, i.e.
. This is exactly the case when
deformation of :$X$ is areduced curve with one node.
$X_{2}=\Psi^{-1}(0,t)$
is asmooth
In which case,\triangle \dag er\rightarrow \triangle
fiber (in fact, is atomic by Theorem 2.0.2). Except this case, I:$X$, which splits
intoand, whereis asplitting deformation of :$X$
by replacing theis obtained fromis areduced curve with one node, and5.1.2,
withwe have thethisresultCriterion
Combined
reduced node by an annulus.
following criterion.
$\mathcal{M}$$\cross$$\Psi$$\pi$$\mathcal{M}$$p$$arrow\triangle\cross\triangle^{\mathrm{t}}$$Marrow\triangle$$X\backslash \{p\}$$\mathcal{M}$

$\pi$

$\pi$

$Marrow\triangle$$X_{1}$$\cross$$X_{2}$$X_{2}$$X_{1}$

is normally minimal such that the singular fiber
Criterion 5.1.3 Let :$X$ contains a multiple nodeis atomic if{of multiplicity $m\geq 1$ ). Then :
$X$
node.
and only if is a reduced curve with one$\pi$$Marrow\triangle$$\pi$$Marrow\triangle$

38

$X$

$X_{1}$

Figure 4: An example for Criterion 5.1.2
We digress to give atopological remark. Taking areal number $(0<\epsilon <1)$ , weconsider agermof the multiple node of multiplicity$,n$ .
Its boundary is areal 3-manifold, which is adisjoint union of two solid tori
$T_{x}:=\{|x|=1, |y|\leq\epsilon^{1/m}\}$ and $T_{y}:=\{|y|=1, |x|\leq\epsilon^{1/m}\}$ . In
Figure 5,
andare respectively described by the gray and black bold lines (in the real 2-dimensi0nal
figure, two gray lines are disconnected, but they are in fact connected; the same fortwo black lines).
$\epsilon$$\{(x, y)\in \mathbb{C}^{2} :|x^{m}y^{m}|\leq\epsilon\}$$T_{x}$$T_{y}$$\leq\epsilon$$|x|=$

Figure 5:
Remark 5.1.4 In the construction of ] in Criterion 5.1.3, we only used one multiple node. When $X$ has multiple nodes $(i=1,2, \ldots,n)$ of multiplicity , we cangeneralize the construction in Criterion 5.1.3 to construct asplitting deformation of, such that ::contains singular fibers X. $(i=1,2, \ldots,n)$ ,$X$
which is obtained from by replacing the multiple node :by the multiple annulus
of multiplicity .
$n$

$\pi$

$Marrow\triangle$$p_{i}$$\pi_{t}$$m_{i}$$M_{t}arrow\triangle_{t}$$\cdot$$p$$\mathrm{m}$39
Criterion in terms of plane curve singularities

5.2

is relatively minimal (notIn this subsection, we always sqppose that ’ :criterion in terms of plane
asplitting
necessarily normally minimal). We will exhibit
curve singularities on $X$ . We begin by introducing some terminology. Assume thatthe origin of $V:=\{(x, y)\in \mathbb{C}^{2} : F(x,y)=0\}$ is aplane curve singularity. (In thispaper, a plane curve singularity always means a reduced one.) For apositive integer$m$ , setting$V_{m}:=\{(x, y)\in \mathbb{C}^{2} : F(x, y)^{m}=0\}$ ,$\tau$$Marrow\triangle$

is amultiple plane curve singularity of multiplicity $m$ . (We also use
we say that)the notation $mV$ for
Proposition 5.2.1 Suppose that there exists a point $p\in X$ such that a germ ofin $X$ is a multiple of a plane curve singularity and the multiplicity $m$ is at least 2.
admits a splitting deformation.
Then :$V_{m}$$V_{m}.$

$p$

$\pi$

$Marrow\triangle$

Proof. We choose an open covering

is agerm$mV$$M_{1}\cap
X$. (See‘outside’
is
(2)
and
,
Then
, where (1)$M=\mathrm{J}/I_{0}\cup \mathrm{M}\mathrm{i}$of the multiple plane curve singularity
Figure 6.) We take local coordinates$mV$$\mathrm{J}/I_{0}\cap X$$\pi(z\beta, \zeta\beta)=F(z\rho, \zeta\beta)^{m}$$(z\beta, \zeta\beta)\in M\circ\cdot$$M$

Figure 6:

defines the plane curve singularity . Next, we take local
where
for some nonnear , thencoordinates.
by , we havevanishing holomorphic function . Rewritingof
Criterion
proof. As in the, we have
Since. So the gluing map
wehave
5.1.2, possibly after coordinate change,
is of the form
andofnear ,,$V$$F(z_{\beta}, \zeta\beta)=0$$(z_{\alpha}, \zeta_{\alpha})\in M_{1}$$\pi(z_{\alpha}, \zeta_{\alpha})=\zeta_{\alpha}^{m}u_{\alpha}(z_{\alpha}, \zeta_{\alpha})^{m}$$p$$\pi(z_{\alpha}, \zeta_{\alpha})=\zeta_{\alpha}$$\zeta_{\alpha}$$\zeta_{\alpha}u_{\alpha}$$u_{\alpha}$$\zeta_{\alpha}^{m}=F(z_{\beta}, \zeta_{\beta})^{m}$$\pi(z_{\alpha}, \zeta_{\alpha})=\pi(z_{\beta}, \zeta_{\beta})$$\zeta_{\alpha}=F(z_{\beta}, \zeta_{\beta})$$M_{1}$$M_{0}$

where$V_{t}$:

is holomorphic. Next, we take anon-equisingular deformation of :, where is holomorphic and $G(z\beta, \zeta\beta, 0)=0$ .
$V$$\phi_{\alpha\beta}$$G$$F(z_{\beta}, (_{\beta})+G(z_{\beta}, \zeta_{\beta},t)=0$

For example, if is anode ( -singularity), takeof , i.e. $V_{t}(t\neq 0)$ has only nodes (take a, byin
we define asmooth hypersurface$V$$\mathrm{J}/Iorsification^{19}$, and otherwise
singularities Next,$G(z_{\beta}, \zeta_{\beta}, t):=t$$A_{1}$$V$$\mathcal{M}0$$A_{1}$$M_{0}\cross\triangle\cross\triangle^{\mathrm{t}}$$\{(z_{\beta}, \zeta_{\beta}, s,t)\in M_{0}\cross\triangle\cross\triangle\dagger :$19\mathrm{A}\mathrm{n}$$p$$\zeta_{\alpha}=\mathrm{F}(\mathrm{z}\mathrm{p}, \zeta_{\beta})$$z_{\alpha}=\phi_{\alpha\beta}(z_{\beta}, \zeta_{\beta})$.

(F(z_{\beta}, \zeta_{\beta})+G(z_{\beta}, \zeta_{\beta}, t))^{m}-s=0\}$

isolated hypersurface singularity always admits aMorsification. See, for example Dimc$\mathrm{a}$40Similarly, we define asmooth hypersurface$\mathcal{M}_{1}$in$\Lambda f_{1}\cross\triangle\cross\triangle^{\mathrm{t}}$$\{(x,s, t)\in M_{1}\cross\triangle\cross\triangle\dagger :We glue$\mathcal{M}_{0}$with$\mathcal{M}_{1}$\pi(x)-s=0\}$

, by

.

by

$z_{\alpha}=\phi\alpha\beta(z\beta, \zeta\beta)$,

near ,$(_{\alpha}=F(z\beta, \zeta\rho)+G(z\rho, \zeta\beta,t)$$p$which yields acomplex 3-manifold. Letting :be the natural projection, the fiber $X_{1}:=\Psi^{-1}(0, t)$ is asingular fiber, which is obtained from$X$ byreplacing the multiple plane curve singularity $mV$ with. (To describe
other singular fibers, it is necessary to compute the discriminant of$G(z\beta, \zeta\rho,t))^{m}-s=0.)$ Since
is relatively minimal, I:
isasplitting deformation.
$\Psi$$\mathcal{M}$$\mathcal{M}$$arrow\triangle$$\cross\triangle^{\mathrm{t}}$$mV_{t}$$(\mathrm{F}(\mathrm{z}\mathrm{p}, \zeta_{\beta})+$$\pi:Marrow\triangle$$\mathcal{M}$$arrow\triangle\cross\triangle^{\mathrm{t}}$$\square$In the assumption of the above proposition, if we replace $m\geq 2$ by $m=1$ , what
can we say about the splittability of :?Also in this case, the above
construction works, and we obtain asplitting deformation, except the case whereis anode andis smooth (this is an atomic case). Combined with Proposition
5.2.1, we have the following results.
$\pi$$Marrow\triangle$$p$$X\backslash p$

Criterion 5.2.2 Let : M
be relatively minimal. Suppose that the singular
X
point
has
a
p,
such
that
a
germ
fiber
of p in X is either$arrow\triangle$$\pi$(1) a multiple of a plane curve singularity of multiplicity at least 2, or(2) a plane curve singularity such that if it is a node, then

Then

$\pi$

:

$Marrow\triangle$$X\backslash p$

is not smooth.

admits a splitting deformation.

6‘ Splitting criteria via configurations, II
In this section, we shall present another type of splitting criteria in terms of existenceof an irreducible component of multiplicity 1satisfying acertain property,

Criterion in terms of connected components

6.1

Criterion 6.1.1 Let:
be normally minimal. Suppose that the singularfiber contains an reducible component of multiplicity 1such thatis
(topologically) disconnected. Denote by , ,,all connected compOnents of. Then :admits a splitting deformation which splits $X$
into, , , where$(i=1,2, \ldots, l)$ is obtained
from $X$ by ‘smoothing’, Y2, , , , (see Figure 7for example). Hereis the omission of .
$X$

$\pi$

$Marrow\triangle$$i$$\ominus_{0}$$\mathrm{Y}_{1}$$X\backslash \ominus_{0}$$\mathrm{Y}_{2}$$\ldots$$X\backslash \ominus_{0}$$X_{1},X_{2}$$\ldots$$\pi$$X_{l}$$\mathrm{Y}_{l}(l\geq 2)$$Marrow\triangle$$X\dot{.}$$\check{\mathrm{Y}}_{\dot{2}}$$\mathrm{Y}_{1}$$\mathrm{Y}_{l}$$\ldots$$\check{\mathrm{Y}}_{k}$$\ldots$$\mathrm{Y}_{i}$Proof. To avoid complicated notation, we only show the statement for the casehere andintersects only at one point
the general case.) We take an open covering$\mathrm{Y}_{i}$$\mathrm{w}_{1}$$\ominus_{0}$$p:$. (The construction below works for$M=\hslasharrow f_{0}\cup M_{1}\cup\cdots\cup\Lambda f_{l}$, such that

41

$X$

$X_{1}$$X_{2}$

Figure 7: An example for Criterion 6.1.1, where(1)$M_{i}\cap X=\mathrm{Y}_{i}\cup D_{i}$(2)$M_{0}\cap X=\ominus_{0}\backslash \{D_{1}’\cup D_{2}’\cup\cdots\cup D_{l}’\}$$D_{i}\subset\ominus_{0}$is adisk around, where$D_{i}’$$p_{i}$,

is adisk satisfying$p_{i}\in D_{i}’\subset D_{i}$.

(See Figure 8.)
Here, we choose$M_{i}$

so that(and so$D_{i}$12$M_{1}$

1232$D_{i}’$) are sufficiently small. For simplicity, we$\mathrm{J}/I_{0}$11 01$M_{2}$$M$

1$22\Leftrightarrow$

Figure 8:. See Figure 9.
andNow we shall construct asplitting deformation of in the following steps: First,$(i=0,1, \ldots, l)$ with proper holomorphic maps
construct complex 3-manifolds
together to construct acomplex 3-manifold lit so. Secondly, glueon
. Finally, we show$(i=0,1, \ldots, l)$ determine aholomorphic map Ion
that
.
of
i asplitting deformationthat :
set$\mathrm{Y}_{i}^{+}:=\mathrm{Y}_{i}\cup D_{i}$$\ominus_{0}:=\ominus_{0}\backslash \{D_{1}’\cup D_{2}’\cup\cdots\cup D_{l}’\}$$\pi$$\mathcal{M}_{i}$$\Psi_{i}$$\mathcal{M}_{i}$$\mathcal{M}_{i}$$\mathcal{M}$$\Psi_{i}$$\Psi$

Step 1.

$\mathcal{M}$$arrow\triangle\cross\triangle^{\mathrm{t}}$$\pi$$\mathrm{s}$Construction ofplex 3-manifolds$\mathrm{c}\mathrm{o}\mathrm{n}\dot{\mathrm{n}}$$\mathcal{M}_{0}$,$\mathcal{M}_{1}$,$\mathrm{A}\mathrm{t}_{l}$\ldots ,42$\mathrm{Y}_{2}^{+}$$\ominus_{0}^{-}$

Figure 9:

We put$\mu:=e^{2\pi \mathrm{i}/l}$$(i=1,2, \ldots, l)$$\{(x, s, t)\in\Lambda f_{i}\cross\triangle\cross\triangle^{\mathrm{t}} ::$\Psi_{:}$$\mathcal{M}\dot{.}$$arrow\triangle\cross\triangle^{\mathrm{t}}$$\Psi_{i}^{-1}(s, t)=\{$$\mathcal{M}_{0}$in\pi(x)-s+\mu^{:}t=0\}$$M_{0}\cross\triangle\cross\triangle^{\mathrm{t}}$.

\pi(x)-s=0\}$

defined by.

$s=0$ ,,
smooth,$\ominus_{0}-$(6.1.1)$\Psi_{0}^{-1}(s, t)=\{$$\ominus_{0}^{-}$\triangle \dag er

be the natural projection. Then for $t\neq 0$ , we have$\mathcal{M}_{0}arrow\triangle\cross\triangle^{\mathrm{t}}$(Note that$\cross$$s=\mu^{i}t$$\{(x,s,t)\in M_{0}\cross\triangle\cross\triangle^{\mathrm{t}} ::$M_{i}\cross\triangle$,
otherwise.

,smooth,Next, we consider asmooth hypersurface$\Psi_{0}$inbe the natural projection. Then for $t\neq 0$ , we have$\mathrm{Y}^{+}.\cdot$(6.1.1)Let$\mathcal{M}_{i}$defined by(6.1.1)Let, and consider asmooth hypersurface

otherwise.

is also smooth!), ,togetherGluingStep 2.
$M$
andofaround . Let
take
local
coordinates
Now wethe multiplicity of the irreducible
be local coordinates around . Denote byat . Then we havecomponent intersecting$\mathcal{M}_{0},\mathcal{M}_{1}$$\mathcal{M}_{l}$$\ldots$$p:$$M\dot{.}$$p:$$\ominus_{0}$$(z_{\beta}, (\rho)\in$$(z_{\alpha}, \zeta_{\alpha})\in M_{0}$$m_{i}$$p_{i}$$\pi(z_{\alpha}, \zeta_{\alpha})=\zeta_{\alpha}f_{\alpha}(z_{\alpha}, \zeta_{\alpha})$,$\pi(z\rho, \zeta_{\beta})=z_{\beta}^{m:}\zeta\rho g_{\beta}(z\rho, \zeta_{\beta})$,are non-vanishing holomorphic functions. We shall change coorand
where. Likewise, rewriting
by , we havedinates. Rewriting, we obtain arelation. Sinceby , we have
. Hence the gluing map of andaround :is of the form
$f_{\alpha}$$g\rho$$\zeta_{\alpha}f_{\alpha}$$\pi(z_{\alpha}, \zeta_{\alpha})=\zeta_{\alpha}$$\zeta_{\alpha}$$\pi(z_{\alpha}, \zeta_{\alpha})=\pi(z\rho, \zeta\rho)$$\pi(z_{\beta}, \zeta_{\beta})=z_{\beta}^{m:}\zeta_{\beta}$$\zeta_{\beta}$$z_{\alpha}=\phi_{\alpha\beta}(z_{\beta}, \zeta_{\beta})$where$\psi_{\alpha\beta}$$M_{i}$$M_{0}$$(_{\alpha}=z_{\beta}^{m}\zeta:\rho$

is holomorphic. Next, we glue$z_{\alpha}=\phi_{\alpha\beta}(z\rho, \zeta_{\beta})$,$\zeta_{\beta}g_{\beta}$,$\mathcal{M}_{0}$$p$$\zeta_{\alpha}=z_{\beta}^{m:}\zeta\rho$,with Ad; $(i=1,2, \ldots, l)$ around$\zeta_{\alpha}=z_{\beta}^{m:}\zeta\rho+\mu^{:}t$around$p:$,$p.\cdot$by43. Note that the above map transforms thewhich yields acomplex 3-manifold, $(x, s, t)\mapsto$
to
that
of
. Let I:near
defining equation of$(s, t)$ , be the natural projection. From (6.1.2) and (6.1.3), for $t\neq 0$ ,$\mathcal{M}$$\mathcal{M}_{i}$$X_{i}$$\Psi^{-1}(s, t)=\{$$arrow\triangle\cross\triangle^{\mathrm{t}}$$\mathcal{M}$$\mathcal{M}_{0}$$p_{i}$,,
otherwise,$s=\mu^{i}t$smooth,

,
,
, ,
,
. As
is
is obtained from $X$ by smoothing
where\triangle \dag er is asplitting\rightarrow \triangle
normally minimal, it follows from Lemma 1.0.1 that :,, . This verifies our assertion.
deformation which splits $X$ into
i $\prod_{i=1}^{l}(s-\mu^{i}t)=0.$ )(Note: the discriminant of :$\check{\mathrm{Y}}_{i}^{+}$$X_{i}$$\mathrm{Y}_{1}^{+}$$\mathrm{Y}_{2}^{+}$$\Psi$$X_{1},X_{2}$$\Psi$$X_{i}$$\mathrm{Y}_{l}^{+}$$\ldots$$\ldots$$\mathcal{M}$$\cross$$X_{l}$$\ldots$$arrow\triangle\cross\triangle^{\mathrm{t}}$$\mathcal{M}$$\square$$\mathrm{s}$From the above construction, we can deduce some property of topological mon, andbe the:odromies. Let be the topological monodromy of:. Then we have arelationintopological monodromy around. Moreover, the following holds.
$Marrow\triangle$$\pi$$\gamma$$X_{i}$$\gamma_{i}$$M_{t}arrow\triangle_{t}$$\pi_{t}$$\gamma=\gamma_{1}\gamma_{2}\cdots\gamma_{l}$Proposition 6.1.2 The topological monodromies$\gamma_{1}$,$\gamma_{2}$,$\ldots$,$\gamma_{l}$commute.

;let be an arslightly modify the above construction of : Atas follows, . Instead of, we define
bitrary permutation of the set 1, 2,,
while
we take(cf. (6.1.1)):. Then we
as in the above construction:$(i=1,2, \ldots, l)$ by
, and
and
with
glue
\
t
r
i
a
n
g
l
e
\
d
a
g
e
r
is also: M\sigma \rightarrow \triangle. The natural projection
obtain acomplex 3-manifold$X$,
appears
,
,
,
.
,
But
into
splitting deformation which splits.
, hence we have arelation,,in the order, commute.
Since is an arbitrary permutation, it follows that , ,Proof.

$arrow\triangle\cross\triangle^{\mathrm{t}}$$\Psi$$\mathrm{t}\mathrm{V}\mathrm{e}$$\{$$\ldots$$l\}$$\sigma$$\mathcal{M}_{i}$$\mathcal{M}_{\sigma,i}$$\mathcal{M}_{\sigma,i}:=\{(\mathrm{x}, s,t)\in iVI_{i} \cross\triangle\cross\triangle\dagger :\pi(x)-s+\mu^{\sigma(i)}t=0\}$$\{(x, s,t)\in NI_{0}\cross\triangle\cross\triangle^{\mathrm{t}} :$\mathcal{M}_{0}$$\mathcal{M}_{0}$$\Psi_{\sigma}$$\mathcal{M}_{\sigma}$$X_{1}$$X_{\sigma(1)},X_{\sigma(2)}$$\ldots$\pi(x)-s=0\}$$\zeta_{\alpha}=z_{\beta}^{m:}\zeta\beta+\mu^{\sigma(:)}t$$z_{\alpha}=\phi_{\alpha}\rho(z\beta, \zeta_{\beta})$$\mathcal{M}_{\sigma,i}$$X_{2}$$\ldots$$\cross$$X_{1},X_{2}$$X_{l}$$X_{\sigma(l)}$$\ldots$$X_{l}$$\gamma=\gamma\sigma(1)\gamma\sigma(2)\ldots$$\gamma\sigma(l)$$\square$$\sigma$$\gamma_{1}$$\gamma_{2}$$\ldots$$\gamma_{l}$in Criterion 6.1.1, we used only one irRemark 6.1.3 In the construction of
reducible component of multiplicity 1. As is clear from the construction, we cansimilarly construct a splitting deformation by using several irreducible componentof multiplicity 1simultaneously, provided that, ,,is disconnected. More generally, in some cases, we can construct asplitting deformation, by ‘mixing up’ all constructions in this paper.
$\Psi$$\Theta_{0}^{(1)}$$X\backslash \{\ominus_{0^{1)}}^{\mathrm{t}}\cup\ominus_{0}(2)\cup$$\Theta_{0}^{(n)}$$\Theta_{0}^{(2)}$$\ldots$$\ldots\cup\Theta_{0}^{(n)}\}$

Inductive criterion

6.2

be normally minimal, such that its singular fiber $X$ contains an:
Letis connected.
of multiplicity 1. We suppose that
irreducible component
results,
we need to
our
state
To
criterion.
Also in this case, we have some splitting,, be the intersection, andintroduce some notation. Let$X$with other irreducible components of . Take an open covering $M=$
points of, such that$\pi$$Marrow\triangle$$X\backslash \ominus_{0}$$\Theta_{0}$$\mathrm{Y}:=X\backslash \ominus_{0}$$p_{1},p_{2}$$\ldots$$p_{n}$$\ominus_{0}$$\Lambda’I_{0}\cup M_{1}$(1) $M_{1}\cap X=\mathrm{Y}\cup D_{1}\cup D_{2}\cup\cdots\cup D_{n}$ , where$D_{i}\subset\ominus_{0}$is adisk around$p_{i}$

,

44
(2)

$M_{0}F^{\ovalbox{\tt\small REJECT}}IX\ovalbox{\tt\small REJECT}$$0_{0}\mathrm{y}${D9$\mathrm{U}\mathrm{D}\ovalbox{\tt\small REJECT} \mathrm{U}\cdots \mathrm{U}D4\}$, where \yen is adisk satisfying

ED” CD;.

$p\ovalbox{\tt\small REJECT}$(See Figure 10.)Figure 10:
Here, we chooseset$M_{1}$

so that$D_{i}$$\mathrm{Y}^{+}:=\mathrm{Y}\cup D_{1}\cup D_{2}\cup\cdots\cup D_{n}$(and so,$D_{i}’$) are sufficiently small. For simplicity, we$\ominus_{0}^{-}:=\ominus_{0}\backslash \{D_{1}’\cup D_{2}’\cup\cdots\cup D_{n}’\}$(Figure 11).

Figure 11:

Criterion 6.2.1 Let :be nor mally minimal s.uch that the singular fiber $X$
contains an irreducible component
be the reof multiplicity 1. Let :striction of to a tubular neighborhoodinM.
Supposethat :
ofadmits a splitting defor notion
which splits
,
, ,
into
. Then$X$:admits a splitting defor notion
which splits
, ,
, ,
into
where
is obtained fromby gluingalong the boundary.
$\pi$$Marrow\triangle$$\ominus_{0}$$M_{1}|$$\pi$$\triangle$$\pi_{1}$$M_{1}arrow\Delta$$\mathrm{Y}_{1}^{+}$$\mathrm{Y}_{2}^{+}$$X\backslash \ominus_{0}$$\mathrm{Y}^{+}$$\Psi_{1}$$\pi_{1}$$M_{1}arrow$$\mathrm{Y}_{l}^{+}$$\ldots$$\pi$$Marrow\triangle$$\Psi$$X_{i}$$\mathrm{Y}_{i}^{+}$$X_{1}$$X_{2}$$\ldots$$X_{l}$$\ominus_{0}^{-}$

Note: We note that:is adegeneration of curves with boundary, forwhich we may also define the notion of splitting deformations in the same way as
for degenerations of compact curves$\pi_{1}$$M_{1}arrow\triangle$

45
Proof. As in the proof of Criterion 6.1.1, we take local coordinates$p_{i}$, and local coordinateswitharoundandsuch that the gluing map ofnear withis of the form$(z\rho, \zeta\beta)\in NI_{1}$$\pi(z_{\alpha},\zeta_{\alpha})=(_{\alpha}$$M_{}$$il’I_{0}$$z_{\beta}^{m_{j}}\zeta_{\beta}$$z_{\alpha}=\phi_{\alpha\beta}(z\rho, \zeta_{\beta})$,$p\dot{.}$$\zeta_{\alpha}=z_{\beta}^{m:}\zeta\rho$:
is holomorphic. Now, letting
wherein the assumption, we consider amap
deformation ofas the graph of :and then realize$p_{i}$near$\pi(z_{\beta}, \zeta_{\beta})=$,$\mathcal{M}_{1}arrow \mathrm{I}\mathrm{S}$$\Psi_{1}$$\phi_{\alpha\beta}$$\in M_{0}$$(z_{\alpha}, \langle_{\alpha})$be the splitting:,$\cross\triangle^{\mathrm{t}}$$\mathcal{M}$$\tilde{\pi}_{1}:=\mathrm{p}\mathrm{r}_{1}\mathrm{o}\Psi_{1}$$\pi_{1}$$arrow\triangle$$\tilde{\pi}_{1}$$\mathcal{M}_{1}$$\mathcal{M}_{1}=\{(x, s,t)\in M_{1}\cross\triangle\cross\triangle^{\mathrm{t}} :\tilde{\pi}_{1}(x,t)-s=0\}$.

, hence we may express $\tilde{\pi}_{1}(x,t)=\pi_{1}(x)+h_{1}(x,t)$ , where
Notice thatis aholomorphic function satisfying $h_{1}(x, 0)=0$ . Next, we define asmoothbyinhypersurface$\tilde{\pi}_{1}(x, 0)=\pi_{1}(x)$$h_{1}$$M_{0}\cross\triangle\cross\triangle^{\mathrm{t}}$$\mathcal{M}_{0}$$\mathcal{M}_{0}=\{(x, s,t)\in M_{0}\cross\triangle\cross\triangle\dagger :Finally, we glue$\mathcal{M}_{0}$with$\mathcal{M}_{1}$around$z_{\alpha}=\phi_{\alpha\beta}(z_{\beta}, \zeta_{\beta})$.

by$p_{i}$,\pi(x)-s=0\}$$\zeta_{\alpha}=z_{\beta}^{m:}\zeta_{\beta}+h_{1}(z_{\beta}, \zeta_{\beta})$,. Then the natural projection 1! :and we obtain acomplex 3-manifoldofis asplitting deformation of . In fact, assuming that the fiber
byobtainedisgluingconstruction,byis singular,
over the point
along the boundary.
with$\mathcal{M}$$\mathcal{M}$$\triangle\cross\triangle^{\mathrm{t}}$$\mathrm{Y}_{k}^{+}$$\pi$$arrow$$\Psi_{1}$$\Psi^{-1}(x_{k})$$x_{k}\in\triangle_{t}$$\square$$\mathrm{Y}_{k}^{+}$$\ominus_{0}^{-}$:
in Criterion 6.2.1, we shall construct adegeneration$X’$
is obtained by replacing theof compact curves, whose singular fiber$(i=1,2, \ldots, n)$ by aprojective line (see Figure 12), after that, we will
disk
with
restate Criterion 6.2.1 in terms of this degeneration. First, we glueby,,
. Then we obis coordinates near , and
where$M’arrow\triangle$$\pi’|_{M_{1}}=\pi$
, and
by
:
tain acomplex surface $M’$ . Define amap
byreobtained
is
of
fiber
. By construction, the singular$(i=1,2, \ldots, n)$ by aprojective line.
placing
Then Criterion 6.2.1 is restated as follows:

From$\pi_{1}$:$\pi’$$M_{1}arrow\triangle$$M’arrow\triangle$$D_{i}$$D_{i}\cross\triangle$$M_{1}$$z_{\alpha}=\phi_{\alpha\beta}(z_{\beta}, \zeta_{\beta})$$(z_{\alpha}, \zeta_{\alpha})\in M_{1}$$\zeta_{\alpha}=z_{\beta}^{h:}\zeta_{\beta}$$(z\rho, \zeta\beta)\in D_{i}\cross\triangle$$p_{i}$$\pi’$$\pi’$$\pi’|_{D_{i}\mathrm{x}\triangle}(z_{\beta}, \zeta_{\beta})=\zeta_{\beta}$$D_{i}$

admits a splitting deformation, then :also admits a splitting deformation. (Note: By construction, the converse is true.)

Criterion 6.2.1’ If$\pi’$:$M’arrow\triangle$$\pi$$Marrow\triangle$(resp. ’ :(resp. ) be the genus of asmooth fiber of :is aprojective line and intersects other irreducible. Except the case whereis adegeneration
components at only one point, we have $g’<g$ , and so :intersect other irreducible components atof curves of lower genus. Indeed, letpoints. By atopological consideration, it is easy to see that

Let

$g$

$\triangle)$$g’$$Marrow\triangle$$\tau_{\mathrm{I}}$$l\mathcal{V}I’arrow$$\ominus_{0}$$M’arrow\triangle$$\pi’$$n$$_{0}$(6.2.1)$\pi$$g=g’+(n-1)+\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{u}\mathrm{s}(\ominus_{0})$.

Hence we have $g’<g$ , unless Oo is aprojective line and $n=1$ .

46

$X’$

Figure 12:

Consequence of splitting criteria

6.3

As before, in this subsection, we assume that any degeneration is normally minimal.
The splitting criteria obtained in this paper altogether imply that if the singularfiber $X$ is not star-shaped, then in many cases,: $Marrow\triangle$ admits asplitting
deformation. Taking into account Theorem 2.0.2, it is interesting to know whether
the following
is true or not (cf. Conjecture 6.3.1’ below):$\pi$$\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{e}^{20}$Conjecture 6.3.1 A degeneration is atomic if and only if its singular fiber is either
a reduced curve with one node, or a multiple of a smooth curve.
, [Ta] for results on this conjecture. Next, we deduce auseful theorem
Seefrom our splitting criteria. Let
be aset of degenerations :of curves ofgenus such that$[\mathrm{T}\mathrm{a},\mathrm{I}\mathrm{I}\mathrm{I}]$$\pi$$\Lambda_{g}$$Marrow\triangle$$g$(1) the singular fiber $X$ has amultiple node (here we exclude the case where $X$
is areduced curve with only one node), or(2) $X$ contains an irreducible componentof multiplicity 1satisfying the following:ifis connected, then either genus(00), oris aprojective line intersecting other irreducible components at at least two$\ominus_{0}$$\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}^{21}$$\geq 1$

$X\backslash \ominus_{0}$$\ominus_{0}$points.

As aconsequence of our splitting criteria, we obtain the following.

Theorem 6.3.2 Suppose that Conjecture 6.3.1 is valid for genus $\leq g-1$ .
is a degeneration in , then is not atomic.
$Marrow\triangle$$\Lambda_{g}$

If

$\pi$

:

$\pi$

Proof First, by Criterion 5.1.3, if the singular fiber contains amultiple node, thenadmits asplitting deformation. Next, suppose that $X$ contains an irreducible
componentof multiplicity 1. if
is not connected, then :has asplitting deformation (Criterion 6.1.1). On the other hand, if
is connected,
then under the assumption of this theorem, we can apply Criterion 6.2.1’, and seethat :
admits asplitting deformation, except the case where
is a
$\pi$

$\mathrm{O}_{0}$$X\backslash \ominus_{0}$$\pi$$Marrow\triangle$$X\backslash \ominus_{0}$$\pi$$Marrow\triangle$$\ominus_{0}$conjecture is valid for the genus 1and 2cases: for the genus 1case, any atomic fiber iseither arational curve with one node, or amultiple of asmooth elliptic curves by [Mo], and forthe genus 2case, any atomic fiber is areduced curve with one node by [Ho].
21If
is not connected, we pose no condition$20\mathrm{T}\mathrm{h}\mathrm{i}\mathrm{s}$$X\backslash \ominus_{0}$47intersects other irreducible components at only one point
projective line, andCl(cf. (6.2.1)). Hence the assertion follows.
$0_{0}$

Thus if the assumption of this theorem is fulfilled (for example, $g=3$ ), to
determine atomic degenerations of curves of genus , it is enough to investigate thesuch that either
splittability for degenerations :
$g$

$\pi$

$Marrow\triangle$(A) $X=\pi^{-1}(0)$ is star-shaped, or(B) $X$ is not star-shaped and (B.I) $X$ has no multiple node and (B.2) if $X$ has an
is aprojective line, andof multiplicity 1, thenirreducible component$X$
intersects other irreducible components of only at one point.
$\mathrm{O}_{0}$$\mathrm{O}_{0}$, the singular fibers of adegeneration in (B) is obtained
In the terminology ofby ‘bonding’ star-shaped singular fibers such that any bonding of two branches iseither (-1)-bonding or 0-bonding of two branches with the same multiplicity at
and also [MM2]. For these cases, we can apply another method
least 2. See(construction of splitting deformations via barkable sub-divisors), which is developed.
in$[\mathrm{T}\mathrm{a},\mathrm{I}\mathrm{I}]$$[\mathrm{T}\mathrm{a},\mathrm{I}\mathrm{I}]$$[\mathrm{T}\mathrm{a},\mathrm{I}\mathrm{I}\mathrm{I}]$

Discussion and open problems
For higher genus cases, Conjecture 6.3.1 seems too optimistic. It is more reasonable:
to replace ‘atomic’ with ‘absolutely atomic’, where adegeneration
is called absolutely atomic if all degenerations with the same topological type as
are atomic (for example, when $X$ is areduced curve with one node or:amultiple of asmooth curve. See Theorem 2.0.2).
$Marrow\triangle$$\tau_{1}$$\pi$$Marrow\triangle$

Conjecture 6.3.1’ A degeneration is absolutely atomic if and only if its singularfiber is either a reduced curve with one node, or a multiple of a smooth curve.

Accordingly, we can show an analogous statement to Theorem 6.3.2 by the same
argument.
Theorem 6.3.2’ Suppose that Conjecture 6.3.1’ is valid for genus $\leq g-1$ .
is a degeneration in , then is not absolutely atomic.
:$\pi$$Marrow\triangle$$\Lambda_{g}$

If

$\pi$

It is plausible that for higher genus cases, there may be an atomic degeneration
which is not absolutely atomic. However, no examples are known, and so we ask:
Problem 6.3.3 Do there exist two degenerationsis atomic whilewith the same topological type such that$\pi_{1}$$\pi_{1}$andis not?

$NI_{1}arrow\triangle$$\pi_{2}$$\pi_{2}$:$\mathrm{J}/I_{2}arrow\triangle$

case, there are degenerations with the same singular
Note that for the genus
fiber, but with different topological types [MM2]. Taking this into account, it isnatural ask the following problem analogous to Problem 6.3.3$\geq 2$

48
Problem 6.3.4 Do there exist two degenerations:
and:with the same singular fiber but with different topological types such that is atomic
while
is not ?
$\pi_{1}$$\Lambda f_{1}arrow\triangle$$\pi_{2}$$\pi_{1}$$\pi_{2}$$\Lambda f_{2}arrow\triangle$

49
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