Towards the classification of atoms of degenerations, I-Splitting criteria via configurations of singular fibers
Towards the classification of atoms of degenerations, I-Splitting criteria via configurations of singular fibers
复制标题
通过奇异纤维的配置对简并原子进行分类，I-分裂标准
DOI：
10.2969/jmsj/1191418698
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发表时间：
2001
影响因子：
0.7
通讯作者：
Shigeru Takamura
中科院分区：
文献类型：
--
作者：
Shigeru Takamura
Motivated by the classification problem of atomic degenerations, in our series of papers, we make asystematic study for splitting deformations of degenerations of complex curves. We provide various new methods to construct splitting deformations, and deduce many splitting criteria of degenerations, which $\mathrm{w}\mathrm{i}\mathrm{U}$ be applied to the classification of atomic degenerations. Roughly, our criteria are separated into two tyPes; in the first type the criteria are expressed 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. thematical Subject Classification: Primary $14\mathrm{D}05,14\mathrm{J}15$ ;Secondary $14\mathrm{H}15,32\mathrm{S}30$ iwords: Degeneration of complex curves, Complex surface, Singular fiber, Riemann surf formation of complex structures, Splittings of singular fibers, Atomic degeneration, Monodr 数理解析研究所講究録 1233巻 2001年 18-50 18 Introduction This paper constitutes one part of our series of papers on degenerations. By a degeneration, we mean aproper surjective map $\pi$ : $Marrow\triangle$ from asmooth complex surface $l\vee I$ to the unit disk $\triangle$ such that the fiber over the origin is singular and any other fiber is asmooth curve of genus $g(g\geq 1)$ . Adeformation of adegeneration is called 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 to classify 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 used the 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 cases, all curves are hyperelliptic, but this is not the case for genus $\geq 3$ ). In our series of papers we develop completely different methods for constructing splitting deformations, and apply them to the classification of atomic degenerations for the genus 3,4 and 5cases [$\mathrm{T}\mathrm{a},\mathrm{I}\mathrm{I}\mathrm{I}$ , Ta]. The aim of this paper is to study the relation between the configurations of singular fibers and the existence of splitting deformations. We first show that two types of degenerations are atomic. Theorem 2.0.2 Let $\pi$ : $Marrow\triangle$ be a degeneration of curves such that the singular fiber $X$ is either (I) a reduced curve with one node, or (II) a multiple of a smooth curve of multiplicity at least 2. Then $\pi$ : $Marrow\triangle$ is atomic. 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 $\pi$ : $Marrow\triangle$ be $\mathrm{a}$ degeneration of compact complex manifolds of arbitrary dimension, if the singular fiber $X$ is amultiple of asmooth complex manifold, then $\pi$ : $Marrow\triangle$ is atomic. Next, we shall state results on existence of splitting deformations. We demonstrate 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 with $\mathrm{n}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{s}\mathrm{h}\mathrm{a}\mathrm{p}\mathrm{e}\mathrm{d}^{1}\sin-$ gular fibers always admit splitting deformations. Together with Theorem 2.0.2 it is lSee \S 4. 19