论文标题
积极特征i中基本曲线基本组的模量空间
Moduli spaces of fundamental groups of curves in positive characteristic I
论文作者
论文摘要
在这一系列论文中，我们研究了一种曲线的新的曲线现象，该现象是积极特征的代数封闭场。令$ \ overline m_ {g，n} $是$ \ overline {\ mathbb {f}} _ p $ over $（g，n）$ type $（g，n）$的模量空间。我们引入了一个拓扑空间$ \叠加π_{g，n} $，可以从理论上从可允许的基本组组成的基本组来确定$（g，n）$的尖稳定曲线的基本组。通过在基础拓扑空间$ | \ overline m_ {g，n} | $ of $ \ OVERLINE m_ {g，n} $上引入一定的等价关系$ \ sim_ {fe} $，我们获得了拓扑空间$ \ overline $ \ overline { n}} |/\ sim_ {fe} $。此外，还有一个自然的连续映射$π_{g，n}^{\ rm adm adm}：\ edropline {\ mathfrak {\ mathfrak {m}} _ {g，n} \ n} \ rightArrow \ rightArrow \ rightArrow \rightlineistlineipounlineπ_{g，n}。 $π_{g，n}^{\ rm adm} $是同构。同态猜想概括了在特征$ p $的代数封闭场上曲线的曲线几何学理论中的所有猜想。本系列论文的主要结果之一表明，同构猜想在$ \ text {dim}（\ overline m_ {g，n}）= 1 $（即$（g，n）=（g，n）=（0,4）$或$（g，n）=（g，n）=（1,1,1）$）。在本文中，我们建立了两个基本工具，以分析可允许基本群体的开放式同态同态的几何行为，这些曲线在一系列论文中发表的理论中起着核心作用。此外，我们证明$π_{0，n}^{\ rm Adm}（[q]）$是$ \ OVERLINEπ_{0，n} $的封闭点，当$ [q] $是$ \ overline {\ mathfrak {m}} _ {0，n} $的关闭点。特别是，我们得到同态构态猜想在$（g，n）=（0，4）$时。
In this series of papers, we investigate a new anabelian phenomenon of curves over algebraically closed fields of positive characteristic. Let $\overline M_{g, n}$ be the moduli space of curves of type $(g, n)$ over $\overline {\mathbb{F}}_p$. We introduce a topological space $\overline Π_{g, n}$ which can be determined group-theoretically from admissible fundamental groups of pointed stable curves of type $(g, n)$. By introducing a certain equivalence relation $\sim_{fe}$ on the underlying topological space $|\overline M_{g, n}|$ of $\overline M_{g, n}$, we obtain a topological space $\overline {\mathfrak{M}_{g, n}}:= |\overline {M_{g, n}}|/\sim_{fe}$. Moreover, there is a natural continuous map $π_{g,n}^{\rm adm}: \overline {\mathfrak{M}}_{g, n} \rightarrow \overline Π_{g, n}.$ Furthermore, we pose a conjecture (=the Homeomorphism Conjecture) which says that $π_{g,n}^{\rm adm}$ is a homeomorphism. The Homeomorphism Conjecture generalizes all the conjectures in the theory of anableian geometry of curves over algebraically closed fields of characteristic $p$. One of main results of the present series of papers says that the Homeomorphism Conjecture holds when $\text{dim}(\overline M_{g, n})=1$ (i.e., $(g, n)=(0,4)$ or $(g, n)=(1,1)$). In the present paper, we establish two fundamental tools to analyze the geometric behavior of curves from open continuous homomorphisms of admissible fundamental groups, which play central roles in the theory developed in the series of papers. Moreover, we prove that $π_{0,n}^{\rm adm}([q])$ is a closed point of $\overline Π_{0,n}$ when $[q]$ is a closed point of $\overline {\mathfrak{M}}_{0, n}$. In particular, we obtain that the Homeomorphism Conjecture holds when $(g, n)=(0, 4)$.
