Fundamental groups of curves in positive characteristic
Fundamental groups of curves in positive characteristic
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正特征曲线的基本组
DOI：
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发表时间：
2007
期刊：
影响因子：
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通讯作者：
Akio Tamagawa
中科院分区：
文献类型：
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作者：
Akio Tamagawa
We present some recent results (mainly of the author) concerning fundamental groups of curves over algebraically closed fields of positive characteristic. \S 1. Main question (cf. $[\mathrm{H}2,4][\mathrm{T}3]$). In this \S , we introduce our main question. Throughout this article, we let $k$ denote an algebraically closed field of characteristic $p>0$ , $X$ asmooth curve over $k$ , $X^{*}$ the smooth compactification of $X$ and $\Sigma=X^{*}-X\mathrm{d}\mathrm{e}\mathrm{f}$ . We define non-negative integers $g$ and $n$ to be the genus of the proper, smooth curve $X^{*}$ and the cardinality of the point set $\Sigma$ , respectively. Note that $X$ is hyperbolic (resp. affine, resp. projective) if and only if $2-2g-n<0$ , $\mathrm{i}.\mathrm{e}.$ , $(g, n)\neq(0, 0)$ , $(0, 1)$ , $(0, 2)$ , $(1, 0)$ (resp. $n>0$ , resp. $n=0$). We denote by $k(X)^{\sim}$ the maximal separable algebraic extension of $k(X)$ in which the discrete valuation ring $\mathcal{O}_{X,x}$ is unramified for all $x\in X$ , and by $k(X)^{\sim \mathrm{t}}$ the maximal separable algebraic extension of $k(X)$ in which $\mathcal{O}_{X,x}$ is unramified for aU $x\in X$ and at most tamely ramified for all $x\in\Sigma$ . Then, the fundamental group $\pi_{1}(X)$ (resp. the tame fundamental group $\pi_{1}^{\mathrm{t}}(X)$ ) of $X$ is nothing but the Galois group Gal(k(X)” $/k(X)$ ) (resp. Gal(k$(X)^{\sim \mathrm{t}}/k(X))$ ). Definition. (i) For a(discrete) group $\Gamma$ , we denote by $\Gamma^{\wedge}$ its profinite completion $\mathrm{N}\triangleleft \mathrm{r},\lim_{(\Gamma\mathrm{N})<\infty}\Gamma/\mathrm{N}arrow$ . (ii) For aprofinite group $G$ , we denote by $G^{p’}$ its maximal pr0-ptamet0-p quotient $\lim_{arrow}$ $G/N$. $N\triangleleft G$ closed, $p\uparrow(G:N)<\infty$ (iii) For non-negative integers $g$ and $n$ , we denote by $\Pi_{g,n}$ the topological fundamental group of acompact orientable surface of genus $g$ with $n$ points deleted. More concretely, $\Pi_{g,n}=\langle\alpha_{1}$ , $\ldots$ , $\alpha_{g}$ , $\beta_{1}$ , $\ldots$ , $\beta_{g},$ $\gamma_{1}$ , $\ldots$ , $\gamma_{n}|\alpha_{1}\beta_{1}\alpha_{1}^{-1}\beta_{1}^{-1}\ldots\alpha_{g}\beta_{g}\alpha_{g}^{-1}\beta_{g}^{-1}\gamma_{1}\ldots\gamma_{n}=1)$ In particular, if $n>0$ , $\Pi_{g,n}$ is afree group of rank $2g+n-1$ . Typeset by $A\lambda \mathit{4}S\mathrm{I}\mathrm{E}\mathrm{K}$ 数理解析研究所講究録 1267巻 2002年 142-146
具有规定基本群的正特征曲线同构类的有限性。
DOI：
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发表时间：
2004
期刊：
J. Algebraic Geom 13, no.4
影响因子：
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作者：
Ohyama;Y.;et. al.;A.Tamagawa
通讯作者：
A.Tamagawa