Étale endomorphisms of algebraic surfaces with $G_m$-actions
Étale endomorphisms of algebraic surfaces with
$G_m$-actions
复制标题
带有 $G_m$-actions 的代数曲面的 Étale 自同态
DOI：
10.1007/pl00004445
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发表时间：
2001
影响因子：
1.4
通讯作者：
M. Miyanishi
中科院分区：
文献类型：
--
作者：
K. Masuda;M. Miyanishi
抽象的。设 X 是在复数域上定义的代数曲面
${\vec C}$ 并赋予
${\vec A}^1_*$-纤维化。作为这样的表面，我们有一个柏拉图式的
${\vec A}^1_*$-纤维空间，一个删除了奇点的加权超曲面，更一般地说，一个具有未混合的仿射代数曲面
$G_m$-动作及其固定点被删除。我们考虑一个 etale 自同态
$\varphi : X \to X$ 并表明
大多数情况下 $\varphi$ 是自同构。特别有趣的是柏拉图式的情况
${\vec A}^1_*$-光纤空间，其中
$\varphi$ 是自同构，与仿射平面的雅可比问题密切相关
${\vec A}^2$。我们还研究了此类曲面的自同构群。
Abstract. Let X be an algebraic surface defined over the complex field
${\vec C}$ and endowed with an
${\vec A}^1_*$-fibration. As such a surface we have a Platonic
${\vec A}^1_*$-fiber space, a weighted hypersurface with its singular point deleted off and, more generally, an affine algebraic surface with an unmixed
$G_m$-action and its fixpoint deleted off. We consider an étale endomorphism
$\varphi : X \to X$ and show that
$\varphi$ is an automorphism in most cases. Of particular interest is the case of a Platonic
${\vec A}^1_*$-fiber space, for which
$\varphi$ being an automorphism is closely related to the Jacobian Problem for the affine plane
${\vec A}^2$. We also investigate the automorphism group of such surfaces.