On the deformations of cyclic Galoi coverings of algebraic curves

关于代数曲线循环伽罗伊覆盖的变形

• 批准号: 62540066
• 负责人: SEKIGUCHI Tsutomu
• 金额: $ 1.54万
• 依托单位: Chuo University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for General Scientific Research (C)
• 财政年份: 1987
• 资助国家: 日本
• 起止时间: 1987 至 1989
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-62540066/
• 关键词: Witt group; Artin-Schreier; Kummer; Algebraic curve; Extension; Group scheme; 群スキーム; 群スキームの拡大; N〓ron blow-up

Our aim of this research is to construct a lifting of a couple (C,sigma) of a complete non-singular curve C over a field of characteristic p (>O), and its automorphism sigma of order p^n (n<greater than or equal>1). For the case of n = 1, we have obtained an affirmative result by using Lang's class field theory. In the argument, one of the important ideas if to construct a deformation of the Artin-Schreier theory to the Kummer theory, i.e., to construct a deformation of an additive group to a multiplicative group. Therefore when we try to solve the problem for n<greater than or equal>2, first of all, we must construct the deformations of the Witt group W_n to a torus (G_m)^n. We decided Completely such deformations for n=2 in 1988 and 1989. In 1990, we discovered a vanishing theorem for extensions of additive groups by a torus over an Artin local ring, and usinit we treated the extensions of group schemes over a Artin local ring. Moreover, by using the vanishing theorem, we succeeded in constructing the deformations of the Artin-Schreier-Witt exact sequence to an exact sequence of Kummer type. In future, we must construct a unified theory of the Artin-Schreier-Witt and the Kummer theories, and apply it to our original problem.

我们这项研究的目的是构造特征p（&gt;O）域上完全非奇异曲线C的偶（C，sigma）的提升，及其阶为p^n（n 1）的自同构sigma<greater than or equal>。对于n = 1的情形，我们利用Lang的类域理论得到了一个肯定的结果。在论证中，一个重要的思想是构造一个由Artin-Schreier理论到库默理论的变形，即，来构造加法群到乘法群的变形。因此，当我们试图解决n = 2的问题时<greater than or equal>，首先必须构造Witt群W_n到环面（G_m）^n的变形.我们在1988年和1989年确定了n=2的完全这样的变形。1990年，我们发现了Artin局部环上加群的环面扩张的消失定理，并利用它处理了Artin局部环上群概型的扩张.此外，利用消失定理，我们成功地构造了将Artin-Schreier-Witt正合列变形为库默型正合列的方法.今后，我们必须建立一个统一的理论，阿廷-施赖埃-维特和库默理论，并应用于我们原来的问题。

项目成果

"How coarse the coarse moduli spaces for curves are!" Algebraic Geometry and Commutative Algebra. 693-712 (1987)

“曲线的粗模空间是多么粗糙啊！”

"Some cases of extensions of group schemes over a discrete valuation ring I" Chuo-Math.Preprint Series. 8. (1989)

“在离散评估环上扩展群体方案的一些案例 I”中央数学.预印本系列。

関口力: Algebraic and topological theories to the memory of Dr.Miyata. 283-298 (1985)

Riki Sekiguchi：纪念宫田博士的代数和拓扑理论 283-298 (1985)。

関口力: Chuo Univ.Preprint series No.1.(1988)

关口力树：中央大学预印本系列第 1 号（1988 年）

K. Sekino: "On the fields of definition for a curve and its Jacobian variety" Bull. Facul. Sci. & Eng. Chuo Univ., 31, 29-31(1988).

K. Sekino：“关于曲线及其雅可比变体的定义领域”Bull。