On the compactification of Witt group schemes and the deformation of Art theory

论维特群方案的紧化与艺术理论的变形

• 批准号: 11640045
• 负责人: SEKIGUCHI Tsutomu
• 金额: $ 2.24万
• 依托单位: Chuo University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1999
• 资助国家: 日本
• 起止时间: 1999 至 2001
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640045/
• 关键词: group scheme; Kummer theory; Artin-Scheme-Witt theory; Witt vector; Artin-hasse exponential series; extension of group schemes; Cartier module; Arti-Hasse指数関数; Canties加群; 群スキームの変形

Already we have showed the existence of group schemes which gave the deformations of the group schemes of Witt vectors to tori. Using those group schemes we could contract the unified Kummer-Artin-Schreier-Witt theory. But, when we want to apply the theory to some problems, for example, the lifting problem of cyclic coverings of algebraic curves, partially solved by Green-Matignon, we need more explict description of the group schemes. In 1999 and 2000, we devoted ourselves to construct concretely the group schemes giving the deformations of the group schemes of Witt vectors to tori, and we succeeded to descrive such group schemes by using several Witt vectors. In the background, there is the Cartier thory, and our thory is given by the representation of that by virtue of deformed Artin-Hasse exponential series.To descrive the ramifications of cyclic coverings, we need to compactfy such group schemes. In positive characteristic case. Garuti gave a nice compactifications of group schemes of Witt vectors by means of ruled surfaces. We tried to give compactifications of the deformed group schemes also, even it is in two-dimensional case, and we are on the way to investigate the description of ramification locuses geometrically.

我们已经证明了群格式的存在性，它给出了Witt向量群格式在环面上的变形。利用这些群格式，我们可以缩并统一的Kummer-Artin-Schreier-Witt理论。但是，当我们要将该理论应用于某些问题时，例如由Green-Matignon部分解决的代数曲线的循环覆盖的提升问题，我们需要更明确地描述群格式。在1999年和2000年，我们致力于构造具体的群格式，给出Witt向量的群格式对环面的变形，并成功地用几个Witt向量来描述这种群格式。背景是Cartier理论，我们的理论是通过变形的Artin-Hasse指数级数来表示的。为了描述循环覆盖的分支，我们需要压缩这样的群方案。在正特征情况下。Garuti利用直纹曲面给出了Witt向量群格式的一个很好的紧化。我们也试图给出变形群格式的紧化，即使是在二维的情况下，我们正在探索分枝点的几何描述。

项目成果

T. Sekiguchi & N. Suwa: "A note on extensions of algebraic and formal groups V"(Preprint). (2001)

T·关口

T. Sekiguchi & N. Suwa: "On the unified Kummer-Artin-Schreier- Witt theory"Mathematiques Pures de Bordeaux C.N.R.S., Prepublication. n^0 111. 1-90 (1999)

T·关口

Tsutomu Sekiguchi: "On the unified Kummer-Artin-Schreier-Witt theory"Mathematiques Pures de Bordeaux C.N.R.S., Prepublication. 111. 1-90 (1999)

Tsutomu Sekiguchi：“论统一的 Kummer-Artin-Schreier-Witt 理论”Mathematiques Pures de Bordeaux C.N.R.S.，预出版。

Tsutomu Sekiguchi: "A note on extensions of algebraie and formal groups, V"Preprint. 1-58 (2001)

Tsutomu Sekiguchi：“关于代数和形式群的扩展的注释，V”预印本。

Tsutomu Sekiguchi: "On the unification of Kummer and Artin-Schreier-Witt theories"KIMS Kokyuriku, Alyelorau runnier theoy and related topics. 1200. 1-12 (2001)

Tsutomu Sekiguchi：“关于 Kummer 和 Artin-Schreier-Witt 理论的统一”KIMS Kokyuriku、Alyelorau runnier 理论及相关主题。