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Fundamental groups of algebraic varieties

代数簇的基本群

基本信息

批准号：
14540053
负责人：
SHIMADA Ichiro
金额：
$ 2.18万
依托单位：
HOKKAIDO UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2002
资助国家：
日本
起止时间：
2002 至 2004
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14540053/
关键词：
tundamental group singularity lattice code K3 surface algebraic surface Galocs cover 柊子代数多様体 6次曲線

项目摘要

We generalized the classical Zariski-van Kampen theorem on the fundamental group of an open algebraic variety. As an application, we obtained a hyperplane-section theorem of Zariski type for Grassmannian varieties, and revealed a subtle relation between fundamental groups and Chow forms. As another application, we calculated the fundamental group of the complement to the discriminant variety of a sufficiently ample line bundle on a compact Riemann surface.Hoping to verify the generalized Hodge conjecture for certain varieties, we investigated the cylinder homomorphism associated with a family of algebraic cycles, and gave a sufficient condition for the image of the cylinder homomorphism contains the module of vanishing cycles. Using Grobner bases, we proved the generalized Hodge conjecture for certain Fano complete intersections.We made the complete list of maximal configurations of rational double points on supersingular K3 surfaces by means of lattice theory and heavy use of computers. The complete list of extremal (quasi-) elliptic fibrations on supersingular K3 surfaces was also obtained. As a corollary, it was shown that every supersingular K3 surface in characteristic 2 is birational to a purely inseparable double cover of a projective plane, which has 21 ordinary nodes. We described the configuration of the 21 nodes in terms of certain binary codes of length 21. By the same method, we also showed that every supersingular K3 surface in odd characteristic is also birational to a double cover of a projective plane.

本文推广了经典的关于开代数簇的基本群的Zapriki-van坎彭定理。作为应用，我们得到了Grassmannian簇的Zapriki型超平面截口定理，并揭示了基本群与Chow型之间的微妙关系。作为另一个应用，我们计算了紧致Riemann曲面上充分充量线丛的判别簇的补的基本群，为了验证某些簇的广义Hodge猜想，我们研究了与代数圈族相关的柱同态，给出了柱同态的象包含消失圈模的一个充分条件.利用Grobner基证明了某些Fano完全交的广义Hodge猜想，并利用格理论和大量的计算机，给出了超奇异K3曲面上有理二重点的极大构型的完整列表.并得到了超奇异K3曲面上极值（拟）椭圆纤维化的完整列表。作为推论，证明了特征为2的超奇异K3曲面对于射影平面的一个纯不可分的二重覆盖是双有理的，该二重覆盖有21个普通结点。我们用长度为21的某些二进制码来描述21个节点的配置。用同样的方法，我们还证明了每一个奇特征的超奇异K3曲面对于射影平面的双复盖也是双有理的。

项目成果

期刊论文数量（27）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

Vanishing cycles, the generalized Hodge conjecture and Grobner bases.

消失循环、广义霍奇猜想和格罗布纳基础。

DOI：
发表时间：
2004
期刊：
Geometric singularity theory, Banach Center Publ. (Polish Acad.Sci., Warsaw) 65
影响因子：
0
作者：
Hara;Tomio;Yuji Kobayashi;Ichiro Shimada
通讯作者：
Ichiro Shimada

Supersingular K3 surfaces in characteristic 2 as double covers of projective plane

特征2中的超奇异K3曲面作为射影平面的双覆盖

DOI：
发表时间：
2004
期刊：
Asian J.Math. 8
影响因子：
0
作者：
Tanaka;Ryuichi;Y.Kobayashi;Ichiro Shimada;Yoshiaki Shoji;Y.Kobayashi;Tamio Hara;Ichiro Shimada;Ichiro Shimada
通讯作者：
Ichiro Shimada

On the Zariski-van Kampen theorem.

关于 Zariski-van Kampen 定理。