Birational geometry: subgroups of the Cremona groups and their generators

双有理几何：克雷莫纳群的子群及其生成元

• 批准号: 15F15751
• 负责人: 向井 茂
• 金额: $ 0.96万
• 依托单位: Kyoto University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for JSPS Fellows
• 财政年份: 2015
• 资助国家: 日本
• 起止时间: 2015-10-09 至 2018-03-31
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-15F15751/
• 关键词: クレモナ変換群; アフィン代数幾何学; 代数学

The decomposition group Dec(C) of a curve C, i.e. the subgroup of the Cremona group Bir(P^2) which preserve the curve C, is generated by quadratic elements in case C is a plane rational curve of degree 1,2 or 3. For every d which is at least 4, there is a plane rational curve C of degree d such that Dec(C) is not generated by its quadratic elements. This joint work with T. Ducat and S. Zimmermann has been accepted for publication in Mathematical Research Letters.In my joint work with A. Dubouloz and T. Kishimoto, we establish basic properties of Ga-threefolds whose algebraic quotient morphism is of a particularly simple form. Here Ga denotes the additive group over the base field, and a Ga-threefold is a variety of dimension 3 with a Ga-action. In particular we give a complete classification of the subclass of Ga-threefolds consisting of threefolds X endowed with proper Ga-actions, whose algebraic quotient morphisms are surjective with degenerate fibres isomorphic to the affine plane A^2 when equipped with their reduced structures. This work has been submitted to the journal Annali della Scuola Normale Superiore di Pisa (on October 2, 2017), and is currently under review.We (Heden and Mukai) studied the decomposition group of 5 lines in the projective plane and found 15 quadratic transformations in the group. I (Heden) later found new ones. By this discovery the solution becomes much harder than the case of 6 lines, for which Mukai determined the decomposition group completely.

曲线C的分解群Dec（C），即Cremona群Bir（P^2）的保持曲线C的子群，在C为1、2或3次平面有理曲线的情况下，由二次元生成.对于任何d至少为4，存在一条d次平面有理曲线C，使得Dec（C）不是由它的二次元素生成的。这与T。Ducat和S.齐默尔曼已被接受发表在数学研究快报。Dubouloz和T. Kishimoto，我们建立了Ga-三重的基本性质，其代数商态射是一个特别简单的形式。这里Ga表示基场上的加性群，Ga-三重是具有Ga-作用的3维变化。特别地，我们给出了由赋予适当Ga作用的三重X构成的Ga-三重子类的完全分类，当赋予它们的约化结构时，它们的代数商态射是满射的，并且退化纤维同构于仿射平面A^2。这项工作已提交给Annali della Scuola Normale Quarterore di比萨杂志（2017年10月2日），目前正在审查中。我们（Heden和Mukai）研究了5条直线在射影平面上的分解群，在群中找到了15个二次变换。 后来，我又找到了新的。 通过这个发现，解决方案变得比6条线的情况困难得多，Mukai完全确定了分解群。

项目成果

ウプサラ大学数学教室(スウェーデン)

乌普萨拉大学数学系（瑞典）

Algebraic additive actions on threefolds

三重的代数加法作用

• 发表时间: 2016
• 作者: Heden;Isac
• 通讯作者: Isac

Russell’s hypersurface from a geometric point of view

从几何角度看罗素超曲面

• 发表时间: 2016
• 期刊: Osaka J. Math.
• 作者: Heden;Isac
• 通讯作者: Isac

On the Makar-Limanov invariant of certain affine hypersurfaces

关于某些仿射超曲面的 Makar-Limanov 不变量

• 发表时间: 2017
• 作者: Heden;Isac
• 通讯作者: Isac

The group of Cremona transformations generated by the standard and linear maps

由标准映射和线性映射生成的克雷莫纳变换组

• 发表时间: 2015
• 作者: Heden;Isac;Isac Heden;Isac Heden
• 通讯作者: Isac Heden