Studies on Diophantine Geometry and Arakelov geometry

丢番图几何与阿拉克洛夫几何研究

• 批准号: 17F17730
• 负责人: 森脇 淳
• 金额: $ 1.41万
• 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: 2017
• 资助国家: 日本
• 起止时间: 2017-10-13 至 2020-03-31
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-17F17730/
• 关键词: Diophantine Geometry; Arakelov Geometry; ディオファントス幾何; アラケロフ幾何

Chunhui Liu obtained fruitful results on counting rational points by the determinant method. During the participation of the thematic activity "Reinventing rational points" in IHP during May and June 2019, he had lots of effective communications with some experts on rational points, and finally he had a significant improvement on his understand to the density of rational points in arithmetic varieties. In his preprint "Determinant method and the pseudo-effective threshold" (arxiv: arxiv:1910.00306), he explicated a connection between the positivity of certain line bundles and the density of rational points, which seems to have a large potential application in the future. For example, we have known a lot on the pseudo-effective threshold on certain line bundles on some particular varieties, and these results can be applied to study their density of rational points. He was also writing another paper on the similar area, which will be available quite soon. Besides these, he also realized that the study of certain heights of points on Hilbert schemes will be useful to the application of determinant. Some auxiliar results were also obtained during the research process, and they deserve to be published as several small papers. I believe that once he accomplishes these subjects, we will have a novel understand to the quantitative arithmetic and rational points. Besides the work on rational points, he went on studying the Diophantine approximation over arithmetic function fields. I believe he will produce an excellent result in this area.

刘春辉用行列式方法计算有理点，取得了丰硕的成果。 2019年5月、6月参加IHP“重塑理性点”主题活动期间，与一些专家就理性点进行了多次有效的交流，最终使他对理性点密度的理解有了明显的提高。算术品种。在他的预印本“行列式方法和伪有效阈值”（arxiv：arxiv：1910.00306）中，他阐述了某些线束的正性与有理点密度之间的联系，这似乎在未来有很大的潜在应用。例如，我们对某些特定品种的某些线束的伪有效阈值已经了解很多，这些结果可以用来研究它们的有理点密度。他还在撰写另一篇关于类似领域的论文，该论文很快就会发布。除此之外，他还认识到研究希尔伯特格式上某些点的高度对于行列式的应用将是有益的。研究过程中也取得了一些辅助成果，值得以几篇小论文的形式发表。我相信，一旦他完成了这些科目，我们将对定量算术和理性点有一个新的理解。除了有理点方面的工作外，他还继续研究算术函数域上的丢番图近似。相信他会在这方面取得优异的成绩。

项目成果

Extreme-Scale Stochastic Particle Tracing for Uncertain Unsteady Flow Visualization and Analysis

• DOI: 10.1109/tvcg.2018.2856772
• 发表时间: 2019-09
• 期刊: IEEE Transactions on Visualization and Computer Graphics
• 影响因子: 5.2
• 作者: Hanqi Guo;Wenbin He;Sangmin Seo;Han-Wei Shen;E. Constantinescu;Chunhui Liu;T. Peterka

Counting rational points in arithmetic varieties by the determinant method

用行列式方法计算算术簇中有理点

• 发表时间: 2019
• 作者: Hanqi Guo;Wenbin He;Sangmin Seo;Han-Wei Shen;Emil Mihai Constantinescu;Chunhui Liu and Tom Peterka;Chunhui Liu
• 通讯作者: Chunhui Liu