Studies on Diophantine Geometry and Arakelov geometry

刘春晖利用行列式方法计算有理点，取得了丰硕的成果。在2019年5月至6月参加国际水文计划“重塑有理点”主题活动期间，他与有理点专家进行了大量有效的交流，最终对算术品种中有理点密度的理解有了明显的提高。在他的预印本《行列式方法和伪有效阈值》（arxiv：arxiv：1910.00306）中，他解释了某些线丛的正性和有理点的密度之间的联系，这似乎在未来有很大的潜在应用。例如，我们已经知道了一些特定品种上某些线丛的伪有效阈值，这些结果可以用来研究它们的有理点密度。他还在撰写关于类似领域的另一份文件，该文件将很快发表。除此之外，他还认识到研究Hilbert格式上点的某些高度对行列式的应用是有益的。在研究过程中也取得了一些有益的成果，值得作为几篇小论文发表。我相信，一旦他完成了这些科目，我们将有一个新的理解，以数量的算术和理性的观点。除了工作的合理点，他继续研究丢番图逼近算术函数领域。我相信他会在这方面取得出色的成绩。

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.