The explicit Bombieri-Lang conjecture for surfaces

本课题主要研究数域K上一般型变种X上K个有理点的集合X(K)。如果X是曲线，则当且仅当X是至少为2属的曲线时，则X是一般型的。对于这些曲线，Mordell猜想和Faltings证明了X(K)总是有限的。这个命题是（Bombieri-）Lang猜想弱形式的一维情况：如果X是定义在数字域K上的各种一般类型，那么X(K)在X中不是Zariski密集的。根据Faltings的结果，它们适用于曲线，更普遍地适用于阿贝尔变量的子变种。然而，即使在这些情况下，也远不清楚是否可以确定X(K)的合适的明确描述。就曲线而言，近年来已经取得了相当大的进展，但对于曲面，情况还相当开放。对于一般型曲面，该猜想断言在不超过1条几何属的有限条曲线之外，只有有限个有理点。本项目旨在一方面证明某些类型表面的猜想，另一方面开发确定X(K)的显式方法。Bauer是一般型曲面及其模的知名专家，而Stoll是曲线的算术和算法方法的权威专家。本项目旨在结合这些专业知识，在pi的共同监督下，为开展与一般类型表面有理点研究相关的博士和博士后研究项目提供理想的环境。

The main focus of this project is the study of the set X(K) of K-rational points on a variety X of general type over a number field K. If X is a curve, then X is of general type if and only if X is a curve of genus at least 2. For these curves, Mordell conjectured and Faltings proved that X(K) is always finite. This statement is the one-dimensional case of the weak form of the (Bombieri-)Lang Conjecture:If X is a variety of general type defined over a number field K, then X(K) is not Zariski dense in X.In general this conjectures is wide open. By results of Faltings, they hold for curves and more generally for subvarieties of abelian varieties.However, even in these cases it is far from clear that a suitable explicit description of X(K) can be determined. In the case of curves, there has been considerable progress in recent years, but for surfaces the situation is quite open. For a surface of general type, the conjecture asserts that outside a finite number of curves of geometric genus at most 1, there are only finitely many rational points.This project aims at proving the conjecture for certain types of surfaces on the one hand and at the development of explicit methods to determine X(K) on the other hand.Bauer is a well-known expert on surfaces of general type and their moduli, whereas Stoll is a leading expert on the arithmetic of curves and algorithmic methods. The proposed project aims at combining this expertise to provide an ideal environment for carrying out PhD and postdoctoral research projects related to the study of rational points on surfaces of general type under the joint supervision of the PIs.