Symmetries of singular del Pezzo surfaces in algebraic and arithmetic geometry

代数几何的一个中心任务是对代数簇进行分类，即多项式方程组的解集。数论中最古老的问题之一是丢番图方程的有理解问题；在代数几何的语言中，这意味着研究代数簇上有理点的存在和分布问题。本课题研究了(可能是奇异的)del Pezzo曲面这两个问题；这些曲面在代数簇的分类理论中扮演着重要的角色。从数论的观点来看，它们很有趣，因为有理点的分布是由马宁的猜想准确预测的。我们将考察del Pezzo曲面的对称性，发展确定其自同构群的方法，对具有丰富对称性的del Pezzo曲面进行分类，并研究其有理点。Manin猜想方法的选择取决于对称性：在有限自同构群的情况下，调和分析的使用是有希望的；否则，通过由Cox环明确描述的泛躯干结合解析数论的方法是合适的。

A central task of algebraic geometry is the classification of algebraic varieties, i.e., solution sets of systems of polynomial equations. One of the oldest problems in number theory is the question of rational solutions of diophantine equations; in the language of algebraic geometry, this meansstudying the question of existence and distribution of rational points on algebraic varieties. This project takes up both questions for the class of (possibly singular) del Pezzo surfaces; these play a central role as basic objects in the classification theory of algebraic varieties. They are interesting from a number theoretic point of view since the distribution of rational points is precisely predicted by Manin's conjecture. We will examine symmetries of del Pezzo surfaces, develop methods to determine their automorphism groups, classify del Pezzo surfaces with rich symmetries and study their rational points.Our approach to determine and describe automorphism groups is based on Cox rings. The choice of approach to Manin's conjecture depends on the symmetries: In case of in finite automorphism groups, the use of harmonic analysis is promising; otherwise an approach via universal torsos, which are described explicitly by Cox rings, in combination with analytic number theory is suitable.