Symmetries of singular del Pezzo surfaces in algebraic and arithmetic geometry

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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.