Arithmetic of Del Pezzo surfaces

Abstract

This survey is an introduction to the birational, qualitative arithmetic of del Pezzo surfaces over global fields: existence of rational points, weak approximation, Zariski density of points. We begin by reviewing the geometry of these surfaces over separably closed fields. We then show that del Pezzo surfaces of degree at least 5 that have a rational point are rational over their ground field, thereby proving they satisfy weak approximation and have a dense set of rational points. Finally, we discuss Brauer-Manin obstructions and give an example of one on a del Pezzo surface of degree 1.