## Conference "Arithmetic, Geometry and Coding Theory"## March 14 - 18, 2011## CIRM, Marseille, France |

**Organisers:** Yves Aubry *(Université de la Méditerranée, IML)*, Christophe Ritzenthaler *(Université de la Méditerranée, IML)*, Alexey Zykin *(Mathematical Department of HSE, Laboratoire Poncelet, IITP)*

Let k be a non-algebraically closed field and X be a surface defined over k. An interesting problem is to know whether the set of k-rational points X(k) is Zariski dense in X. A lot of research is done in this field but, surprisingly, this problem is not completely solved for the simplest class of surfaces, the rational, where one expects a positive answer. In this lecture I will focus on del Pezzo surfaces, a important subclass of rational surfaces. I will talk about the cases already treated (mainly by Manin), as well as the two cases left open, the del Pezzo surfaces of degrees one and two, presenting recent results (in progress) in the field.