# Counting rational points on a cubic surface

# Counting rational points on a cubic surface

A conjecture of Manin predicts precise asymptotic for the density of rational points on del Pezzo surfaces. This has been satisfactorily settled for del Pezzo surfaces of higher degree. But for lower degree not much is known. At present, most research in the field is concentrated around del Pezzo surfaces of degree four (intersections of two quadratics in $\mathbb P4$), and degree three (cubic surfaces). Although many special cases of singular cubic and quartic surfaces have been successfully dealt with, it is generally believed that the problem is much harder for smooth surfaces and for surfaces with `mild' singularities. In this talk I will describe a joint work with Iwaniec, where we use sieves and methods from analytic number theory to prove an almost sharp lower bound for the density of rational points in case of a family of cubic surfaces.