Ritabrata Munshi (Rutgers)
Rational Points on Surfaces
In late 1980's Manin et al put forward a precise conjecture
about the density of rational points on Fano varieties. Over the last two
decades some progress has been made towards proving this conjecture. But
the conjecture is far from being proved even for the case of two
dimensional Fano varieties or del Pezzo surfaces. These surfaces are
geometrically
classified according to `degree', and the geometric, as well as, the
arithmetic complexity increases as the degree drops. The most interesting
cases of Manin's conjecture for surfaces are degrees four and lower. In
this talk I will mainly focus on the arithmetic of these del Pezzo
surfaces, and report some of my own results (partly joint with Henryk
Iwaniec). I will also talk about some other problems which apparently have
a different flavor but, nonetheless, are directly related with the problem
of rational points on surfaces.