Number Theory Seminar

Title: Density of the rational points on a family of del Pezzo surfaces of degree 1

Abstract: Let X be an algebraic variety over a number field k. We want to study the set of k-rational points X(k). For example, is X(k) empty? If not, is it dense with respect to the Zariski topology? Del Pezzo surfaces are classified by their degrees d, an integer between 1 and 9. Manin and various authors proved that for all del Pezzo surfaces of degree >1 is dense provided that the surface has a k-rational point (that lies outside a specific subset of the surface for d=2). For d=1, the del Pezzo surface always has a rational point. However, we don't know in general if the set of rational points is Zariski-dense. In this talk, I present a result, joint with Rosa Winter, in which we prove the density of rational points for an interesting family of del Pezzo surfaces of degree 1 over k.