# Number Theory Seminar

Speaker: Rosa Winter (Leiden)

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

Abstract:

Let X be an algebraic variety. We want to study the set of rational points $X(\mathbb{Q})$. For example, is $X(\mathbb{Q})$ empty? And if not, is it dense with respect to the Zariski topology? Del Pezzo surfaces are surfaces that are classified by their degree d (for d different from 8, over an algebraically closed field they are isomorphic to the blow up of $\mathbb{P}^2$ in 9-d points). For all del Pezzo surfaces of degree at least 2, we know that the set of rational points is dense provided that the surface has one rational point to start with (that lies outside a specific subset of the surface for degree 2). But for del Pezzo surfaces of degree 1, even though we know that they always contain at least one rational point, we do not know if the set of rational points is dense. In this talk I will focus on one of my results, which states that for a specific family of del Pezzo surfaces of degree 1, under a mild condition, the rational points are dense with repsect to the Zariski topology. I will compare this to previous results. This is joint work with Julie Desjardins.