Rosa Winter, Max Planck Institute for Mathematics in the Sciences (Leipzig)

February 19th, 2021 4:00-5:00 pm CET

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

Abstract : Del Pezzo surfaces are surfaces classified by their degree $d$, which is an integer between 1 and 9 (for $d$≥$3$, these are the smooth surfaces of degree $d$ in ℙ$\stackrel{d}{}$). For del Pezzo surfaces of degree at least $2$ over a field $k$, we know that the set of $k$-rational points is Zariski dense provided that the surface has one $k$-rational point to start with (that lies outside a specific subset of the surface for degree $2$). However, for del Pezzo surfaces of degree 1 over a field $k$, even though we know that they always contain at least one $k$-rational point, we do not know if the set of $k$-rational points is Zariski dense in general. I will talk about a result that is joint work with Julie Desjardins, in which we give sufficient conditions for the set of $k$-rational points on a specific family of del Pezzo surfaces of degree 1 to be Zariski dense, where $k$ is any infinite field of characteristic 0. These conditions are necessary if $k$ is finitely generated over ℚ. I will compare this to previous results.