Density of rational points on 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 $d\geq3$, these are the smooth surfaces of degree $d$ in $\mathbb{P}^d$). 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 $\mathbb{Q}$. I will compare this to previous results.

number theory

Audience: researchers in the topic

London number theory seminar

Series comments: For reminders, join the (very low traffic) mailing list at mailman.ic.ac.uk/mailman/listinfo/london-number-theory-seminar

For a record of talks predating this website see: wwwf.imperial.ac.uk/~buzzard/LNTS/numbtheo_past.html