Rosa Winter

MPI Leipzig

## Time:

Thursday, February 25, 2021 - 10:00am to 11:00am

## Location:

Zoom: https://uci.zoom.us/j/94683355687
Del Pezzo surfaces are surfaces that are classified by their degree $d$, which is an integer between 1 and 9; well-known examples (when $d=3$) are the smooth cubic surfaces in $\mathbb{P}^3$. For del Pezzo surfaces with $d\geq2$ 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 we do not know if the set of $k$-rational points is Zariski dense in general, even though these surfaces always contain a $k$-rational point. This makes del Pezzo surfaces of degree 1 challenging objects. In this talk I will first explain what del Pezzo surfaces are, and show some of their geometric features, focussing on del Pezzo surfaces of degree 1. I will then talk about a result that is joint work with Julie Desjardins, in which we give necessary and 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 field that is finitely generated over $\mathbb{Q}$.