Modul: MAT076 Arbeitsgemeinschaft in Codierungstheorie und Kryptographie

## Rational points and configurations of lines on del Pezzo surfaces of degree one

Talk by Dr. Rosa Winter

**Date:** 21.09.22 **Time:** 16.30 - 17.30 **Room:**

Let $X$ be an algebraic variety over a field $k$. In arithmetic geometry we are interested in the set $X(k)$ of $k$-rational points on $X$. For example, is $X(k)$ empty? And if not, how `large' is $X(k)$? If $k$ is infinite, is $X(k)$ dense in $X$ with respect to the Zariski topology?
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~$\mathbb{P}^d$. They contain a fixed number of `lines' (exceptional curves), depending on their degree. The lower the degree, the more complicated these surfaces become, and the more open questions there are.
I will talk about the density of the set of rational points and the configurations of lines on del Pezzo surfaces of degree 1. I will show why these two topics are interesting to study, what they have to do with each other, and talk about results in both directions.

(**This eSeminar will also be live-streamed on Zoom, using the same meeting details as previous seminars. If you do not have meeting details, please contact simran.tinani@math.uzh.ch **)