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Session I.2 - Computational Number Theory

Monday, June 12, 14:30 ~ 15:00

Rational points and intersecting lines on del Pezzo surfaces

Rosa Winter

King's College London, United Kingdom   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

Del Pezzo surfaces are 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 $\mathbb{P}^d$). Over algebraically closed fields they are rational, and contain a fixed number of `lines' (exceptional curves), depending on $d$. The set of rational points over non-algebraically closed fields is not fully understood, with more open questions as $d$ goes down. A long-standing open problem is whether every del Pezzo surface of degree 1 has a dense set of rational points. Partial results are known, and often, the configuration of the lines on the surface plays a role in these results. In this talk I will show how the lines come in to play, and go over several computational results on the configuration of the 240 lines on a del Pezzo surface of degree 1. This is based on joint results, as well as work in progress, with Julie Desjardins, Yu Fu, Kelly Isham, and Ronald van Luijk.

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