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Verifying Zariski density of rational points on del Pezzo surfaces of degree 1 van Luijk, Ronald


Let $S$ be a del Pezzo surface of degree $1$ over a number field $k$. The main goal of this talk is to give easily verifiable sufficient conditions under which its set $S(k)$ of rational points is Zariski dense. It is well known that almost all fibers of the anticanonical map $\varphi \colon S \dashrightarrow \mathbb{P}^1$ are elliptic curves with the unique base point of $\varphi$ as zero. Suppose that $P \in S(k)$ is a point of finite order $n>1$ on its fiber. Then there is another elliptic fibration on the blow-up of $S$ at $P$. We will see where it comes from and how it can be used to define a proper closed subset $Z \subset S$ such that (1) it is easy to verify for any point on $S$ whether it lies in $Z$, and (2) the set $S(k)$ contains a point outside $Z$ if and only if $S(k)$ is Zariski dense. In other words, if $S(k)$ is Zariski dense, then we can prove this by exhibiting a rational point outside $Z$. We will also compare this to previous work. This is joint work with Jelle Bulthuis inspired by an example of Noam Elkies.

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