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Verifying Zariski density of rational points on del Pezzo surfaces of degree 1 van Luijk, Ronald
Description
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.
Item Metadata
Title |
Verifying Zariski density of rational points on del Pezzo surfaces of degree 1
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Creator | |
Publisher |
Banff International Research Station for Mathematical Innovation and Discovery
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Date Issued |
2018-05-30T09:00
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Description |
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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Extent |
32.0
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Type | |
File Format |
video/mp4
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Language |
eng
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Notes |
Author affiliation: Universiteit Leiden
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Series | |
Date Available |
2019-03-14
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Provider |
Vancouver : University of British Columbia Library
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Rights |
Attribution-NonCommercial-NoDerivatives 4.0 International
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DOI |
10.14288/1.0376868
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URI | |
Affiliation | |
Peer Review Status |
Unreviewed
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Scholarly Level |
Researcher
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Rights URI | |
Aggregated Source Repository |
DSpace
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Rights
Attribution-NonCommercial-NoDerivatives 4.0 International