Density of rational points on a family of del Pezzo surfaces of degree $1$

Abstract: Del Pezzo surfaces are classified by their degree d, which is an integer between $1$ and $9$ (for $d ≥ 3$, these are the smooth surfaces of degree $d$ in $\mathbb{P}^d$). For del Pezzo surfaces of degree at least $2$ 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 $1$ over a field k, even though we know that they always contain at least one $k$-rational point, we do not know if the set of $k$-rational points is Zariski dense in general. I will 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 a number field. I will compare this to previous results.

algebraic geometrynumber theory

Audience: researchers in the topic

( slides )

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SFU NT-AG seminar

Series comments: The Number Theory and Algebraic Geometry (NT-AG) seminar is a research seminar dedicated to topics related to number theory and algebraic geometry hosted by the NT-AG group (Nils Bruin, Imin Chen, Stephen Choi, Katrina Honigs, Nathan Ilten, Marni Mishna).

We acknowledge the support of PIMS, NSERC, and SFU.

For Fall 2025, the organizers are Katrina Honigs and Peter McDonald.

We normally meet in-person in the indicated room. For online editions, we use Zoom and distribute the link through the mailing list. If you wish to be put on the mailing list, please subscribe to ntag-external using lists.sfu.ca

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