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: Description: Research/departmental seminar

Seminar usually meets in-person. For online editions, the Zoom link is shared via mailing list.

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