Density of rational points on a family of del Pezzo surface of degree 1

Abstract: Let $k$ be a number field and $X$ an algebraic variety over $k$. We want to study the set of $k$-rational points $X(k)$. For example, is $X(k)$ empty? If not, is it dense with respect to the Zariski topology? Del Pezzo surfaces are classified by their degrees $d$ (an integer between 1 and 9). Manin and various authors proved that for all del Pezzo surfaces of degree $d>1$, $X(k)$ is dense provided that the surface has a $k$-rational point (that lies outside a specific subset of the surface for $d=2$). For $d=1$, the del Pezzo surface always has a rational point. However, we don't know if the set of rational points is Zariski-dense. In this talk, I present a result that is joint with Rosa Winter in which we prove the density of rational points for a specific family of del Pezzo surfaces of degree 1 over $k$.

number theory

Audience: researchers in the discipline

POINT: New Developments in Number Theory

Series comments: There will be two 20-minute contributed talks during each meeting aimed at a general number theory audience, including graduate students.

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