AbstractThis thesis concerns two topics on the frontiers of research in arithmetic geometry, namely thin sets and Campana points, and their intersection.
To begin with, we investigate the Hilbert property, a geometric indicator of the abundance of rational points on an algebraic variety. In this area, we prove that certain conic bundles and del Pezzo surfaces (along with their higher-dimensional analogues) satisfy the Hilbert property, meaning that their set of rational points is not thin. These results support a conjecture of Colliot-Thélène that unirational varieties over number fields have the Hilbert property.
We proceed (in joint work with Julian Demeio) to explore the stronger notion of weak approximation for del Pezzo surfaces of low degree. We combine our geometric
methods for verifying the Hilbert property with results on arithmetic surjectivity, illustrating the connections not only between geometry and arithmetic, but between the so-called "local" and "global" realms.
We then move on to the study of Campana points, an idea which brings together rational and integral points in one cohesive framework and admits applications to
problems of number-theoretic interest. We establish asymptotic formulae for Campana points on certain toric varieties, and in doing so we derive asymptotics for powerful values of norm forms for extensions of number fields.
Finally, in joint work with Masahiro Nakahara, we combine these theories and initiate the study of weak approximation and the Hilbert property for Campana points,
both of which are topics of great significance in relation to Manin-type conjectures for Campana points, where one seeks asymptotics outside a thin set.
|Date of Award
|1 Nov 2021
|Daniel Loughran (Supervisor) & Alastair King (Supervisor)