Research

My research focuses on rational points, a topic at the crossroads of algebraic geometry and number theory. I study both geometric and analytic notions of abundance for rational points on varieties. In my research, I employ tools from algebraic geometry, topology, analysis, Galois theory and number theory.

Hilbert property: I began my PhD by studying the Hilbert property, which is possessed by varieties whose sets of rational points are not thin. In a sense, this means that their rational points come from solving non-linear equations over the ground field, which is rare over arithmetically interesting fields.

In [1], I showed that double conic bundles and certain del Pezzo surfaces (along with some higher-dimensional analogues) possess this property. This is connected to the inverse Galois problem, which asks if any finite group can be recognised as the Galois group of some field extension.

In [9] (joint work with Julian Demeio and Rosa Winter), we show that a non-thin family of del Pezzo surfaces of degree one and Picard rank one each satisfy the Hilbert property. These are in some sense the most arithmetically complicated del Pezzo surfaces and admit a close connection to elliptic surfaces; indeed, our proof proceeds via a result on the Hilbert property for so-called "elliptic families".

Weak approximation: Another geometric indicator of the abundance of rational points on a variety is weak approximation, a property possessed by varieties whose rational points over a number field are dense in the local points over all completions of the ground field. This property may be viewed as a geometric generalisation of the Chinese remainder theorem, and results in this area provide striking examples of the interplay between the "local" and "global" realms.

In [3] (joint work with Julian Demeio) we extended the methods used in [1] and combined them with a result of Denef on arithmetic surjectivity to prove that many del Pezzo surfaces of low degree satisfy a weak version of weak approximation, appropriately named weak weak approximation (WWA). This result offers support to a conjecture of Colliot-Thélène that all smooth unirational varieties satisfy WWA.

In [6] (joint work with Julian Demeio and Rosa Winter), we extended geometric constructions used to prove unirationality of del Pezzo surfaces in order to prove WWA for all degree-two del Pezzo surfaces with a general rational point.

Semi-integral points: I am particularly interested in the theory of semi-integral (Campana and Darmon) points, which forms a bridge between integral and rational points and provides a geometric perspective on solutions to equations with restricted prime factorisations. A Manin-type conjecture for Campana points of bounded height, henceforth the PSTVA conjecture, was put forward in 2019 by Pieropan, Smeets, Tanimoto and Várilly-Alvarado.

In [2], I established PSTVA-type asymptotics on orbifolds associated to norm forms. This can be interpreted as an asymptotic for the number of powerful values of a norm form. The height zeta function approach I employed, modeled on the work of Batyrev—Tschinkel and Loughran, draws on both harmonic analysis and the theory of L-functions associated to Hecke characters.

In [4] (joint work with Masahiro Nakahara), we explored the Hilbert property and weak approximation for Campana points, bringing together the seemingly disparate topics studied in my first two papers. Our work was the first to explore this topic in detail, despite its relevance to the PSTVA conjecture, and we prove some foundational results.

In [5] (joint work with Masahiro Nakahara and Vlad Mitankin), we developed a semi-integral analogue of the Brauer-Manin obstruction and determined the status of local-global principles for orbifolds associated to quadric hypersurfaces. 

In [7] (joint work with Alec Shute), we verified the PSTVA conjecture (upon replacing the leading constant by a later prediction of Chow, Loughran, Takloo-Bighash and Tanimoto) for smooth toric orbifolds.

Geometry of del Pezzo surfaces: Like my former advisor Daniel Loughran, I have a soft spot for del Pezzo surfaces.

In [8] (joint work with Enis Kaya, Stephen McKean and H. Uppal), we are working on generalising the famous Cayley—Salmon theorem that any smooth projective cubic surface contains exactly 27 lines. We generalise to arbitrary fields, del Pezzo surfaces of arbitrary degree and extend to conic families.