Research project

Rational Points: New Dimensions

The 21st century is witnessing a revolution in our understanding of rational points on surfaces and higher-dimensional varieties, and it is in this field that this project lies.

Duration: 2023 - 2028
Contact: Martin Bright
Funding: NWO Open Competition ENW-XL 2021
Partners: Valentijn Karemaker and Marta Pieropan, Utrecht University
Cecília Salgado and Steffen Müller, University of Groningen

A cubic surface with rational points and lines - © Martin Bright

The project concerns solutions to systems of polynomial equations or, equivalently, rational points on algebraic varieties. The project has three interrelated research themes comprising three projects each.

From curves to surfaces and beyond

This theme consists of projects which take established techniques from the study of rational points on curves and extend them to the substantially more difficult setting of higher-dimensional varieties.

Chabauty’s method has proved very fruitful in the algorithmic study of rational points on curves, and we investigate its application to certain classes of surfaces of general type.
Algebraic geometry codes arising from curves are well established; we build on recent constructions in the geometry of surfaces to produce good locally recoverable codes on surfaces.
In the third project we investigate jumping of Mordell–Weil ranks in families of abelian varieties, generalising existing results on families of elliptic curves.

From characteristic zero to characteristic p, and back

This theme looks at various settings in which geometry in characteristic p is related to arithmetic.

The Brauer–Manin obstruction is an important tool for understanding rational points on a variety; we deepen our understanding of it by relating it to the geometry of the variety when reduced modulo primes.
Abelian varieties in characteristic p have consistently received a lot of attention for their theoretical relevance and real-world applications; we study the reduction and lifting properties of abelian varieties of dimension at least two.
In this third project we study the density of rational points within the p-adic or even adelic points on del Pezzo and K3 surfaces.

From rational points to Campana points

Campana points are an emerging area of research in Diophantine geometry, linking rational and integral points on varieties.

Manin’s conjecture was originally conceived for rational points and recently extended to Campana points; we develop a toolbox to test the conjecture in fundamental examples.
Secondly, we develop a theory of local-global principles and Brauer–Manin obstructions for Campana points, bringing together the theories for rational and integral points.
Finally, we investigate the Hilbert property for Campana points, building on the latest covering techniques.

For more information, go to the Rational Points: New Dimensions website.