This proposal concerns solutions to systems of polynomial equations or, equivalently, rational points on algebraic varieties. This area of mathematics has seen exciting developments in recent years, and the Netherlands is home to a group of energetic young researchers at the forefront of these developments. This proposal will kickstart a strong and durable research community around these individual scientists. 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 our proposal lies. We have identified three interrelated research themes comprising three projects each. The first theme, “from curves to surfaces and beyond”, 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 will investigate its application to certain classes of surfaces of general type. Algebraic geometry codes arising from curves are well established; we will build on recent constructions in the geometry of surfaces to produce good locally recoverable codes on surfaces. The third project is to investigate jumping of Mordell–Weil ranks in families of abelian varieties, generalising existing results on families of elliptic curves. The second theme, “from characteristic zero to characteristic p, and back”, 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 will 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 will study the reduction and lifting properties of abelian varieties of dimension at least two. The third project under this theme is to study the density of rational points within the p-adic or even adelic points on del Pezzo and K3 surfaces. Our third theme is “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 will develop a toolbox to test the conjecture in fundamental examples. Secondly, we will 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 will investigate the Hilbert property for Campana points, building on the latest covering techniques. These proposed projects are to be undertaken by six PhD students and three postdocs. While the projects themselves are independent, they share mathematical background and technical tools and will therefore benefit from extensive interaction and collaboration. To further stimulate co-operation we will hold monthly meetings for learning, discussion and collaboration. We will also organise an instructional workshop early in the programme, and an international conference towards the end of the programme.