The explicit Bombieri-Lang conjecture for surfaces
Final Report Abstract
This is a project in the context of rational points on algebraic varieties. An algebraic variety is the set of points whose coordinates satisfy one or several algebraic equations. Such a point is rational, if its coordinates are rational numbers. Algebraic varieties can be roughly classified by their dimension: 1 for curves, 2 for surfaces, and so on. While we have a pretty good understanding of the structure of the set of rational points on algebraic curves, much less is known about the set of rational points on a surface. There is a conjectural description that applies to so-called surfaces of general type, which is known to hold in some cases, but not generally. However, the proof that applies to these cases does not lead to an algorithm that would provide a concrete description of the set of rational points for a given surface. The goal of the project is to come up with a method that can give such a concrete description at least in some cases, for surfaces from a certain family. This description has two parts. One part lists finitely many curves on the surface (that can contain infinitely many rational points); the other part lists the remaining finitely many points. We have a complete solution for the first part (for the family of surfaces we studied) and a method for the second part that works in many cases, leading to a number of concrete examples, for which we are able to completely determine their rational points.
