The project will focus on three related areas of research in algebraic geometry over finite fields. First of all we intend to attack Artin's conjecture that the Brauer group of a projective surface over a finite field is finite. Here do Jong will study especially elliptic surfaces and surfaces with ample cotangent bundle. Secondly, de Jong will study the arithmetic fundamental groups of curves over finite fields. Here the main questions are concerning the dynamic of the Verschiebung on the moduli spaces of bundles, the growth of the fundamental group, and the distribution of the Frobenius elements in the fundamental group. And thirdly, in an ongoing collaboration with others (Starr, Hassett, Tschinkel, et al) de jong will study the geometry of moduli spaces of rational curves on higher dimensional varieties. Here of particular interested are in finding applications to other areas of research. Meanwhile, with the goal of making it easier for graduate students and newcomers to work on these problems, de jong intends to run a web-site where collaborative development of introductory texts on the topics is done.
The area of mathematics that this proposal finds itself in has seen a lot of progress in the last decade. Nonetheless there are many important problems outstanding. Perhaps the most exiting of these is Artin's conjecture mentioned above. Among other things it implies the Birch-Swinnerton-Dyer conjecture for elliptic curves over function fields of curves over finite fields. An example of such a field is the field of rational functions in one variable over a finite field. Many famous classical number theoretical questions have their analogue for such function fields, and a number of these, such as the Riemann Hypothesis, have been shown to be true in the function field case. The reason for this is that people can study curves and more generally do algebraic geometry over finite fields, to prove the conjectures. In this project we will study geometric approaches to Artin's conjecture, for example by thinking about moduli of vector bundles over curves and surfaces over finite fields.
