Algebraic geometry over finite fields

点击翻译按钮获取中文摘要

The project will focus on three related areas of research in algebraicgeometry over finite fields. First of all we intend to attack Artin'sconjecture that the Brauer group of a projective surface over a finitefield is finite. Here do Jong will study especially elliptic surfaces andsurfaces with ample cotangent bundle. Secondly, de Jong will study thearithmetic fundamental groups of curves over finite fields. Here themain questions are concerning the dynamic of the Verschiebung on themoduli spaces of bundles, the growth of the fundamental group, and thedistribution of the Frobenius elements in the fundamental group. Andthirdly, in an ongoing collaboration with others (Starr, Hassett,Tschinkel, et al) de jong will study the geometry of moduli spaces ofrational curves on higher dimensional varieties. Here ofparticular interested are in finding applications to other areas ofresearch. Meanwhile, with the goal of making it easier for graduatestudents and newcomers to work on these problems, de jong intends to run aweb-site where collaborative development of introductory texts on thetopics is done.The area of mathematics that this proposal finds itself in has seen alot of progress in the last decade. Nonetheless there are manyimportant problems outstanding. Perhaps the most exiting of these isArtin's conjecture mentioned above. Among other things it implies theBirch-Swinnerton-Dyer conjecture for elliptic curves over functionfields of curves over finite fields. An example of such a field is thefield of rational functions in one variable over a finite field. Manyfamous classical number theoretical questions have their analogue forsuch function fields, and a number of these, such as the RiemannHypothesis, have been shown to be true in the function field case. Thereason for this is that people can study curves and more generally doalgebraic geometry over finite fields, to prove the conjectures. Inthis project we will study geometric approaches to Artin's conjecture,for example by thinking about moduli of vector bundles over curves andsurfaces over finite fields.