Algebraic geometry over finite fields

该项目将集中于有限域上代数几何研究的三个相关领域。首先，我们要反驳Artin关于有限域上射影曲面的Brauer群是有限的猜想。在这里，Do Jong将特别研究椭圆曲面和具有充足余切丛的曲面。其次，De Jong将研究有限域上曲线的算术基本群。这里的主要问题是丛的模空间上Verschiebung的动态，基本群的增长，以及基本群中Frobenius元的分布。第三，在与其他人(Starr，Hassett，Tschinkel等)的持续合作中，De Jong将研究高维簇上有理曲线的模空间的几何。这里特别感兴趣的是寻找在其他研究领域的应用。与此同时，为了让毕业生和新人更容易解决这些问题，德容打算运营一个网站，在那里合作开发关于拓扑学的导论文本。这项提议所在的数学领域在过去十年里取得了很大进展。尽管如此，仍然有许多重要的问题悬而未决。其中最令人兴奋的可能是上面提到的Artin猜想。此外，它还包含有限域上曲线的函数域上的椭圆曲线的Birch-Swinnerton-Dyer猜想。这类域的一个例子是有限域上的一元有理函数域。许多著名的经典数论问题都有类似于这样的函数域，其中一些问题，如黎曼假设，已被证明在函数域的情况下是正确的。其原因是，人们可以研究有限域上的曲线和更一般的代数几何，以证明猜想。在这个项目中，我们将研究Artin猜想的几何方法，例如，通过考虑有限域上曲线和曲面上向量丛的模数。

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.