Algebraic geometry over finite fields

该项目将侧重于有限域上代数几何的三个相关研究领域。首先，我们打算攻击Artin关于有限域上射影曲面的Brauer群是有限的猜想。在这里做钟将研究特别是椭圆曲面和曲面与丰富的余切丛。其次，de Jong将研究有限域上曲线的算术基本群。这里的主要问题是关于丛的模空间上的Verbebung的动力学，基本群的增长，以及Frobenius元素在基本群中的分布。第三，在一个正在进行的合作与他人（斯塔尔，哈西特，Tschinkel，等）德容将研究几何模空间的分数曲线的高维品种。在这里，我们特别感兴趣的是在其他研究领域的应用。与此同时，为了让研究生和新来者更容易解决这些问题，德容打算建立一个网站，在那里合作开发关于这些问题的介绍性文本。这个提议所涉及的数学领域在过去十年中已经取得了很大的进展。尽管如此，仍有许多重要问题悬而未决。也许最令人兴奋的这些isArtin的猜想上述。除其他事项外，它意味着Birch-Swinnerton-Dyer猜想的椭圆曲线上的函数域的曲线在有限域。一个这样的领域的例子是域的有理函数在一个变量在有限域。许多著名的经典数论问题都有类似的功能领域，其中一些，如黎曼假设，已被证明是真实的功能领域的情况。这样做的原因是，人们可以研究曲线，更一般地做代数几何在有限域上，以证明这些结构。在这个项目中，我们将研究几何方法阿廷的猜想，例如通过考虑模向量丛曲线和曲面上有限域。

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.