Algebraic Stacks

由该奖项资助的研究将针对几个密切相关的问题和活动。项目的第一部分是从零开始建立代数堆栈理论，目标是用最少量的基本方案假设来编写代数堆栈的基础，没有代数空间的分离公理，也没有代数堆栈的分离公理。这将在堆栈项目中持续在线记录，请参阅http://math.columbia.edu/algebraic_geometry/stacks-git。许多代数堆栈是商堆栈，但情况并非总是如此。该项目的第二个组成部分涉及到是否每个代数堆栈在局部总是一个商堆栈的问题。这涉及到域上分离光滑代数空间的上同布劳尔群和经典布劳尔群是否重合的问题。这就引出了其他问题，特别是Brauer类的周期和指数之间的关系。该项目的第三部分是查看git堆栈上的零环组上是否存在自然度映射。最后，本课题包括第四部分，旨在研究叠曲线的模，这是一个自然显式的高等代数叠的例子。在整个PI中，将对上述堆栈项目进行调节。长期以来，物理学家和哲学家们一直着迷于一个问题：什么是空间？对于数学家来说，三维流形似乎是空间的一个很好的近似。在学习了狭义相对论之后，一个带有洛伦兹度规的四维空间似乎更接近了。在学习了标准模型之后，空间似乎被赋予了某些向量束。等等......。在代数几何中，学生们很早就被教导有许多不同的空间，事实上，所有空间的集合形成了某种空间。一个著名的例子是g属曲线（黎曼曲面）的模空间，它被物理学家用于弦理论。Deligne和Mumford在一篇关于代数几何中曲线模的基础论文中指出曲线空间具有附加结构，曲线空间中的点被赋予一定的有限群（即相应曲线的自同构群）。他们创造了“代数堆栈”这个词来表示这种空间。事实证明，代数堆栈语言在研究非常经典的对象，如曲线和面上的向量束、椭圆曲线的模和阿贝尔变分等方面是一个非常有用的工具。该项目将部分地在非常一般的环境中发展代数堆栈的基础，部分地发现这些空间的新性质，例如是否以某种自然方式连接到代数堆栈点上的有限群都包含在单个更大的群中。

The research funded by the award will be directed towards several closely related questions and activities. A first part of the project is to build the theory of algebraic stacks from scratch, with the goal of writing foundations for algebraic stacks with a minimal amount of assumptions on the base scheme, no separation axioms for algebraic spaces, and no separation axioms for algebraic stacks. This will be continually documented online in the stacks project, see http://math.columbia.edu/algebraic_geometry/stacks-git. Many algebraic stacks are quotient stacks, but this is not always the case. A second component of the project involves the question of whether every algebraic stack is always etale locally a quotient stack. This is related to the question of whether the cohomological Brauer group and the classical Brauer group coincide for separated smooth algebraic spaces over a field. This leads into other questions, especially the relation between period and index for Brauer classes. A third part of the project is to see whether there exists a natural degree map on the group of zero cycles on a GIT-stack. And finally, the project includes a fourth part aimed at studying moduli of stacky curves, which is a natural explicit example of a higher algebraic stack. Throughout the PI will moderate the stacks project mentioned above.A question that has long fascinated physicists and philosophers is:What is space? For a mathematician a 3-dimensional manifold seems a good first approximation to space. After learning about special relativity a four dimensional space with a Lorentz metric seems closer. After learning about the standard model it seems that space comes endowed with certain vector bundles. And so on. In algebraic geometry students are taught early on that there are many different spaces, and that in fact the collection of all spaces forms itself some kind of space. A famous example is the moduli space of curves of genus g (Riemann surfaces) which is used by physicists in string theory. In a fundamental paper on moduli of curves in algebraic geometry, Deligne and Mumford pointed out that the space of curves has additional structure in that its points come endowed with certain finite groups (namely the automorphism groups of the corresponding curves). They coined the phrase "algebraic stack" to denote this type of space. It turns out that the language of algebraic stacks is an extremely useful tool in studying very classical objects such as vector bundles on curves and surfaces, moduli of elliptic curves and abelian varieties, etc, etc. The project will partly develop the foundations of algebraic stacks in a very general setting, and partly find new properties of these spaces, such as whether the finite groups attached to the points of an algebraic stack all in some natural way are contained in a single bigger group.