Algebraic Stacks

该奖项资助的研究将针对几个密切相关的问题和活动。该项目的第一部分是从头开始构建代数堆栈理论，目标是编写代数堆栈的基础，其中对基本方案的假设量最少，代数空间没有分离公理，代数堆栈没有分离公理。这将在stacks项目中持续在线记录，请参阅http://math.columbia.edu/algebraic_geometry/stacks-git。许多代数栈是商栈，但情况并非总是如此。该项目的第二个组成部分涉及的问题，是否每个代数栈总是局部etale商栈。这涉及到域上分离光滑代数空间的上同调Brauer群和经典Brauer群是否重合的问题。这导致了其他问题，特别是布劳尔类的周期和指数之间的关系。该项目的第三部分是查看GIT堆栈上的零循环组是否存在自然度映射。最后，该项目包括第四部分，旨在研究stacky曲线的模量，这是一个自然的显式例子，一个更高的代数堆栈。整个PI将主持上述堆栈项目。一个长期吸引物理学家和哲学家的问题是：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，具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丛。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，所有空间的集合本身就形成了某种空间。一个著名的例子是亏格g（黎曼曲面）曲线的模空间，物理学家在弦理论中使用它。在一个基本文件模的曲线在代数几何，德利涅和芒福德指出，空间的曲线有额外的结构，它的点来赋予某些有限的群体（即自同构群的相应曲线）。他们创造了“代数栈”这个短语来表示这种类型的空间。事实证明，代数栈的语言是一个非常有用的工具，在研究非常经典的对象，如曲线和曲面上的向量丛，椭圆曲线和阿贝尔簇的模等，该项目将部分发展代数栈的基础在一个非常一般的设置，并部分发现这些空间的新性质，例如，以某种自然方式连接到代数堆栈点的有限群是否包含在一个更大的群中。

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.