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栈理论，目标是在基本方案上使用最少的假设、没有代数空间的分离公理、也没有代数栈的分离公理来编写代数栈的基础。这将在 stacks 项目中不断在线记录，请参阅 http://math.columbia.edu/algebraic_geometry/stacks-git。许多代数栈都是商栈，但情况并非总是如此。该项目的第二个组成部分涉及以下问题：每个代数堆栈是否始终是局部商堆栈。这与上同调布劳尔群和经典布劳尔群对于域上的分离光滑代数空间是否重合的问题有关。这导致了其他问题，特别是布劳尔类的周期和索引之间的关系。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图。最后，该项目包括旨在研究堆叠曲线模的第四部分，这是高级代数堆叠的自然显式示例。 PI 将全程主持上述堆栈项目。 长期以来令物理学家和哲学家着迷的一个问题是：什么是空间？对于数学家来说，3 维流形似乎是空间的良好初步近似。了解狭义相对论后，具有洛伦兹度量的四维空间似乎更接近。在了解了标准模型之后，空间似乎被赋予了某些向量丛。等等。在代数几何中，学生很早就被教导存在许多不同的空间，并且事实上所有空间的集合本身就形成了某种空间。一个著名的例子是物理学家在弦理论中使用的 g 曲线模空间（黎曼曲面）。在一篇关于代数几何中曲线模的基础论文中，德利涅和芒福德指出，曲线空间具有附加结构，因为它的点被赋予某些有限群（即相应曲线的自同构群）。他们创造了“代数栈”一词来表示这种类型的空间。事实证明，代数栈语言是研究非常经典的对象（例如曲线和曲面上的向量丛、椭圆曲线的模和阿贝尔簇等）的非常有用的工具。该项目将部分地在非常一般的环境中开发代数栈的基础，部分地发现这些空间的新属性，例如以某种自然方式附加到代数栈的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.