Moduli spaces and blow-up constructions for stacks

堆栈的模数空间和爆炸结构

• 批准号: 2099935
• 金额: --
• 依托单位: University of Oxford
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2018
• 资助国家: 英国
• 起止时间: 2018 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2099935
• 关键词: Moduli; spaces; blow; up; constructions

Moduli spaces arise naturally in classification problems in algebraic geometry, and form a central ingredient in enumerative geometry (where we want to count objects of different types). A typical such problem, for example the classification of nonsingular projective curves up to isomorphism, can be resolved into two basic steps. The first step is to find as many discrete invariants as possible. The second step is to fix the values of all the discrete invariants and try to construct a moduli space; that is, an algebraic variety (or some more general geometric object) whose points correspond in a natural way to the equivalence classes of the objects to be classified. In enumerative geometry the aim is then to construct virtual fundamental classes on suitable moduli spaces and use them to evaluate cohomology classes.What is meant by 'natural' here can be made precise given suitable notions of families of objects parametrised by base spaces and of equivalence of families. A fine moduli space is a base space for a universal family of the objects to be classified, but typically the existence of a fine moduli space is too much to hope for, unless we are willing to replace spaces by stacks. A moduli stack may have an associated coarse moduli space, satisfying slightly weaker conditions, but it is often the case that not even a coarse moduli space will exist. Typically, 'unstable' objects must be left out in order for a moduli space to exist. The notion of stability here can depend on parameters which may be varied, and in enumerative geometry there are 'wall-crossing formulas' which describe the dependence on such parameters. Typically there is constancy on 'chambers' but things change in a prescribed way as walls between chambers are crossed.In order to study moduli stacks, construct coarse moduli spaces and define and calculate enumerative invariants, an important tool is to have analogues for stacks of the blow-up construction for schemes which is used throughout algebraic geometry. A recent paper by Edidin and Rydh gives a blow-up construction which applies to an Artin stack with a stable good moduli space. Frances Kirwan is working with former students Victoria Hoskins and Joshua Jackson and current student Eloise Hamilton on a generalisation of the Edidin-Rydh construction which applies to much more general Artin stacks, while Dominic Joyce is working on a generalisation which allows the construction of the virtual fundamental classes needed in enumerative geometry. The aim of this research project is to extend these constructions and to exploit them, in particular in the case of surfaces. This research is likely to have impact in the study of moduli stacks and wall-crossing formulas in enumerative geometry. It is not directly related to any of the EPSRC's priority research areas. The research methodology using blow-up constructions for stacks is very new, although based on well established constructions for algebraic varieties and schemes. No companies or collaborators are involved.This project falls within the EPSRC Geometry and Topology research area

模空间自然地出现在代数几何的分类问题中，并形成了计数几何(我们想要对不同类型的对象进行计数)的核心成分。一个典型的此类问题，例如同构的非奇异射影曲线的分类，可以分解为两个基本步骤。第一步是找到尽可能多的离散不变量。第二步是确定所有离散不变量的值，并试图构造一个模空间；即，一个代数簇(或更一般的几何对象)，其点以一种自然的方式对应于待分类对象的等价类。在计数几何中，目的是在适当的模空间上构造虚拟的基本类，并用它们来评估上同调类。在给定由基空间参数化的对象族和族的等价性的适当概念的情况下，这里所指的自然的含义可以变得精确。精细模空间是要分类的对象的普适族的基本空间，但通常情况下，精细模空间的存在太过奢望，除非我们愿意用堆栈来取代空间。模堆栈可以具有相关联的粗模空间，满足稍弱的条件，但通常的情况是甚至不存在粗模空间。通常，要想存在模空间，就必须去掉“不稳定的”对象。在这里，稳定性的概念可以依赖于可能变化的参数，在计数几何中，有描述对这些参数的依赖的“跨墙公式”。通常“室”是恒定的，但当室与室之间的墙交叉时，事物会以规定的方式变化。为了研究模堆，构造粗模空间，并定义和计算计数不变量，一个重要的工具是对整个代数几何中使用的方案的爆破结构的堆栈有类似的东西。Edidin和Rydh最近的一篇论文给出了一个爆破结构，它适用于具有稳定良模空间的Artin堆栈。France Kirwan正在与以前的学生Victoria Hoskins和Joshua Jackson以及现在的学生Eloise Hamilton一起研究适用于更一般的Artin堆栈的Edidin-Rydh结构的推广，而Dominic Joyce正在研究允许构造枚举几何中所需的虚拟基础类的推广。这个研究项目的目的是扩展和开发这些结构，特别是在曲面的情况下。这一研究可能对计数几何中的模堆叠和跨壁公式的研究产生影响。它与EPSRC的任何优先研究领域都没有直接关系。对堆栈使用爆破结构的研究方法是非常新的，尽管是基于对代数簇和方案的成熟的构造。本项目属于EPSRC几何与拓扑学研究领域