Moduli spaces and blow-up constructions for stacks

模空间在代数几何中的分类问题中自然出现，并形成枚举几何（我们想要计算不同类型的对象）的核心成分。一个典型的此类问题，例如非奇异投影曲线直至同构的分类，可以分为两个基本步骤。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;也就是说，一个代数簇（或一些更一般的几何对象），其点以自然的方式对应于要分类的对象的等价类。在枚举几何中，目标是在适当的模空间上构造虚基本类，并用它们来计算上同调类。这里的“自然”的意思可以精确地给出由基空间参数化的对象族和族的等价性的适当概念。细模空间是要分类的对象的泛族的基空间，但通常细模空间的存在是太多的希望，除非我们愿意用堆栈来代替空间。一个模栈可能有一个相关的粗模空间，满足稍微弱一点的条件，但通常情况下甚至不存在粗模空间。通常，为了使模空间存在，必须排除“不稳定”对象。这里的稳定性的概念可以依赖于参数，这些参数可以是变化的，并且在枚举几何中有“跨壁公式”来描述对这些参数的依赖性。通常情况下，有恒定的“室”，但事情的变化，在规定的方式作为室之间的墙壁是crossed. To研究模栈，构建粗模空间和定义和计算枚举不变量，一个重要的工具是有类似物的堆栈的爆破建设的计划，这是整个代数几何。Edidin和Rydh最近的一篇论文给出了一个爆破构造，它适用于具有稳定好模空间的Artin栈。弗朗西斯Kirwan是与以前的学生维多利亚霍斯金斯和约书亚杰克逊和目前的学生埃洛伊塞汉密尔顿的概括埃迪丁-Rydh建设适用于更一般的阿丁堆栈，而多米尼克乔伊斯是工作的概括，使建设的虚拟基本类需要枚举几何。本研究项目的目的是扩展这些结构，并利用它们，特别是在曲面的情况下。这一研究对枚举几何中的模量叠加和跨壁公式的研究具有一定的参考价值。它与EPSRC的任何优先研究领域都没有直接关系。使用爆破结构的堆栈的研究方法是非常新的，虽然基于良好的代数簇和计划的建设。没有公司或合作者参与。这个项目属于EPSRC几何和拓扑研究领域的福尔斯

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