Singularities of punctual Hilbert schemes

这篇论文将探索一个著名的几何空间，称为点的希尔伯特格式-与一些被广泛研究的奇异曲面有关。光滑曲面上的点的Hilbert格式是代数几何和几何表示理论中最美丽的几何对象之一。关于这一领域的许多最有力的结果可以追溯到福格蒂在60年代末和70年代初的工作。简单地说，希尔伯特格式表面上可以是任意复杂的，当曲面本身是光滑的时，它被证明是光滑的空间。在接下来的几十年里，这项工作支撑了许多进展。另一方面，人们对与奇异曲面相关的点的希尔伯特方案知之甚少。Craw及其合作者最近的工作[Craw-Gamelgaard-Gyenge-Szendroi，Alg.天啊。2021]建立了与最简单的奇异曲面族有关的点的希尔伯特格式的许多关键性质；这些曲面的研究可以追溯到20世纪30年代，有许多名称，包括克莱因奇点。他们的突破是通过应用[Bellamy-Craw，发明]的相对较新的工作，实现了将这些希尔伯特方案作为箭袋品种的例子。数学课。2018年]。这篇论文的具体目标是将这些技术推广到研究一族奇异曲面，即Kleian奇点的可信性部分分解。我们的期望是，人们可以形成一个共同的陈述来描述Kleian奇点上的点的Hilbert格式、它们的所有部分可分解以及与Kleian曲面相关的自然光滑曲面；简而言之，我们的目标是将Fogarty的经典描述与Craw等人最近的工作统一起来，导致对Kleian奇点的所有部分可分解族上的点的Hilbert格式的完整和统一的描述。最初，Ruth With开始研究Mori Dream空间和箭图变种，使她能够接受[Craw-Gamelgaard-Gyenger-Szendroi，Alg]的工作。天啊。2021年]。然后，挑战是采用工作中引入的参数，并允许它更自由地变化，从而不仅探索Kleian奇点上的点的希尔伯特格式，而且更一般地，探索该奇异空间的所有部分初等分辨率。Ruth将并行考虑的一个关键目标是确定Kleian奇点上的点的Hilbert格式是否正常(这大致意味着它不是太严重的奇点)。这个问题仍然悬而未决，人们可以进行一些自然的思维实验来帮助解决这个问题。

This thesis will explore a celebrated geometric space, called the Hilbert scheme of points - associated to some much studied singular surfaces. The Hilbert scheme of points associated to a smooth surface is one of the most beautiful geometric objects to be studied in algebraic geometry and geometric representation theory. Many of the strongest results about this space go back to the work of Fogarty from the late 60s and early 70s. Put simply, the Hilbert scheme, which on the face of it can be arbitrary complicated, turns out to be a smooth space when the surface itself is smooth. That work underpinned many advances in the subsequent decades.On the other hand, remarkably little is known for the Hilbert scheme of points associated to singular surfaces. Recent work of Craw and collaborators [Craw-Gamelgaard-Gyenge-Szendroi, Alg. Geom. 2021] established many key properties for the Hilbert scheme of points associated to the simplest family of singular surfaces; these surfaces, whose study goes back to the 1930s, have many names, including Kleinian singularities. Their breakthrough came in realising these Hilbert schemes as examples of quiver varieties by applying relatively recent work of [Bellamy-Craw, Invent. Math. 2018]. The concrete goal of the thesis is to generalise these techniques to study a family of singular surfaces, known as the crepant partial resolutions of the Kleinian singularities. The expectation is that one can formulate a common statement that describes the Hilbert scheme of points on the Kleinian singularities, all of their partial crepant resolutions, and a natural smooth surface associated to the Kleinian surface; put simply, we aim to unify the classical description of Fogarty with the recent work of Craw et al., leading to a complete and uniform description of the Hilbert schemes of points on the family of all partial crepant resolutions of Kleinian singularities.Initially, Ruth with get to grips with the study of Mori Dream Spaces and quiver varieties, putting her in a position to come to terms with the work of [Craw-Gamelgaard-Gyenge-Szendroi, Alg. Geom. 2021]. The challenge then is to take a parameter introduced in that work and allow it to vary more freely, thereby probing not only the Hilbert scheme of points on the Kleinian singularity, but more generally, all partial crepant resolutions of that singular space. A key goal that Ruth will consider in parallel is to decide whether or not the Hilbert scheme of points on a Kleinian singularity is normal (which means roughly that it is not too badly singular). This question is still open, and there are some natural thought experiments that one can carry out to help settle this question.