Singularities of punctual Hilbert schemes

本文将探讨一个著名的几何空间，称为希尔伯特方案的点-与一些研究很多奇异曲面。光滑曲面上的点的希尔伯特格式是代数几何和几何表示论中最美丽的几何对象之一。关于这个空间的许多最强有力的结果可以追溯到Fogarty在60年代末和70年代初的工作。简而言之，表面上可以是任意复杂的希尔伯特方案，当表面本身是光滑的时，它就变成了一个光滑空间。这项工作在随后的几十年里支撑了许多进展。另一方面，对与奇异曲面相关的点的希尔伯特方案知之甚少。Craw和合作者的最新工作[Craw-Gamelgaard-Gyenge-Szendroi，Alg. Geom. 2021]建立了与最简单的奇异曲面族相关联的点的希尔伯特方案的许多关键性质;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Kleinian奇点。他们的突破是通过应用[Bellamy-Craw，Invent. Math.2018]。论文的具体目标是推广这些技术来研究一个家庭的奇异表面，被称为crepant部分决议的Kleinian奇点。期望的是，人们可以制定一个共同的声明，描述了克莱因奇点上的点的希尔伯特方案，所有它们的部分crepant分辨率，以及与克莱因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等人的最近工作统一起来，导致一个完整的和统一的描述的希尔伯特计划的点的家庭的所有部分crepant决议的Kleinian奇点。最初，露丝与得到掌握的研究森梦空间和mri品种，把她放在一个位置来条款的工作[Craw-Gamelgaard-Gyenge-Szendroi，Alg. 2021]。然后，挑战是采取一个参数中引入的工作，并允许它更自由地变化，从而探测不仅是希尔伯特计划的点的Kleinian奇点，但更普遍的是，所有部分crepant决议的奇异空间。露丝将同时考虑的一个关键目标是决定克莱因奇点上的希尔伯特点集是否是正规的（这大致意味着它不是太严重的奇异）。这个问题仍然是开放的，有一些自然的思想实验，人们可以进行，以帮助解决这个问题。

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.