AI 中文总结
研究具有典范奇点的曲面\(S\)上\(n\)个点的希尔伯特概型，推广福加蒂定理，证明其基础约化子概型的性质，包括维数、奇点类型等，还探讨了\(S\)有辛奇点时的情况，结果基于前人工作。
AI 中文摘要
对于\(n\geq1\)，我们研究具有典范奇点的曲面\(S\)上\(n\)个点的希尔伯特概型。我们推广了著名的福加蒂定理，证明\(\text{Hilb}^n(S)\)的基础约化子概型是维数为\(2n\)且具有典范奇点的正规簇，对于\(n\leq7\)，我们证明\(\text{Hilb}^n(S)\)是约化的。当\(S\)在\(\mathbb{C}\)上具有辛奇点时，我们证明\(\text{Hilb}^n(S)\)的基础约化子概型也具有辛奇点，从而推广了博韦尔的一个结果。我们的结果建立在第一作者与延盖、甘梅尔加德和森德罗伊的工作基础上，他们试图将克莱因奇点上\(n\)个点的希尔伯特概型的基础约化子概型与中岛箭图簇进行等同。
英文摘要
For $n\geq 1$, we investigate the Hilbert scheme of $n$-points on a surface $S$ with canonical singularities. We generalise the well-known theorem of Fogarty by showing that the underlying reduced subscheme of $\operatorname{Hilb}^n(S)$ is a normal variety of dimension $2n$ with canonical singularities, and for $n\leq 7$, we show that $\operatorname{Hilb}^n(S)$ is reduced. When $S$ has symplectic singularities over $\mathbb{C}$, we show that the underlying reduced subscheme of $\operatorname{Hilb}^n(S)$ also has symplectic singularities, thereby generalising a result of Beauville. Our results build on work of the first author with Gyenge, Gammelgaard and Szendrői that sought to identify the underlying reduced subscheme of the Hilbert scheme of $n$-points on a Kleinian singularity with a Nakajima quiver variety.
Comments33 pages. This paper contains the geometric results from v1 of our arXiv:2312.08527 preprint, though we establish reducedness only for n\leq 7
