曲面X的Hilbert概形描述了x上n个(不必相异的)点的集合，更准确地说，它是X的长为n的0维子概形的模空间。人们最近意识到，最初在代数几何中研究的Hilbert概形与数学的多个分支紧密相关，诸如奇点、辛几何、表示论，甚至理论物理。中岛平著的《曲面上点的Hilbert概形讲义(英文版)(精)》中的讨论反映了Hilbert概形这方面的特性。
这个学科近期的研究热点之一，是无限维Heisenberg代数表示的构造，即Fock空间。这种表示在文献中被广泛研究，它与仿射Lie代数、共形场论等有关。但是，本书给出的构造是独一无二的，它给出几何与表示论之间一种未曾考虑过的关联。
本书精彩概述了这个快速发展学科的近期进展，适合用作研究生高年级教材。

Pretace
Introduction
Chapter 1.Hilbert scheme of points
1.1.General Results on the Hilbert scheme
1.2.Hilbert scheme of points on the plane
1.3.Hilbert scheme of points on a surface
1.4.Symplectic structure
1.5.The Douady space
Chapter 2.Framed moduli space of torsion free sheaves on p2
2.1.Monad
2.2.Rank 1 case
Chapter 3.Hyper—Kahler metric on(C2)[n]
3.1.Geometric invariant theory and the moment map
3.2.Hyper—Kghler quotients
Chapter 4.Resolution of simple singularities
4.1.General Statement
4.2.Dynkin diagrams
4.3.A geometric realization of the McKay correspondence
Chapter 5.Poinca％polynomials of the Hilbert schemes(1)
5.1.Perfectness of the Morse function arising from the moment map
5.2.Poincar~polynomial of f(C2)[n]
Chapter 6.Poinca％polynomials of Hilbert schemes(2)
6.1.Results on intersection cohomology
6.2.Proof of the formula
Chapter 7.Hilbert scheme on the cotangent bundle of a Riemann surface
7.1.Morse theory on holomorphic symplectic manifolds
7.2.Hilbert scheme of T*∑
7.3.Analogy with the moduli space of Higgs bundles
Chapter 8.Homology group of the Hilbert schemes and the Heisenberg algebra
8.1.Heisenberg algebra and Clifford algebra
8.2.CorresDondences
8.3.Main construction
8.4.Proof of Theorem 8.13
Chapter 9.Symmetric products of an embedded curve，symmetric flunctions
and vertex operators
9.1.Symmetric functions and symmetric groups
9.2. Grojnowski’S formulation
9.3.Symmetric products of an embedded CUrve
9.4.Vertex algebra
9.5. Moduli space of rank 1 sheaves
Bibliography
Index