The Hilbert scheme $X^{[n]}$ of a surface $X$ describes collections of $n$ (not necessarily distinct) points on $X$. More precisely, it is the moduli space for $0$-dimensional subschemes of $X$ of length $n$. Recently it was realized that Hilbert schemes originally studied in algebraic geometry are closely related to several branches of mathematics, such as singularities, symplectic geometry, representation theory-even theoretical physics. The discussion in the book reflects this feature of Hilbert schemes.

For example, a construction of the representation of the infinite dimensional Heisenberg algebra (i.e., Fock space) is presented. This representation has been studied extensively in the literature in connection with affine Lie algebras, conformal field theory, etc. However, the construction presented in this volume is completely unique and provides the unexplored link between geometry and representation theory.

The book offers a nice survey of current developments in this rapidly growing subject. It is suitable as a text at the advanced graduate level.

Readership

Graduate students and research mathematicians interested in algebraic geometry, topology, or representation theory.

Reviews

"This beautifully written book deals with one shining example: the Hilbert schemes of points on algebraic surfaces ... The topics are carefully and tastefully chosen ... The young person will profit from reading this book."

-- Mathematical Reviews

Table of Contents

• Introduction
• Hilbert scheme of points
• Framed moduli space of torsion free sheaves on $\mathbb{P}^2$
• Hyper-Kähler metric on $(\mathbb{C}^2)^{[n]}$
• Resolution of simple singularities
• Poincaré polynomials of the Hilbert schemes (1)
• Poincaré polynomials of Hilbert schemes (2)
• Hilbert scheme on the cotangent bundle of a Riemann surface
• Homology group of the Hilbert schemes and the Heisenberg algebra
• Symmetric products of an embedded curve, symmetric functions and vertex operators
• Bibliography
• Index