**University Lecture Series**

Volume: 18;
1999;
132 pp;
Softcover

MSC: Primary 14;
Secondary 17; 16; 53; 81

Print ISBN: 978-0-8218-1956-2

Product Code: ULECT/18

List Price: $26.00

AMS Member Price: $20.80

MAA Member Price: $23.40

**Electronic ISBN: 978-1-4704-1834-2
Product Code: ULECT/18.E**

List Price: $26.00

AMS Member Price: $20.80

MAA Member Price: $23.40

# Lectures on Hilbert Schemes of Points on Surfaces

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*Hiraku Nakajima*

The Hilbert scheme 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.

One example of the modern, broader interest in the subject is a
construction of the representation of the infinite-dimensional
Heisenberg algebra, i.e., Fock space. 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 an
unexplored link between geometry and representation theory.

The book offers an attractive 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 & Endorsements

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

# Table of Contents

## Lectures on Hilbert Schemes of Points on Surfaces

- Cover Cover11 free
- Other titles in this series i2 free
- Title page v6 free
- Contents vii8 free
- Preface ix10 free
- Interdependence of the chapters xi12 free
- Introduction 114 free
- Hilbert scheme of points 518 free
- Framed moduli space of torsion free sheaves on ℙ² 1730
- Hyper-Kähler metric on (ℂ²)^{[𝕟]} 2942
- Resolution of simple singularities 4760
- Poincaré polynomials of the Hilbert schemes (1) 5972
- Poincaré polynomials of Hilbert schemes (2) 7386
- Hilbert scheme on the cotangent bundle of a Riemann surface 7992
- Homology group of the Hilbert schemes and the Heisenberg algebra 89102
- Symmetric products of an embedded curve, symmetric functions and vertex operators 105118
- Bibliography 125138
- Index 131144 free
- Back Cover Back Cover1146