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Lectures on Hilbert Schemes of Points on Surfaces

Hiraku Nakajima Kyoto University, Japan
Available Formats:
Softcover ISBN: 978-0-8218-1956-2
Product Code: ULECT/18
List Price: $28.00 MAA Member Price:$25.20
AMS Member Price: $22.40 Electronic ISBN: 978-1-4704-1834-2 Product Code: ULECT/18.E List Price:$26.00
MAA Member Price: $23.40 AMS Member Price:$20.80
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List Price: $42.00 MAA Member Price:$37.80
AMS Member Price: $33.60 Click above image for expanded view Lectures on Hilbert Schemes of Points on Surfaces Hiraku Nakajima Kyoto University, Japan Available Formats:  Softcover ISBN: 978-0-8218-1956-2 Product Code: ULECT/18  List Price:$28.00 MAA Member Price: $25.20 AMS Member Price:$22.40
 Electronic ISBN: 978-1-4704-1834-2 Product Code: ULECT/18.E
 List Price: $26.00 MAA Member Price:$23.40 AMS Member Price: $20.80 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$42.00 MAA Member Price: $37.80 AMS Member Price:$33.60
• Book Details

University Lecture Series
Volume: 181999; 132 pp
MSC: Primary 14; Secondary 17; 16; 53; 81;

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.

• Table of Contents

• Chapters
• Introduction
• Chapter 1. Hilbert scheme of points
• Chapter 2. Framed moduli space of torsion free sheaves on $\mathbb {P}^2$
• Chapter 3. Hyper-Kähler metric on $(\mathbb {C}^2)^{[n]}$
• Chapter 4. Resolution of simple singularities
• Chapter 5. Poincaré polynomials of the Hilbert schemes (1)
• Chapter 6. Poincaré polynomials of Hilbert schemes (2)
• Chapter 7. Hilbert scheme on the cotangent bundle of a Riemann surface
• Chapter 8. Homology group of the Hilbert schemes and the Heisenberg algebra
• Chapter 9. Symmetric products of an embedded curve, symmetric functions and vertex operators
• Additional Material

• 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
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Volume: 181999; 132 pp
MSC: Primary 14; Secondary 17; 16; 53; 81;

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.

• Chapters
• Introduction
• Chapter 1. Hilbert scheme of points
• Chapter 2. Framed moduli space of torsion free sheaves on $\mathbb {P}^2$
• Chapter 3. Hyper-Kähler metric on $(\mathbb {C}^2)^{[n]}$
• Chapter 4. Resolution of simple singularities
• Chapter 5. Poincaré polynomials of the Hilbert schemes (1)
• Chapter 6. Poincaré polynomials of Hilbert schemes (2)
• Chapter 7. Hilbert scheme on the cotangent bundle of a Riemann surface
• Chapter 8. Homology group of the Hilbert schemes and the Heisenberg algebra
• Chapter 9. Symmetric products of an embedded curve, symmetric functions and vertex operators
• 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
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