Lectures on Hilbert Schemes of Points on Surfaces

Overview

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 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...

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Overview

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 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.

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Editorial Reviews

Booknews
The author discusses the Hilbert scheme of points X[n] on a complex surface X from various points of view. It inherits structures of X, e.g. it is a nonsingular complex manifold, it has a holomorphic symplectic form if X has one, and so on. Annotation c. Book News, Inc., Portland, OR (booknews.com)
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Product Details

  • ISBN-13: 9780821819562
  • Publisher: American Mathematical Society
  • Publication date: 1/1/1999
  • Series: University Lecture Series , #18
  • Edition number: 1
  • Pages: 132
  • Product dimensions: 7.98 (w) x 8.06 (h) x 0.24 (d)

Table of Contents

Preface
Introduction 1
Ch. 1 Hilbert scheme of points 5
Ch. 2 Framed moduli space of torsion free sheaves on P[superscript 2] 17
Ch. 3 Hyper-Kahler metric on (C[superscript 2])[superscript [n]] 29
Ch. 4 Resolution of simple singularities 47
Ch. 5 Poincare polynomials of the Hilbert schemes (1) 59
Ch. 6 Poincare polynomials of Hilbert schemes (2) 73
Ch. 7 Hilbert scheme on the cotangent bundle of a Riemann surface 79
Ch. 8 Homology group of the Hilbert schemes and the Heisenberg algebra 89
Ch. 9 Symmetric products of an embedded curve, symmetric functions and vertex operators 105
Bibliography 125
Index 131
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