Lectures on Hilbert Schemes of Points on Surfaces (University Lecture Series)

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. 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 Cover 1 Other titles in this series 2 Title page 6 Contents 8 Preface 10 Interdependence of the chapters 12 Introduction 14 Hilbert scheme of points 18 Framed moduli space of torsion free sheaves on P2 30 Hyper-Kähler metric on (C2)^{[n]} 42 Resolution of simple singularities 60 Poincaré polynomials of the Hilbert schemes (1) 72 Poincaré polynomials of Hilbert schemes (2) 86 Hilbert scheme on the cotangent bundle of a Riemann surface 92 Homology group of the Hilbert schemes and the Heisenberg algebra 102 Symmetric products of an embedded curve, symmetric functions and vertex operators 118 Bibliography 138 Index 144 Back Cover 146