Lectures on Hilbert Schemes of Points on Surfaces

ISBN-13: 9780821819562 / Angielski / Miękka / 1999 / 132 str.

Hiraku Nakajima

Opis

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 $n$. It has been 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 unique and provides the unexplored link between geometry and representation theory.