Lectures on Hilbert scheme of points on surfaces (draft, AMS 1999)

Nakajima H.

In the spring of 1996, I gave a series of lectures on the Hilbert schemes of points on surfaces at Department of Mathematical Sciences, University of Tokyo. The purpose of the lectures was to discuss various properties of the Hilbert schemes of points on surfaces. Although it was not noticed until recently, the Hilbert schemes have relationship with many other branch of mathematics, such as topology, hyper-Kahler geometry, symplectic geometry, singularities, and representation theory. This is reflected to this note: each chapter, which roughly corresponds to one lecture, discusses different topics. These lectures were intended for graduate students who have basic knowledge on algebraic geometry and ordinary topology. The only results which will be used but not proved in this note are Grothendieck’s construction of the Hilbert scheme (Theorem 1.1) and results on intersection cohomology (§6.1). I recommend to the reader to accept these results when he/she is not familiar with them. I have tried to make it possible to read each chapter independently. I believe that it is almost successful. The interdependence of chapters is figured in the next page. The broken arrows mean that we need only the statement of results in the outgoing chapter, and do not need its proof. Sections 9.1,9.3 are based on A. Matsuo’s lectures. His lectures contained Monster and Macdonald polynomials. I regret omitting these subjects. I hope to understand these by Hilbert schemes in future. The note was prepared by T. Gocho and N. Nakamura. I would like to thank them for their efforts. I am also grateful to A. Matsuo and H. Ochiai for their comments throught the lectures. Particular thanks are due to G. Ellingsrud, I. Grojnowski, K. Hasegawa, N. Hitchin, Y. Ito, A. King, G. Kuroki, G. Segal, and S. Str0mme for discussions on results in this note.