I am a theoretical physics major student working on string theory. I want to understand the work of Nakajima, "Lectures on Hilbert Schemes of Points on Surfaces" . What kinds of mathematical background does it need? Hartshorne or Griffiths & Harris? More precisely, which chapters are necessary? (I only learned Nakahara's book on geometry) Thanks in advance.
1 Answer
If by a book of Nakahara you mean "Geometry, Topology and Physics", you are going to need alot from Hartshorone, as based on this book, you don't even know what a sheaf or a scheme precisely is. So at least the first 2 chapters of Hartshorne are needed. But I think reading that book of Nakahara, you almost have all the complex differential geometric tools that one can get from GriffithsHarris and needs to understand Nakajima. By the way, for the concept of moduli, you can consult the books "Moduli of Curves" by HarrisMorrison, the first chapter. the Book "Quasiprojective moduli for polarized manifolds" by Viehweg is a good introduction to the concept. Specially the first chapter has an introduction to Hilbert schemes and moduli problems, but this book is more technical.

$\begingroup$ Actually, one may be able to avoid schemes and sheaves by a quite natural description of a Hilbert scheme as a space of pairs of matrices divided by a certain action. Look at this question: mathoverflow.net/questions/13305/… $\endgroup$ Commented Aug 4, 2013 at 6:47