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.
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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 Griffiths-Harris and needs to understand Nakajima. By the way, for the concept of moduli, you can consult the books "Moduli of Curves" by Harris-Morrison, the first chapter. the Book "Quasi-projective 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.
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$\begingroup$ 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