Reference Request for Hilbert Schemes

Question

I'm a physicist working on Fractional Quantum Hall effect. The mathematical subjects of study are symmetric, translational invariant, homogeneous polynomials on $\mathbb{C}$. Very early in my study I realized that there is some beautiful geometry underneath all of the physics. As I dug deeper, turns out ($\to$Nakajima), other than symmetric polynomials, the Heisenberg algebra has also a representation via Hilbert schemes. So I decided to learn about Hilbert schemes from ZERO knowledge about algebraic geometry.

It has been months that I've been trying to learn more and more algebraic geometry. As fascinating as it is, I realized it is about time I ask for some serious guidance. Assuming I know nothing about algebraic geometry (which is not true), what is the most direct (maybe a 20 step program) to learn at least what Nakajima is doing and how to modify his method to my less well-behaving (translational invariant) polynomials?

The problem is algebraic geometry, as I've seen it, is gigantic. Any text explores lots of different avenues. It is kind of impossible to learn it all in one year; it takes time and I'm prepared to give it time. But in the meantime my research is hurting!

So if someone could be so kind to introduce to me a series of concepts (+maybe good references for those concepts) to get step by step closer to understand specifically what is going on in Hilbert scheme, I would be very grateful. Forgive me if maybe this is not the best place to ask such a thing.

Answer 1

I am not an expert on Hilbert schemes but I would recommend the following references:

• Eisenbud, Harris, "the geometry of schemes". A beautiful, relatively short introduction to scheme theory. Specifically section II.3 will be relevant, giving a concrete feeling for the notion of multiple points by means of concrete examples.

• Nakajima, "Lectures on Hilbert schemes of points on surfaces". As you said, difficult, but some sections are easier than others. For example section 1.4 should be fairly accessible, especially after understanding II.3 of Eisenbud and Harris.

• Stromme, "Elementary introduction to representable functors and hilbert schemes". A more difficult reference than the previous ones but nicely written. Contains a proof of the existence of Hilbert schemes in general.

• Bolognese, Losev, "A general introduction to the Hilbert scheme of points on the plane". In the spirit of Nakajima.

• de Cataldo, Migliorini, "The Douady space of a complex surface". A good reference for the analogue of the Hilbert scheme in complex analytic geometry.

• Voisin, "on the Hilbert scheme of an almost complex four-fould" Technical, but written in more differential geometric terms.

In general I would say it's important to get a good feeling for the notion of multiple points, flat morphisms (which are relatively easy to understand in the case the fibers are all 0-dimensional, see also Fischer, "complex analytic geometry" section 3.13) and representable functors.