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.