Homology of algebraic varieties in Okounkov's paper on enumerating algebraic curves

Question

This is a series of questions in chronological order. I am lately trying to understand Okounkov's Random surfaces enumerating algebraic curves. So he mentions something about virtual fundamental class.

1. Can someone give a little intuition or exposition about what this is in the context of section 2.1.3 of the paper?
2. Another question is how does one define $H_2(X)$ in section 2.2.2 (I apologize for the elementary nature of the question in the eyes of algebraic geometers, but I am only familiar with $H_k$ for topological spaces); I presume one needs some sort of etale cohomology? And what does he mean by "the hyperplane class induced from the ambient $\mathbb P^N$" in the same section?
3. In section 3.1.3, the author says in the second paragraph that "the total width of these infinite rows and columns(2, in this example)", why is it $2$? Also he says hte constant term $\chi$(=9 here), immediately after. Why is that $9$? I presume he is talking about figure 2, but I couldn't see the $2$ or the $9$ in the diagram.

That's it for now, probably will have more questions later. But as usual help is greatly appreciated!

Answer 1

It appears that you want the "rough, introductory answers", so feel fine to ask more detailed questions if anything is unclear/potentially wrong. Also, I only have very general knowledge about enumerations!

1. It looks like the standard moduli stack of marked curves. There are quite a lot of references obtainable by searching Internet, arXiv and Math Overflow, or check out that Wikipedia page.

2. He refers to topological $H_2(X)$ of the space considered as a complex manifold. $H$ can be defined as the divisor of $\mathcal O(1)$, or simply the hyperplane section under that projective embedding. It so happens that the number he's interested in is an intersection number of $C$ and $H$; you could compute it algebro-geometrically as $C\cdot H$ or topologically, doesn't matter.

3. Because it's 1 infinite row above and 1 infinite column left; and then he makes use of the equality $1+1=2$. After I'll read the whole paper I'll probably know this should equal to the degree $d$ :)

4. I'm less sure about how he gets the 9, but one way to renormalize infinity here would be to consider a whole infinite row or column to have zero area. By that logic, we have an infinite row, then an infinite colun without a piece, then 10 pieces, which adds up as $0 + 0 - 1 + 10$.