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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!

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    $\begingroup$ I made your list actually list (I recommend actually looking at the preview; it's very hard for people to read a big block of text like that and separate out your questions) $\endgroup$
    – Ben Webster
    May 2, 2010 at 23:55
  • $\begingroup$ Thanks! Somehow I couldn't edit based on your edited version, but could only access the source code for the original without the list. $\endgroup$
    – John Jiang
    May 3, 2010 at 3:10
  • $\begingroup$ You might consider changing the title of your question to mention Okounkov's paper explicitly. There are plenty of people here who know about homology of algebraic varieties, but unless one happens to have read the actual paper, this question is impossible to answer. On the plus side, there may well be people around who would be enticed to read the question if it mentioned the paper, but who would otherwise easily skip over such a generic title. $\endgroup$
    – Joel Fine
    May 3, 2010 at 13:48
  • $\begingroup$ Just changed. Thanks for the suggestion! $\endgroup$
    – John Jiang
    May 3, 2010 at 17:05
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    $\begingroup$ You also might want to link to the paper, to send a trackback to the Arxiv, so people who read the paper will find your question. $\endgroup$ May 3, 2010 at 17:42

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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 $0 + 0 - 1 + 10$.

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  • $\begingroup$ Hi Ilya, thanks for all the explanations! Regarding question 2, I am still not getting the point of hyperplane sections. Maybe you could explain in a bit more layman's term? For example in a way that an analyst could understand. $\endgroup$
    – John Jiang
    May 3, 2010 at 23:48
  • $\begingroup$ Every time you have $X$ in projective space, you have a divisor $X\cup H$ on $X$ where $H$ is a hyperplane... Take a look at any book on projective varieties. $\endgroup$ May 4, 2010 at 4:13
  • $\begingroup$ That should be $X\cap H$ above. $\endgroup$ May 4, 2010 at 4:13
  • $\begingroup$ So does that mean H can be any divisor of X? $\endgroup$
    – John Jiang
    May 4, 2010 at 4:48
  • $\begingroup$ Not really: you can't get all divisors that way. That's a fascinating story you can search for in courses and textbooks on something like "algebraic geometry" or "geometry of surfaces". $\endgroup$ May 5, 2010 at 19:31

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