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Nakajima gives a definition of the Hilbert polynomial on p. 5 here that I can't explain or find explanation for anywhere else. It goes, $X$ is a projective scheme over algebraically closed field with ample line bundle $\mathcal{O}_X(1)$. We look at closed subschemes $Z$ of the product $X\times U$ such that the projection $\pi: Z\to U$ is flat. Let $Z_u=\pi^{-1}(u)$. Now the Hilbert polynomial in $u$ is $$p_u(m)=\chi(\mathcal{O}_{Z_u}\otimes\mathcal{O}_X(m))$$ where $\chi$ is the Euler characteristic (which just becomes the rank of the global sections for large $m$ by Serre vanishing theorem).

I'm ok with the usual $p(m)=\chi(\mathcal{O}_X(m))$, or $p_\mathscr{F}(m)=\chi(\mathscr{F}(m))$ for quasi-coherent sheaves $\mathscr{F}$ but I dont know what this sheaf $\mathcal{O}_{Z_u}$ is exactly. I see from Hartshorne that it could be the structure sheaf of the fiber product $Z_u=Z\times_U \operatorname{Spec} k(u)$ where $k(u)$ is the residue field of $u$. But I guess I don't unerstand enough about products to know why this can also be a quasi-coherent sheaf on $X$ (?) tensorable (over $\mathcal{O}_X$ ?) with the twisted sheaf $\mathcal{O}_X(m)$.

Thinking again of $Z_u$ as homeomorphic to the subset $\pi^{-1}(u)\subset X$, maybe were throwing the structure sheaf onto this subset and doing something skyscraper-y to get just the $m^{th}$ graded global sections in their restriction? Or is $Z_u$ actually a closed subscheme because we only take closed points $u$ making $\mathcal{O}_{Z_u}$ an honest quotient of the structure sheaf of $X$ by an ideal sheaf?

Something like this would seem to solve my problems, because then $\mathcal{O}_{Z_u}\otimes \mathcal{O}_X(m)=\mathcal{O}_{Z_u}(m)$ and we have the usual Hilbert polynomial, but I don't know why the fibers should be closed.

Please halp.

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