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On page 4 of Nitin Nitsure's paper Construction of Hilbert and Quot Schemes, the author refers to the fact that Hilbert polynomials are indeed polynomials as

a special case of Snapper's Lemma, see "An Intersection Theory for Divisors (preprint 1994)" by Steven Kleiman for a proof.

Kleiman's paper (or book?) mentioned by Nitsure must have either changed its title, never gone into publication, or elsehow disappeared, since I am unable to find it. Does anybody have a link, or an alternative source for the proof?

Thanks in advance.

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2 Answers 2

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A proof by Kleiman can be found in ‘‘Toward a Numerical Theory of Ampleness’’. I suspect it's the intended proof, although the paper is from around 30 years before the cited 1994 preprint.

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Note that Nitsure's paper is part of the book FGA Explained. There is a proof of Snapper's lemma in Theorem B.7 of Appendix B ("Basic intersection theory" by Kleiman) in the same book. Kleiman's part of the book can also be found independently on arXiv: 0504020 and contains the relevant appendix.

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