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my question concerns the Hilbertpolynomial of a coherent sheaf $F$ on a smooth projective variety $X$ with canonical bundle $\omega_X$: does the Hilbertpolynomial always have degree equal to $dim supp(F)$ or does in general only an inequality hold? And is there a connection to ampleness of the canonical sheaf resp. its dual?

Thanks a lot

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up vote 4 down vote accepted

Yes, the degree of the Hilbert polynomial of a coherent sheaf $F$ on a projective variety $X$ over a field $K$ is equal to the dimension of the support of $F$.
This is true even if $X$ is not smooth and has nothing to do with the canonical sheaf of $X$.
Beware however that the actual Hilbert polynomial does depend on the projective embeddding of $X$ in projective space, even if its degree is independent of the embedding.
You can check this by calculating the Hilbert polynomials of the structural sheaf $\mathcal O_{\mathbb P^1_K}$ of the projective line and that of the structural sheaf $\mathcal O_{C}$ of a smooth plane conic $C\subset \mathbb P^2_K$.

And I'll give you the most splendid reference: it is the very last result in Serre's FAC! (one of the most important papers ever published in Mathematics)
More precisely it is Proposition 6, § 81, page 276 in Serre, Faisceaux Algébriques Cohérents. There is an English translation here (as I learned from MathOverflow)

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Great answer, Georges! Thank you very much! – Descartes Aug 15 '11 at 17:54

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