Let $X$ be a projective variety over $\mathbb{C}$. If $L$ is an ample line bundle, then $h\_L$ denotes the Hilbert polynomial.



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> > Is it true that, if $L$ and $L'$ are ample line bundles which are equal in the Neron-Severi group, then $h\_L = h\_{L'}$?
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> > Does this imply, together with finite generation of Neron-Severi groups, that the set of polynomials $\{h\_L \ | \ L $ ample line bundle on $X \}$ is finite?
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Can anyone recommend a text (book or article) in which these things are explained (to some extent)?



Does a projective variety have only finitely many associated Hilbert polynomials?
Yes for the first question, by Riemann--Roch.


No for the second --- even in the simplest case of a projective line, the polynomial $td + 1$ is the Hilbert polynomial (with respect to $L = O(d)$).


