# Boundedness of Hilbert polynomials of hypersurfaces

Let $(X,H)$ be a smooth polarized projective variety of dimension $n$. If $Y \subset X$ is an irreducible hypersurface then its degree is $H^{n-1} \cdot Y$, and its Hilbert polynomial is $p_Y(t) = \chi(\mathcal O_Y(tH))$.

Given a natural number $N$, is the set of Hilbert polynomials of reduced irreducible hypersurfaces of degree at most $N$ finite ?

If $X$ is a surface then the answer is yes, as the Hilbert polynomial of a curve is determined by its degree and its arithmetical genus. The former is bounded by hypothesis, and the latter can be bounded after embedding $X$ into a projective space.

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Yes: for a given $N, d > 0$, the Hilbert polynomials of degree-$d$ geometrically integral subschemes of $\mathbf{P}^N$ (as we vary over all ground fields) have only finitely many possibilities. This is due to Chow, and is an ingredient in Grothendieck's original construction of Hilbert schemes. It can be deduced from the theory of Chow forms (or Chow coordinates), for example. –  BCnrd Oct 7 '10 at 14:17
Thanks. That was helpful. –  jvp Oct 7 '10 at 16:43