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.