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Given a Hilbert scheme $H$ of curves in $\mathbb{P}^3$ satisfying certain Hilbert polynomial, is there any way of understanding the degree or arithmetic genus of an irreducible component of the reduced scheme $H_{red}$? If so can this be generalised to Hilbert scheme parametrizing higher dimensional projective varieties or to Hilbert flag schemes?

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It seems to me that the title of your post does not match with the question. Maybe you meant "Hilbert function of a Hilbert scheme" instead of "Euler characteristic of a Hilbert scheme"? –  Francesco Polizzi Oct 1 '12 at 12:35

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