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In the Proposition 2.1 of the paper 'Nodes and the Hodge conjecture', R.P.THOMAS gives a proof to descending the Hodge conjecture into showing that every (n,n)-Hodge class in a $2n$-dimensional smooth projective complex variety is algebraic.

However, I can't understand the part of relative Hilbert scheme there. Since the Hilbert scheme there seems different from the usual one and I even can't find the definition. I am wondering if anyone could explain to me about this or provide to me with some proper references.

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  • $\begingroup$ As far as I can tell, Grothendieck constructed the Hilbert Scheme in this generality. It's certainly done in the first chapter of Kollár's book on Rational Curves. $\endgroup$ – Tabes Bridges Aug 1 '19 at 19:30
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As R.P.Thomas told me, the relative Hilbert scheme there is the usual one, and the section 4 of the paper A remark on singularities of primitive cohomology classes gives a more explicit refinement.

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