# How to understand the proof of Proposition 2.1 in the paper 'Nodes and the Hodge conjecture'?

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.

• 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. – Tabes Bridges Aug 1 '19 at 19:30