In his book "Deformation Theory", Hartshorne claims:

Let $Y$ be a closed subscheme of the projective space $X=\mathbb{P}^n_k$ over a field $k$. Then there exists a projective scheme $H$, called the Hilbert scheme, parametrizing closed subschemes of $X$ with the same Hilbert polynomial $P$ as $Y$ ...

What does projectivity of $H$ exactly mean here? Does it mean that it is also a closed subscheme of $\mathbb{P}^n_k$? Or does it mean that it is a closed subscheme of $\mathbb{P}^n_Y$? Wikipedia claims that it is projective over $Spec(\mathbb{Z})$, does this imply one of these options?

Actually I would like to know if $H$ is locally of finite type, which would be the case if it is a closed subscheme of $\mathbb{P}^n_k$.

Many thanks.

  • $\begingroup$ Yes it is projective as a closed subsecheme of $\mathbb{P}^d_k$. $\endgroup$ – Ahr Jun 12 at 15:01

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