# Projectivity of the Hilbert scheme

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.

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