Given a Hilbert polynomial $f(x)=\binom{x+r}{r}\in \mathbb{Q}(x)$ where $r>0$. It’s said that Grassmannian scheme $Grass(n+1,r+1)$ over $\mathbb{Z}$ parameterises a tautological family of subscheme of $\mathbb{P}^n_\mathbb{Z}$ with Hilbert polynomial $f$.
By definition, it should mean a closed subscheme $Y\subset \mathbb{P}^n_\mathbb{Z} \times Grass(n+1,r+1)$ such that $Y$ is flat over Grass with Hilbert polynomial $f$. I don't know how to see this. Why is the Hilbert polynomial of the Grassmannian given above $f$? Hope someone could help. Thanks!