Fixed-polarization Hilbert construction over a Noetherian base

On an affine open $U$ of $S$, choose a power $L^d$ giving an embedding $X_U\hookrightarrow {\mathbb{P}}_U^n$ as in [F3]. A polynomial-$P$ family for $L$ has polynomial $Q(t)=P(dt)$ for the ambient $\mathcal{O}(1)$; conversely equality of these substituted polynomials forces equality of the original polynomials. In the ambient representing scheme from [F1], require its universal quotient $\mathcal{O}\to {\mathcal{O}}_{\mathcal{Z}}$ to kill the pullback of the ideal of $X_U$. The zero locus in [F1] is a closed subscheme and universally imposes exactly $\mathcal{Z}\subseteq X_U$. It therefore represents the required functor on all $U$-schemes. The local representatives and universal ideals agree uniquely on overlaps through their functorial descriptions, and hence glue to $H_{X\mathrm{/}S}^{P,L}$ and its family. These schemes are of finite presentation locally on $S$, and the finite cover gives finite presentation globally; separatedness likewise follows from their Grassmannian embeddings on each base open.

A valuative diagram for this scheme is a generic-fibre family inside $X_R$ for some arbitrary valuation ring $R$. The map $\mathrm{Spec}R\to S$ factors through an affine open containing the image of its closed point, since the remaining images are generizations of that point. Use the local embedding of step 1.1 and [F2] to extend the generic family by flat schematic closure. It lies in $X_R$: every local section of the ideal of $X_R$ maps to zero generically, and torsion-freeness of the flat closure's structure sheaf forces it to vanish already over $R$. Uniqueness is that of flat closure. All hypotheses of the properness criterion in [F2] hold by step 1.1, so $H_{X\mathrm{/}S}^{P,L}\to S$ is proper.

For projectivity, use a finite affine cover as in step 1.1, take a common positive multiple $d$ of its embedding powers, and then a single sufficiently large regularity degree $r$ on that finite cover. The universal family's section bundle $V={\pi }_{\ast }({\mathcal{O}}_{\mathcal{Z}}\otimes L^{dr})$, is finite locally free by [F3]. Its determinant is relatively ample: on each base open it is the restriction of the Grassmannian Plücker bundle in the construction, hence relatively very ample there. Apply [F2] to obtain a closed embedding in a coherent projective bundle. For a specified global $X\subseteq {\mathbb{P}}_S^n$ with induced $L$, the same ambient construction is global: properness turns its locally closed Grassmannian immersion into a closed immersion, and its further closed vanishing locus gives the H-projective embedding.