Global Hilbert strata in a coherent projective bundle over a locally Noetherian base

Every affine open $U\subseteq S$ is Noetherian and $E{\mathrm{\mid }}_U$ has finitely many generators. A surjection ${\mathcal{O}}_U^{n+1}\twoheadrightarrow E{\mathrm{\mid }}_U$ embeds $Y_U$ in ${\mathbb{P}}_U^n$, carrying $M$ to the ambient twist. Choose $r\ge b(P)$ as in [F1], so the same degree works for all these local embeddings, even if their $n$ are unbounded. For a flat family on any $T$, the ambient degree-$r$ polynomial sections surject onto ${\pi }_{\ast }{\mathcal{O}}_Z(r)$ by [F1]. The map factors through $W_T=({\mathrm{Sym}}^{r}E{)}_T$, since the degree algebra relations of $Y$ vanish on $Z$. Its target is locally free of rank $P(r)$ and commutes with all base changes, giving a natural map to $G={\mathrm{Gr}}_S(W,P(r))$.

On $G$ let $K$ be the kernel of the universal quotient $W_G\twoheadrightarrow U$. This kernel commutes with arbitrary pullback because the locally free quotient splits the sequence locally. On $Y_G$ form the cokernel of evaluation ${\pi }^{\ast }K\otimes M^{-r}\to {\mathcal{O}}_{Y_G}$, a quotient structure algebra $F$. On every Noetherian affine base open, apply [F2] after embedding $Y$ into the local projective space: the polynomial-$P$ flattening stratum of $F$ represents exactly those Grassmannian quotients that reconstruct a flat family. A family's ideal in the ambient space is generated in degree $r$, so its image ideal on $Y$ is generated by $K$ under evaluation. Conversely when the reconstructed quotient is flat with polynomial $P$, the map $U\to {\pi }_{\ast }F(r)$ is onto and between locally free modules of the same rank, hence an isomorphism. This is the same two-sided reconstruction as [F2]. Thus the local strata represent the identical functor on overlaps, and their universal ideals agree. They glue to $H_P\to G$ and its universal family; this morphism is an immersion on the preimage of every affine base open.

Over a Noetherian affine base open $U$, $G_U$ is a finite-type Noetherian scheme, and its flattening stratum is locally closed and of finite presentation. Thus $H_P\to S$ is finitely presented and separated, these properties being local on the base. For a valuative diagram, the image of the valuation ring's closed point lies in such an affine open and all other images are its generizations. Use the local embedding $Y_U\subseteq {\mathbb{P}}_U^n$ and [F2] to obtain the unique flat closure. It lies in $Y_R$, since its flat structure sheaf is torsion-free and the ideal of $Y_R$ kills it generically. This proves the criterion over every valuation ring, so $H_P\to S$ is proper. Since $G\to S$ is separated, $H_P\to G$ is also proper: use its closed graph in $H_P{\times }_{S}G$ and the proper projection. Its local immersions are therefore closed immersions. Closed immersion is local on the target, so they give a global closed immersion in $G$. The global coherent-source Plücker embedding in [F1] proves the displayed projective-bundle embedding and the relatively very ample line assertion.