Construction of the fixed-polynomial Hilbert scheme of projective space

Choose $r$ large enough that every kernel and quotient of ${\mathcal{O}}_{{\mathbb{P}}_k^n}\twoheadrightarrow {\mathcal{O}}_Z$ with polynomial $P$, over every field, is $r$-regular. For any Hilbert family on $T$, its ideal $I$ and quotient are base-flat: the ambient structure sheaf is base-flat, so the Tor sequence gives this for $I$. They are finitely presented, locally by descent to a flat Noetherian stage as in [F1]. Thus the section sequence is an exact sequence of vector bundles $0\to {\pi }_{\ast }I(r)\to W_T\to {\pi }_{\ast }{\mathcal{O}}_Z(r)\to 0$, with $W=H^0({\mathbb{P}}_S^n,\mathcal{O}(r))$ and quotient rank $P(r)$, compatible with every base change. If the rank is impossible the functor is empty. Otherwise it gives a natural map to $G={\mathrm{Gr}}_S(W,P(r))$.

On $G$, let $K$ be the kernel of its universal quotient. On ${\mathbb{P}}_G^n$ form $F=\mathrm{coker}({\pi }^{\ast }K\otimes \mathcal{O}(-r)\to \mathcal{O})$, with the map obtained by evaluation. Since the image is an ideal, $F$ is the structure sheaf of a closed finitely presented subscheme. Let $H_P\subseteq G$ be its polynomial-$P$ universal flattening stratum from [F2]. For every Hilbert family, evaluation generates $I(r)$ by [F1], so the cokernel reconstruction is exactly its structure sheaf and the map to $G$ factors through $H_P$.

Conversely a map $T\to H_P$ gives a flat family with polynomial $P$. The universal rank-$P(r)$ quotient $J$ of $W_T$ maps to its section module because evaluation kills $K_T$. That map is surjective by [F1] applied to the reconstructed ideal and structure sheaf. Both source and target are locally free of the same rank, so it is an isomorphism. Thus the recovered Grassmannian quotient is the original one. These two constructions are inverse for every $T$, and commute with every pullback. This proves representability and the universal-family assertion, including nonreduced test schemes.