(Hilbert functor)
Let \(S\) be a Noetherian scheme, \(X \to S\) a projective morphism with a relatively very ample invertible sheaf \(\OO_X(1)\), and \(P \in \QQ[m]\). The Hilbert functor \(\operatorname{Hilb}^P_{X/S}\) sends a locally Noetherian \(S\)-scheme \(T\) to the set of closed subschemes \(Z \subseteq X \times_S T\) that are flat over \(T\) and whose fibres \(Z_t \subseteq X_t\), for \(t \in T\), have Hilbert polynomial \(m \mapsto \chi(Z_t, \OO_{Z_t}(m)) = P(m)\); it acts on morphisms \(T' \to T\) by pullback of subschemes. The Hilbert functor \(\operatorname{Hilb}_{X/S}\) is the disjoint union of the \(\operatorname{Hilb}^P_{X/S}\) over all \(P\).
(Grothendieck)
Each \(\operatorname{Hilb}^P_{X/S}\) is represented by a projective \(S\)-scheme, the Hilbert scheme \(\operatorname{Hilb}^P(X/S)\), and \(\operatorname{Hilb}_{X/S}\) is represented by \(\bigsqcup_P \operatorname{Hilb}^P(X/S)\). The universal family is a closed subscheme \(\mathcal{Z} \subseteq X \times_S \operatorname{Hilb}^P(X/S)\), flat over \(\operatorname{Hilb}^P(X/S)\), such that every family \(Z \subseteq X \times_S T\) as above is the pullback of \(\mathcal{Z}\) along a unique morphism \(T \to \operatorname{Hilb}^P(X/S)\).
Let \(k\) be a field and \(X = \PP^n_k\).
For \(P = 1\), a closed subscheme of \(\PP^n_k\) with Hilbert polynomial \(1\) is a \(k\)-point, and \(\operatorname{Hilb}^1(\PP^n_k) \cong \PP^n_k\) with \(\mathcal{Z}\) the diagonal.
For \(d \geq 1\) and \(P(m) = \binom{m+n}{n} - \binom{m-d+n}{n}\), the Hilbert polynomial of a degree-\(d\) hypersurface, from the sequence \(0 \to \OO(m-d) \to \OO(m) \to \OO_H(m) \to 0\), every closed subscheme with Hilbert polynomial \(P\) is a hypersurface \(V(F)\) of degree \(d\), and \(\operatorname{Hilb}^P(\PP^n_k) \cong \PP(H^0(\PP^n_k, \OO(d))) \cong \PP^{N}_k\) with \(N = \binom{n+d}{d} - 1\).
Dependencies
D-L6ERWThe Hilbert polynomial and its coefficientsuses