When $P(t) = n$ is a constant polynomial, $\mathrm{Hilb}^n_{X/k}$ parametrizes 0-dimensional subschemes of length $n$. This is the Hilbert scheme of $n$ points.

For $X = \mathbb{A}^2$ (or any smooth surface), $\mathrm{Hilb}^n(X)$ is a smooth variety of dimension $2n$ (Fogarty's theorem). It comes with a Hilbert-Chow morphism $\mathrm{Hilb}^n(X) \to \mathrm{Sym}^n(X) = X^n / \mathfrak{S}_n$ that is a resolution of singularities.

The Hilbert scheme $\mathrm{Hilb}^2(\mathbb{A}^2)$ parametrizes ideals $I \subset k[x,y]$ with $\dim_k k[x,y]/I = 2$. These are either:

• Two distinct points: $I = (x - a_1, y - b_1) \cap (x - a_2, y - b_2)$, or
• A "fat point" with a tangent direction: $I = (x^2, y - \lambda x)$ at the origin (up to translation).

The scheme $\mathrm{Hilb}^2(\mathbb{A}^2) \cong \mathrm{Bl}_{\Delta}(\mathbb{A}^2 \times \mathbb{A}^2) / \mathfrak{S}_2$, the blowup of the symmetric product along the diagonal. It is smooth of dimension 4.

For the affine line, $\mathrm{Hilb}^n(\mathbb{A}^1) \cong \mathbb{A}^n$. An ideal $I \subset k[x]$ of colength $n$ is generated by a unique monic polynomial of degree $n$: $I = (x^n + a_{n-1}x^{n-1} + \cdots + a_0)$. The coefficients give the isomorphism with $\mathbb{A}^n$.

The Hilbert polynomial of a curve of degree $d$ and arithmetic genus $g$ in $\mathbb{P}^3$ is $P(t) = dt + 1 - g$. The Hilbert scheme $\mathrm{Hilb}^{dt + 1 - g}_{\mathbb{P}^3}$ has components:

• Twisted cubics ($d = 3, g = 0$): $P(t) = 3t + 1$. The component parametrizing smooth twisted cubics has dimension 12.
• Lines ($d = 1, g = 0$): $P(t) = t + 1$. This component is the Grassmannian $\mathrm{Gr}(2, 4) = \mathrm{Gr}(1, \mathbb{P}^3)$, dimension 4.
• Plane conics ($d = 2, g = 0$): $P(t) = 2t + 1$. Dimension 8 (choice of plane + conic in the plane).

A hypersurface of degree $d$ in $\mathbb{P}^n$ is a closed subscheme defined by a single homogeneous polynomial. The Hilbert polynomial is $P(t) = \binom{t+n}{n} - \binom{t+n-d}{n}$. The Hilbert scheme is $\mathrm{Hilb}^P_{\mathbb{P}^n} \cong \mathbb{P}^{\binom{n+d}{d} - 1}$ the projectivization of the space of degree-$d$ forms. For $n = 2, d = 3$: plane cubics form $\mathbb{P}^9$.

The Grassmannian is the Quot scheme: $\mathrm{Gr}(k, n) = \mathrm{Quot}^k_{\mathcal{O}^n_{\mathrm{pt}} / \mathrm{pt} / \mathrm{pt}}$ parametrizing rank-$k$ quotients of $\mathcal{O}^n$. More generally, the flag variety $\mathrm{Fl}(d_1, \ldots, d_r; n)$ can be realized as an iterated Quot scheme.

For a smooth curve $C$ and $\mathcal{E} = \mathcal{O}_C^r$, the Quot scheme $\mathrm{Quot}^n_{\mathcal{O}_C^r / C}$ (quotients of length $n$) has dimension $rn$. For $r = 1$, this is $\mathrm{Hilb}^n(C) = C^{(n)} = \mathrm{Sym}^n(C)$, which is smooth of dimension $n$.

For $Z = \lbrace p \rbrace \subset \mathbb{A}^2$ a single reduced point, $\mathcal{I}_Z / \mathcal{I}_Z^2 \cong \mathfrak{m}_p / \mathfrak{m}_p^2$, the cotangent space. So $T_{[p]} \mathrm{Hilb}^1(\mathbb{A}^2) = \mathrm{Hom}(\mathfrak{m}_p/\mathfrak{m}_p^2, k) = T_p \mathbb{A}^2 \cong k^2,$ consistent with $\mathrm{Hilb}^1(\mathbb{A}^2) \cong \mathbb{A}^2$.

For a smooth hypersurface $Z = V(f) \subset \mathbb{P}^n$ of degree $d$, the ideal sheaf is $\mathcal{I}_Z = \mathcal{O}(-d)$. Then $T_{[Z]} \mathrm{Hilb} = \mathrm{Hom}(\mathcal{O}(-d), \mathcal{O}_Z) = H^0(Z, \mathcal{O}_Z(d)) = H^0(\mathbb{P}^n, \mathcal{O}(d)) / \langle f \rangle$ The dimension is $\binom{n+d}{d} - 1$, confirming that hypersurfaces of degree $d$ form $\mathbb{P}^{\binom{n+d}{d}-1}$.

For a smooth projective surface $S$, $\mathrm{Hilb}^n(S)$ is smooth of dimension $2n$. The proof uses the fact that for a length-$n$ subscheme $Z \subset S$: $\mathrm{Ext}^1(\mathcal{I}_Z, \mathcal{O}_Z) \cong \mathrm{Ext}^2(\mathcal{O}_Z, \mathcal{O}_Z) = 0$ by Serre duality on the surface (since $\mathrm{Hom}(\mathcal{O}_Z, \mathcal{O}_Z \otimes \omega_S) \cong H^0(Z, \omega_S|_Z)$, which relates to the obstruction space, and the vanishing follows from dimension reasons on $Z$).

Let $X \subset \mathbb{P}^4$ be a smooth cubic threefold. The Hilbert scheme of lines (subschemes with Hilbert polynomial $P(t) = t + 1$) on $X$ is the Fano surface $F(X)$. It is a smooth surface of general type with:

• $\dim F(X) = 2$
• $\chi(\mathcal{O}_{F(X)}) = 1$
• The Albanese variety of $F(X)$ is the intermediate Jacobian $J(X)$, a principally polarized abelian variety of dimension 5.

For a smooth cubic fourfold $X \subset \mathbb{P}^5$, the Fano variety of lines $F(X)$ is a smooth projective 4-fold. By Beauville-Donagi, $F(X)$ is a hyperkahler manifold deformation-equivalent to $\mathrm{Hilb}^2(K3)$. It carries a natural symplectic form.

Rank-2 torsion-free sheaves on $\mathbb{P}^2$ with $c_1 = 0$ and $c_2 = n$ have moduli space $M(2, 0, n)$. By the ADHM construction, $M(2, 0, n)$ can be described as a GIT quotient: $M(2, 0, n) \cong \left\lbrace (B_1, B_2, I, J) \;\middle|\; [B_1, B_2] + IJ = 0, \text{ stability} \right\rbrace / \mathrm{GL}_n$ where $B_1, B_2 \in \mathrm{End}(V)$, $I \in \mathrm{Hom}(W, V)$, $J \in \mathrm{Hom}(V, W)$ with $\dim V = n$, $\dim W = 2$.

Let $f: \mathcal{C} \to B$ be a family of smooth curves of genus $g$. The relative Hilbert scheme $\mathrm{Hilb}^n_{\mathcal{C}/B}$ parametrizes effective divisors of degree $n$ on the fibers. Its fiber over $b \in B$ is $\mathrm{Sym}^n(C_b)$. When $n \geq 2g - 1$, the Abel-Jacobi map $\mathrm{Hilb}^n_{\mathcal{C}/B} \to \mathrm{Pic}^n_{\mathcal{C}/B}$ is a $\mathbb{P}^{n-g}$-bundle.

The nested Hilbert scheme $\mathrm{Hilb}^{n, n+1}(S)$ for a surface $S$ parametrizes pairs $(Z, Z')$ where $Z \subset Z'$ are subschemes of lengths $n$ and $n+1$. It fits into a diagram: $\mathrm{Hilb}^{n,n+1}(S) \xrightarrow{p_1} \mathrm{Hilb}^n(S)$ $\phantom{\mathrm{Hilb}^{n,n+1}(S)} \xrightarrow{p_2} \mathrm{Hilb}^{n+1}(S)$ The fiber of $p_1$ over $[Z]$ is the blowup $\mathrm{Bl}_Z(S)$. These nested structures play a role in the theory of Heisenberg algebras acting on $\bigoplus_n H^*(\mathrm{Hilb}^n(S))$ (Nakajima).