The Hilbert functor $\operatorname{Hilb}_{X/S}^P$ (parametrizing closed subschemes of $X/S$ with Hilbert polynomial $P$) satisfies Artin's criteria:

• Limit preservation: A closed subscheme of $X \times \operatorname{Spec}(\varinjlim A_i)$ is determined by a closed subscheme of $X \times \operatorname{Spec} A_i$ for some $i$ (finite presentation of ideals).
• Deformations: $T^1 = \operatorname{Hom}_{\mathcal{O}_Z}(\mathcal{I}/\mathcal{I}^2, \mathcal{O}_Z)$ (normal sheaf).
• Obstructions: $T^2 \subseteq \operatorname{Ext}^1_{\mathcal{O}_Z}(\mathcal{I}/\mathcal{I}^2, \mathcal{O}_Z)$.
• Effectivity: Grothendieck's formal GAGA.

This proves that $\operatorname{Hilb}_{X/S}^P$ is an algebraic space -- in fact, a projective scheme (by Grothendieck's original argument using Castelnuovo--Mumford regularity).

The Picard functor $\operatorname{Pic}_{X/S}$ (parametrizing line bundles on $X/S$) satisfies Artin's criteria:

• Deformations: $T^1 = H^1(X_s, \mathcal{O}_{X_s})$ (first cohomology of the structure sheaf).
• Obstructions: $T^2 = H^2(X_s, \mathcal{O}_{X_s})$.
• Effectivity: Formal GAGA again.

When $H^2(X_s, \mathcal{O}_{X_s}) = 0$ (e.g., for curves, or for abelian varieties), the Picard functor is smooth. The result is that $\operatorname{Pic}_{X/S}$ is an algebraic space (and a group scheme when $X/S$ is proper with geometrically connected fibers).

The moduli stack $\mathcal{M}_g$ satisfies Artin's criteria:

• Deformations of a curve $C$: $T^1 = H^1(C, \mathcal{T}_C)$, where $\mathcal{T}_C$ is the tangent sheaf. For $g \geq 2$, $\dim T^1 = 3g - 3$ (by Riemann--Roch: $\chi(\mathcal{T}_C) = 3 - 3g$ and $H^0(C, \mathcal{T}_C) = 0$).
• Obstructions: $T^2 = H^2(C, \mathcal{T}_C) = 0$ (since $C$ is a curve). So $\mathcal{M}_g$ is smooth.
• Automorphisms: $T^0 = H^0(C, \mathcal{T}_C) = 0$ for $g \geq 2$ (finite automorphism group $\Rightarrow$ no infinitesimal automorphisms).
• Effectivity: Follows from formal GAGA for proper schemes.

This proves $\mathcal{M}_g$ is an algebraic stack, smooth of dimension $3g - 3$.

Let $C$ be a smooth projective curve of genus $g$. For the stack $\operatorname{Bun}_n(C)$ of rank-$n$ vector bundles:

• Deformations of $E$: $T^1 = H^1(C, \operatorname{End}(E)) = \operatorname{Ext}^1(E, E)$, with $\dim T^1 = n^2(g - 1) + \dim H^0(C, \operatorname{End}(E))$.
• Obstructions: $T^2 = H^2(C, \operatorname{End}(E)) = 0$ (curve!). So $\operatorname{Bun}_n(C)$ is smooth.
• Automorphisms: $T^0 = H^0(C, \operatorname{End}(E)) = \operatorname{End}(E)$. For a stable bundle, $\operatorname{End}(E) = k$ (Schur's lemma), so $\dim T^0 = 1$ and $\operatorname{Aut}(E) = \mathbb{G}_m$.

The dimension is $\dim \operatorname{Bun}_n(C) = \dim T^1 - \dim T^0 = n^2(g-1)$ (as a stack).

For a smooth projective surface $X$ and the moduli stack of coherent sheaves with fixed numerical invariants:

• Deformations of $\mathcal{F}$: $T^1 = \operatorname{Ext}^1(\mathcal{F}, \mathcal{F})$.
• Obstructions: $T^2 = \operatorname{Ext}^2(\mathcal{F}, \mathcal{F})$. This is nonzero in general, so the moduli space may be singular.
• By Serre duality, if $\omega_X \cong \mathcal{O}_X$ (K3 or abelian surface), then $\operatorname{Ext}^2(\mathcal{F}, \mathcal{F}) \cong \operatorname{Hom}(\mathcal{F}, \mathcal{F})^{\vee}$, and the trace-free obstructions lie in a space of dimension $\dim \operatorname{Ext}^2_0 = \dim \operatorname{Ext}^0_0$. The expected dimension is $\dim \operatorname{Ext}^1 - \dim \operatorname{Ext}^0 - \dim \operatorname{Ext}^2 + \dim \operatorname{Ext}^0 = 2n - (r^2 - 1)\chi(\mathcal{O}_X)$ by Riemann--Roch.

For proper $X/S$ and separated $Y/S$ of finite presentation, the mapping stack $\operatorname{Map}_S(X, Y)$ (whose $T$-points are morphisms $X_T \to Y_T$) is an algebraic stack:

• Deformations of $f : X \to Y$: $T^1 = H^0(X, f^*\mathcal{T}_Y)$ (sections of the pullback of the tangent bundle).
• Obstructions: $T^2 \subseteq H^1(X, f^*\mathcal{T}_Y)$.
• The expected dimension is $\chi(X, f^*\mathcal{T}_Y) = \deg(f^*\mathcal{T}_Y) + \dim Y \cdot (1-g)$ for $X$ a curve of genus $g$.

Artin's theorem proves algebraicity. Properness (for the Kontsevich compactification with stable maps) requires additional arguments.

The moduli stack $\mathcal{A}_g$ of principally polarized abelian varieties satisfies Artin's criteria:

• Deformations of $(A, \lambda)$: $T^1 = \operatorname{Sym}^2 H^1(A, \mathcal{O}_A) \cong \operatorname{Sym}^2(\operatorname{Lie}(A^{\vee}))$, with $\dim T^1 = g(g+1)/2$.
• Obstructions: Unobstructed (the deformation space is smooth), $T^2 = 0$.
• Automorphisms: $T^0 = \operatorname{Lie}(\operatorname{Aut}(A, \lambda))$. For a general ppav, $\operatorname{Aut}(A, \lambda) = \{\pm 1\}$, so $T^0 = 0$.

Hence $\mathcal{A}_g$ is a smooth algebraic stack of dimension $g(g+1)/2$.

For polarized K3 surfaces $(X, H)$ of degree $H^2 = 2d$:

• Deformations: $T^1 = H^1(X, \mathcal{T}_X) \cong H^1(X, \Omega^1_X)$ (by $\mathcal{T}_X \cong \Omega^1_X$ since $\omega_X \cong \mathcal{O}_X$). By Hodge theory, $\dim T^1 = 20$.
• Obstructions: $T^2 = H^2(X, \mathcal{T}_X) \cong H^0(X, \Omega^1_X) = 0$ (since $h^{1,0} = 0$ for K3). So the moduli is unobstructed and smooth of dimension 20.
• The polarization condition cuts out a divisor, giving $\dim \mathcal{F}_{2d} = 19$.

Artin's theorem proves that the moduli of polarized K3 surfaces is an algebraic stack (in fact a DM stack, since $\operatorname{Aut}(X, H)$ is finite).

In derived algebraic geometry, Lurie proved a derived version of Artin representability: a functor $\mathcal{X} : \mathrm{CAlg}_k^{\mathrm{cn}} \to \mathcal{S}$ (from connective $E_\infty$-algebras to spaces) is a derived algebraic stack if and only if:

1. $\mathcal{X}$ is a sheaf for the etale topology.
2. $\mathcal{X}$ is nilcomplete, integrable, and infinitesimally cohesive.
3. $\mathcal{X}$ has a cotangent complex $L_{\mathcal{X}}$ that is connective and almost of finite presentation.

This dramatically simplifies Artin's criteria: the cotangent complex subsumes conditions (3)--(6) of the classical theorem. Many modern constructions of moduli spaces (e.g., derived moduli of perfect complexes, derived mapping stacks) use Lurie's version.

Consider the functor $F(T) = \{f \in \Gamma(T, \mathcal{O}_T) : f^2 = 0 \text{ everywhere}\}$. This is the functor of points of $\operatorname{Spec} k[x]/(x^2)$, which is representable by a scheme. But if we modify to $F(T) = \{f \in \Gamma(T, \mathcal{O}_T) : f \text{ is nilpotent}\}$ (nilpotent of unspecified order), this fails the finite presentation condition (limit preservation) and is not an algebraic space.

The "formal completion functor" $\hat{F}(T) = \varprojlim F(T/\mathfrak{m}^n)$ (formal deformations without algebraization) satisfies Schlessinger's conditions but fails effectivity, and hence is not algebraic.

The Quot functor $\operatorname{Quot}_{E/X/S}^P$ (parametrizing quotients of a coherent sheaf $E$ on $X/S$ with Hilbert polynomial $P$) satisfies Artin's criteria:

• Deformations of $E \twoheadrightarrow F$: $T^1 = \operatorname{Hom}(K, F)$ where $K = \ker(E \to F)$.
• Obstructions: $T^2 \subseteq \operatorname{Ext}^1(K, F)$.
• Effectivity: Formal GAGA.

Grothendieck proved $\operatorname{Quot}_{E/X/S}^P$ is a projective scheme (stronger than just algebraic).

For a reductive group $G$ over a field $k$ and a smooth projective curve $C$, the moduli stack $\operatorname{Bun}_G(C)$ of principal $G$-bundles on $C$ satisfies Artin's criteria:

• Deformations of $P$: $T^1 = H^1(C, \operatorname{Ad}(P))$, where $\operatorname{Ad}(P) = P \times^G \mathfrak{g}$ is the adjoint bundle.
• Obstructions: $T^2 = H^2(C, \operatorname{Ad}(P)) = 0$.
• Dimension: $\dim \operatorname{Bun}_G(C) = \dim G \cdot (g - 1)$.

This is a smooth algebraic stack, locally of finite type, central to the geometric Langlands program.