A representable moduli functor
The Hilbert functor and the Hilbert scheme (Tag D-HILBSCHEME)
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 \(N = \binom{n+d}{d} - 1\).
Quotients by étale equivalence relations
Étale equivalence relations and algebraic spaces (Tag D-ALGSPACE)
Let \(S\) be a scheme, and regard \(S\)-schemes as sheaves on \((\Sch_{/S})_{\mathrm{et}}\) through their functors of points (D-ETFPPF).
(Étale equivalence relation)
Let \(U\) be an \(S\)-scheme. An étale equivalence relation on \(U\) is an \(S\)-scheme \(R\) with a monomorphism \((s, t) \colon R \to U \times_S U\) such that
for every \(S\)-scheme \(T\), the subset \(R(T) \subseteq U(T) \times U(T)\) is an equivalence relation on \(U(T)\), and
the two projections \(s, t \colon R \to U\) are étale.
Its quotient \(U/R\) is the sheafification, on \((\Sch_{/S})_{\mathrm{et}}\), of the presheaf \(T \mapsto U(T)/R(T)\).
(Algebraic space)
An algebraic space over \(S\) is a sheaf \(X\) on \((\Sch_{/S})_{\mathrm{et}}\) such that
the diagonal \(X \to X \times_S X\) is representable: for every \(S\)-scheme \(T\) and every \(T \to X \times_S X\), the fibre product \(X \times_{X \times_S X} T\) is a scheme, and
there are a scheme \(U\) and a morphism \(U \to X\) that is étale and surjective, meaning that for every scheme \(T \to X\) the base change \(U \times_X T \to T\), a morphism of schemes by condition 1, is étale and surjective.
A morphism \(U \to X\) as in condition 2 is an étale atlas.
If \(U \to X\) is an étale atlas of an algebraic space, then \(R = U \times_X U \to U \times_S U\) is an étale equivalence relation and \(X \cong U/R\). Conversely, the quotient \(U/R\) of any étale equivalence relation on a scheme \(U\) is an algebraic space, with étale atlas \(U \to U/R\).
Every scheme \(X\) is an algebraic space, with atlas \(\id \colon X \to X\) and \(R\) the diagonal. If a finite group \(G\) acts freely on an \(S\)-scheme \(U\), then \(R = U \times G \to U \times_S U\), \((u, g) \mapsto (u, ug)\), is an étale equivalence relation, and \(U/R\) is an algebraic space, the quotient \(U/G\); it is a scheme when, for instance, \(U\) is quasiprojective over a field.
Stacks
Objects with nontrivial automorphisms, such as curves of genus \(g \geq 2\) with automorphisms, obstruct the representability of a moduli functor by a scheme or algebraic space (Coarse versus fine moduli spaces); a stack records the automorphisms as part of the data.
Algebraic stacks, substacks, smooth stacks, and orbifolds (Tag D-ALGSTACK)
Let \(S\) be a scheme and \(\Sch_{/S}\) the category of \(S\)-schemes with the étale topology (D-ETFPPF).
(Stack)
A stack over \(S\) is a category \(\mathcal{X}\) with a functor \(p \colon \mathcal{X} \to \Sch_{/S}\) such that
\(\mathcal{X}\) is fibred in groupoids: for every morphism \(T' \to T\) and object \(\xi\) over \(T\) there is a pullback \(\xi|_{T'}\) over \(T'\), unique up to unique isomorphism, and every morphism lying over an identity is an isomorphism;
for all objects \(\xi, \eta\) over \(T\), the presheaf \(\operatorname{Isom}(\xi, \eta) \colon (T' \to T) \mapsto \{\text{isomorphisms } \xi|_{T'} \to \eta|_{T'} \text{ over } T'\}\) is a sheaf on \((\Sch_{/T})_{\mathrm{et}}\);
every descent datum is effective: for an étale covering \(\{T_i \to T\}\), objects \(\xi_i\) over \(T_i\) with isomorphisms \(\varphi_{ij} \colon \xi_j|_{T_{ij}} \to \xi_i|_{T_{ij}}\) satisfying \(\varphi_{ij} \circ \varphi_{jk} = \varphi_{ik}\) on \(T_{ijk}\) come from an object \(\xi\) over \(T\).
A \(1\)-morphism \(\mathcal{Y} \to \mathcal{X}\) of stacks is representable if for every \(S\)-scheme \(T\) and \(T \to \mathcal{X}\), the \(2\)-fibre product \(\mathcal{Y} \times_{\mathcal{X}} T\) is an algebraic space (D-ALGSPACE). For a class \(P\) of morphisms of algebraic spaces that is stable under base change and étale local on the target (D-MORLOCAL), a representable \(\mathcal{Y} \to \mathcal{X}\) has \(P\) if every such base change \(\mathcal{Y} \times_{\mathcal{X}} T \to T\) has \(P\).
(Algebraic and Deligne--Mumford stacks)
A stack \(\mathcal{X}\) over \(S\) is algebraic if its diagonal \(\mathcal{X} \to \mathcal{X} \times_S \mathcal{X}\) is representable and there are a scheme \(U\) and a representable, smooth, surjective morphism \(U \to \mathcal{X}\), a smooth atlas. It is Deligne–Mumford if moreover it has an atlas \(U \to \mathcal{X}\) that is étale.
(Substacks)
A substack of a stack \(\mathcal{X}\) is a full subcategory \(\mathcal{Z} \subseteq \mathcal{X}\) closed under isomorphisms and pullbacks. It is an open, closed or locally closed substack if the inclusion \(\mathcal{Z} \to \mathcal{X}\) is representable by open immersions, closed immersions or immersions: for every \(S\)-scheme \(T\) and object of \(\mathcal{X}\) over \(T\), the base change \(\mathcal{Z} \times_{\mathcal{X}} T \to T\) is an open, closed or locally closed immersion of schemes.
(Smooth stack)
An algebraic stack \(\mathcal{X}\) over \(S\) is smooth over \(S\) if for some smooth atlas \(U \to \mathcal{X}\) the composite \(U \to S\) is smooth.
Smoothness of \(U \to S\) does not depend on the choice of smooth atlas.
(Orbifold)
An orbifold over \(\CC\) is a smooth, separated Deligne–Mumford stack of finite type over \(\CC\) whose stabilizer group at a general point is trivial.
The weighted projective line \(\PP(1,2)\) is the quotient stack \([(\AA^2 \setminus \{0\}) / \GG_m]\) for the action \(\lambda \cdot (x, y) = (\lambda x, \lambda^2 y)\) over \(\CC\). The atlas \(\AA^2 \setminus \{0\} \to \PP(1,2)\) is smooth, and the stabilizers are finite and reduced: a point \((x, y)\) with \(x \neq 0\) has trivial stabilizer, while \((0, 1)\) has stabilizer \(\{\lambda : \lambda^2 = 1\} = \mu_2\). So \(\PP(1,2)\) is a smooth separated Deligne–Mumford stack with trivial generic stabilizer, an orbifold with one point of stabilizer \(\mu_2\). The moduli stacks \(\mathcal{M}_g\) of smooth curves of genus \(g \geq 2\) are smooth Deligne–Mumford stacks (T-MGSMOOTH).