Local structure of algebraic stacks

Since the inception of algebraic stacks it was expected that they are locally quotient stacks of the form ${\displaystyle [{\text{Spec}}(A)/G]}$ where ${\displaystyle G}$ is a linearly reductive algebraic group. This was recently proved to be the case:^[1] given a quasi-separated algebraic stack ${\displaystyle {\mathfrak {X}}}$ locally of finite type over an algebraically closed field ${\displaystyle k}$ whose stabilizers are affine, and ${\displaystyle x\in {\mathfrak {X}}(k)}$ a smooth and closed point with linearly reductive stabilizer group ${\displaystyle G_{x}}$, there exists an etale cover of the GIT quotient ${\displaystyle (U,u)\to (N_{x}//G_{x},0)}$, where ${\displaystyle N_{x}=(J_{x}/J_{x}^{2})^{\vee }}$, such that the diagram

${\displaystyle {\begin{matrix}([W/G_{x}],w)&\to &([N_{x}/G_{x}],0)\\\downarrow &&\downarrow \\(U,u)&\to &(N_{x}//G_{x},0)\end{matrix}}}$

is cartesian, and there exists an etale morphism

${\displaystyle f:([W/G_{x}],w)\to ({\mathfrak {X}},x)}$

inducing an isomorphism of the stabilizer groups at ${\displaystyle w}$ and ${\displaystyle x}$.

Stack quotients

If ${\displaystyle X}$ is a scheme ${\displaystyle (Sch/S)}$ and ${\displaystyle G}$ is a smooth affine group scheme acting on ${\displaystyle X}$, then there is a quotient algebraic stack ${\displaystyle [X/G]}$,^[2] taking a scheme ${\displaystyle Y\to S}$ to the groupoid of ${\displaystyle G}$-torsors over the ${\displaystyle S}$-scheme ${\displaystyle Y}$ with ${\displaystyle G}$-equivariant maps to ${\displaystyle X}$. Explicitly, given a space ${\displaystyle X}$ with a ${\displaystyle G}$-action, form the stack ${\displaystyle [X/G]}$ which (intuitively speaking) sends a space ${\displaystyle Y}$ to the groupoid of pullback diagrams

${\displaystyle [X/G](Y)={\begin{Bmatrix}Z&{\xrightarrow {\Phi }}&X\\\downarrow &&\downarrow \\Y&{\xrightarrow {\phi }}&[X/G]\end{Bmatrix}}}$

where ${\displaystyle \Phi }$ is a ${\displaystyle G}$-equivariant morphism of spaces and ${\displaystyle Z\to Y}$is a principal ${\displaystyle G}$-bundle. The morphisms in this category are just morphisms of diagrams where the arrows on the right-hand side are equal and the arrows on the left-hand side are morphisms of principal ${\displaystyle G}$-bundles.

Classifying stacks

A special case of this when X is a point gives the classifying stack BG of a smooth affine group scheme G: ${\displaystyle {\textbf {B}}G:=[pt/G].}$ It is named so since the category ${\displaystyle \mathbf {B} G(Y)}$, the fiber over Y, is precisely the category ${\displaystyle \operatorname {Bun} _{G}(Y)}$ of principal ${\displaystyle G}$-bundles over ${\displaystyle Y}$. Note that ${\displaystyle \operatorname {Bun} _{G}(Y)}$ itself can be considered as a stack, the moduli stack of principal G-bundles on Y.

An important subexample from this construction is ${\displaystyle \mathbf {B} GL_{n}}$ which is the moduli stack of principal ${\displaystyle GL_{n}}$-bundles. Since the data of a principal ${\displaystyle GL_{n}}$-bundle is equivalent to the data of a rank ${\displaystyle n}$ vector bundle, this is isomorphic to the moduli stack of rank ${\displaystyle n}$ vector bundles ${\displaystyle Vect_{n}}$.

Gerbes

A gerbe is a stack in groupoids which always has a nonempty category. for example the trivial gerbe ${\displaystyle BG}$ that assigns to each scheme the groupoid of principal ${\displaystyle G}$-bundles over the scheme, for some group ${\displaystyle G}$.

Relative spec and proj

If A is a quasi-coherent sheaf of algebras in an algebraic stack X over a scheme S, then there is a stack Spec(A) generalizing the construction of the spectrum Spec(A) of a commutative ring A. An object of Spec(A) is given by an S-scheme T, an object x of X(T), and a morphism of sheaves of algebras from x*(A) to the coordinate ring O(T) of T.

If A is a quasi-coherent sheaf of graded algebras in an algebraic stack X over a scheme S, then there is a stack Proj(A) generalizing the construction of the projective scheme Proj(A) of a graded ring A.

Moduli of curves

• Mumford (1965) studied the moduli stack M_1,1 of elliptic curves, and showed that its Picard group is cyclic of order 12. For elliptic curves over the complex numbers the corresponding stack is similar to a quotient of the upper half-plane by the action of the modular group.
• The moduli space of algebraic curves ${\displaystyle {\mathcal {M}}_{g}}$ defined as a universal family of smooth curves of given genus ${\displaystyle g}$ does not exist as an algebraic variety because in particular there are curves admitting nontrivial automorphisms. However there is a moduli stack ${\displaystyle {\mathcal {M}}_{g}}$ which is a good substitute for the non-existent fine moduli space of smooth genus ${\displaystyle g}$ curves. More generally there is a moduli stack ${\displaystyle {\mathcal {M}}_{g,n}}$ of genus ${\displaystyle g}$ curves with ${\displaystyle n}$ marked points. In general this is an algebraic stack, and is a Deligne–Mumford stack for ${\displaystyle g\geq 2}$ or ${\displaystyle g=1,n\geq 1}$ or ${\displaystyle g=0,n\geq 3}$ (in other words when the automorphism groups of the curves are finite). This moduli stack has a completion consisting of the moduli stack of stable curves (for given ${\displaystyle g}$ and ${\displaystyle n}$) which is proper over Spec Z. For example, ${\displaystyle {\mathcal {M}}_{0}}$ is the classifying stack ${\displaystyle B{\text{PGL}}(2)}$ of the projective general linear group. (There is a subtlety in defining ${\displaystyle {\mathcal {M}}_{1}}$, as one has to use algebraic spaces rather than schemes to construct it.)

Kontsevich moduli spaces

Another widely studied class of moduli spaces are the Kontsevich moduli spaces parameterizing the space of stable maps between curves of a fixed genus to a fixed space ${\displaystyle X}$ whose image represents a fixed cohomology class. These moduli spaces are denoted^[3]

${\displaystyle {\overline {\mathcal {M}}}_{g,n}(X,\beta )}$

and can have wild behavior, such as being reducible stacks whose components are non-equal dimension. For example,^[3] the moduli stack

${\displaystyle {\overline {\mathcal {M}}}_{1,0}(\mathbb {P} ^{2},3[H])}$

has smooth curves parametrized by an open subset ${\displaystyle U\subset \mathbb {P} ^{9}=\mathbb {P} (\Gamma (\mathbb {P} ^{2},{\mathcal {O}}(3)))}$. On the boundary of the moduli space, where curves may degenerate to reducible curves, there is a substack parametrizing reducible curves with a genus ${\displaystyle 0}$ component and a genus ${\displaystyle 1}$ component intersecting at one point, and the map sends the genus ${\displaystyle 1}$ curve to a point. Since all such genus ${\displaystyle 1}$ curves are parametrized by ${\displaystyle U}$, and there is an additional ${\displaystyle 1}$ dimensional choice of where these curves intersect on the genus ${\displaystyle 1}$ curve, the boundary component has dimension ${\displaystyle 10}$.

Other moduli stacks

• A Picard stack generalizes a Picard variety.
• The moduli stack of formal group laws classifies formal group laws.
• An ind-scheme such as an infinite projective space and a formal scheme is a stack.
• A moduli stack of shtukas is used in geometric Langlands program. (See also shtukas.)

Weighted projective stacks

Constructing weighted projective spaces involves taking the quotient variety of some ${\displaystyle \mathbb {A} ^{n+1}-\{0\}}$ by a ${\displaystyle \mathbb {G} _{m}}$-action. In particular, the action sends a tuple

${\displaystyle g\cdot (x_{0},\ldots ,x_{n})\mapsto (g^{a_{0}}x_{0},\ldots ,g^{a_{n}}x_{n})}$

and the quotient of this action gives the weighted projective space ${\displaystyle \mathbb {WP} (a_{0},\ldots ,a_{n})}$. Since this can instead be taken as a stack quotient, the weighted projective stack^{[4] pg 30} is

${\displaystyle {\textbf {WP}}(a_{0},\ldots ,a_{n}):=[\mathbb {A} ^{n}-\{0\}/\mathbb {G} _{m}]}$

Taking the vanishing locus of a weighted polynomial in a line bundle ${\displaystyle f\in \Gamma ({\textbf {WP}}(a_{0},\ldots ,a_{n}),{\mathcal {O}}(a))}$ gives a stacky weighted projective variety.

Stacky curves

Stacky curves, or orbicurves, can be constructed by taking the stack quotient of a morphism of curves by the monodromy group of the cover over the generic points. For example, take a projective morphism

${\displaystyle {\text{Proj}}(\mathbb {C} [x,y,z]/(x^{5}+y^{5}+z^{5}))\to {\text{Proj}}(\mathbb {C} [x,y])}$

which is generically etale. The stack quotient of the domain by ${\displaystyle \mu _{5}}$ gives a stacky ${\displaystyle \mathbb {P} ^{1}}$ with stacky points that have stabilizer group ${\displaystyle \mathbb {Z} /5}$ at the fifth roots of unity in the ${\displaystyle x/y}$-chart. This is because these are the points where the cover ramifies.

Non-affine stack

An example of a non-affine stack is given by the half-line with two stacky origins. This can be constructed as the colimit of two inclusion of ${\displaystyle [\mathbb {G} _{m}/(\mathbb {Z} /2)]\to [\mathbb {A} ^{1}/(\mathbb {Z} /2)]}$.