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4
votes
1answer
106 views

Descending a monomorphism of stacks

The question is about Proposition 3.8.1 in Laumon and Moret-Bailly book on algebraic stacks. Let $S$ be a scheme and let $F: \mathscr{X} \rightarrow \mathscr{Y}$ be a morphism of $S$-stacks (for the ...
6
votes
1answer
222 views

Singularities of the moduli stack of Calabi-Yau threefolds

Let $M$ be the moduli of polarized Calabi-Yau threefolds over $\mathbb C$ with fixed Euler characteristic. The coarse moduli space is singular (as usual), but what about the stack? In many cases I ...
8
votes
1answer
296 views

Separation condition for higher Deligne-Mumford stacks

Let $X$ be a stack of $n$-groupoids on the site of affine schemes over a fixed base, with the etale topology. If $n=1$ then for $X$ to be Deligne-Mumford, aside from having an etale atlas from an ...
2
votes
0answers
173 views

Finiteness of the connected components of a stack

Let $X$ be an algebaic stack over a scheme $S$, for any $S$-scheme $Y$ we can consider the groupoid $X(Y)$ of $Y$-points. Denote by $\pi_0(X(Y))$ the set of isomorphism classes of the groupoid. Are ...
8
votes
6answers
2k views

Representation of Groupoids

The title is vague, my actuall question is the following: Has the representations of groupoids been systematically studied? Is there any new phenomenon, compare with the representation of groups? ...
11
votes
1answer
632 views

Conditions for “bootstrapping” a smooth DM stack?

In the preprint "Smooth toric DM stacks", Fantechi, Mann and Nironi define the stacks of their title, and show that each of these can be obtained through the following sequence of steps: 1) start ...
6
votes
0answers
145 views

Is there a more general obstruction to the existence of moduli spaces than the existence of automorphisms?

We are taught that, in general: A type of objects that has nontrivial automorphisms cannot have a fine moduli space. The proof generally goes along the lines of: Take an object $X$ with a ...
2
votes
1answer
179 views

Given a map of classifying spaces, can the target be described as a groupoid quotient of the source mod some action of some (co)kernel?

Let $H \to G$ be a homomorphism of affine algebraic groups (over characteristic $0$, if it matters). The case I care most about is when $H \to G$ is an inclusion. There is a corresponding map $f: ...
4
votes
2answers
218 views

Universal curve of stacks of stable curve

Let $\overline{M}_{g,A}$ the moduli stack of pointed genus $g$ stable curves with weights $A = (a_1,...,a_n)$ introduced in Brendan Hassett, Moduli spaces of weighted pointed stable curves, Adv. ...
3
votes
1answer
303 views

What are the automorphisms of $BG$?

Setup: Let's work in the category of schemes over $\mathbb C$. Let $G$ be a finite group. Let $BG=[pt/G]$ be the classifying stack of principal $G$ bundles. This is a fiberd category over the big ...
3
votes
0answers
106 views

Writing down gerbes explicitly over the projective line

Let $X = [\mathbb P^1/(\mathbb Z/2\mathbb Z)]$, where we take the trivial action of $\mathbb Z/2\mathbb Z$ on $\mathbb P^1$. Is this DM stack over $\mathbb C$ a gerbe over $\mathbb P^1$? Is it the ...
5
votes
1answer
198 views

Representability of morphism of stacks

A morphism of Artin stacks $f:X\to Y$ over $\mathbb Q$ is representable by algebraic spaces if and only if its geometric fibres are algebraic spaces. I would like to know if one can use this to prove ...
5
votes
1answer
145 views

Coherent sheaves on $\mathbb C^2$ and commuting matrices

Let $V$ be an $n$-dimensional complex vector space. The stack $Coh^n(\mathbb C^2)$ of coherent sheaves on $\mathbb C^2$ supported on $n$ points (not necessarily distinct) is equivalent to the stack ...
4
votes
1answer
225 views

Do algebraic stacks satisfy fpqc descent?

It is known, thanks to Gabber, that algebraic spaces are sheaves in the fpqc topology: Stacks project 03W8 Is the analogous statement for algebraic (Artin) stacks true? If not, is it true under ...
2
votes
1answer
149 views

Galois cohomology out of the classifying stack

Suppose $G$ is a smooth and abelian $k$-group scheme, for $k$ a field. Is it possible to get back galois cohomology groups $H^*(k,G)$ studying the cohomology of the classifying stack $BG=[*/G]$ ?
2
votes
1answer
111 views

Picard group of classifying stack

Suppose $S$ is a scheme, and $G$ a smooth $S$-group scheme. Then there exists an algebraic stack BG called the classifying stack of $G$, defined as the quotient stack $[S/G]$ where $G$ acts trivially ...
4
votes
0answers
102 views

$\mathcal{M}_{g,n}$ a scheme for $n \gg 0$? [duplicate]

I think that for $n \geq 3$, the Deligne-Mumford moduli stack $\mathcal{M}_{0,n}$ is a scheme. Is it more generally true that for every $g$, the Deligne-Mumford moduli stack $\mathcal{M}_{g,n}$ is a ...
6
votes
2answers
517 views

Connections on principal bundles via stacks?

Let G be a Lie group and M a smooth manifold. Suppose that P is a principal G-bundle over M. Then by Yoneda, this corresponds to a smooth map $p:M \to [G]$, where $[G]$ is the differentiable stack ...
4
votes
2answers
441 views

Based loop groups as stacks?

I have been stuck for some time, thinking about the following question. Let $G$ be a Lie group. Its classifying space $BG$ can be seen as the differentiable stack $[pt/G]$, which is of dimension ...
29
votes
1answer
2k views

D-modules, deRham spaces and microlocalization

Given a variety (or scheme, or stack, or presheaf on the category of rings), some geometers, myself included, like to study D-modules. The usual definition of a D-module is as sheaves of modules over ...
3
votes
1answer
127 views

reference for “curves over S are locally the base change of a curve over S' which is finite type over R”

So recently I heard someone claiming that if $X\rightarrow S$ is a smooth curve (not necessarily proper?) and $S$ is an arbitrary scheme over $\text{Spec }R$ (for $R$ sufficiently nice), then there is ...
5
votes
1answer
465 views

Local Joyal-simplicial presheaves?

It is well known that left Bousfield localizations of the global functor model category $Func(C^{op}, SSet_{standard})$ of functors with values in simplicial sets equipped with the standard model ...
6
votes
0answers
139 views

Is the quotient of a scheme by the free action of an elliptic curve an algebraic space?

Let $X$ be a scheme (I'm happy to assume that $X$ is of finite type, separated, and over $\mathbb{C}$) and let $E$ be an elliptic curve which acts freely on $X$. Does the quotient stack $[X/E]$ have ...
2
votes
1answer
234 views

Algebraic stacks: limit preserving versus locally of finite presentation

I'm wondering what the precise relationship is between an algebraic stack being locally of finite presentation and being limit preserving. Under some mild hypotheses on the diagonal (in force ...
12
votes
1answer
563 views

Do canonical stacks exist over Spec(Z)?

Suppose a scheme $X$ has tame quotient singularities. Does there exist a smooth DM stack $\mathcal X$ with coarse space $X$ so that the coarse space morphism $\mathcal X\to X$ is an isomorphism ...
5
votes
0answers
116 views

Groupoid cardinality of DM stack and point counting on coarse moduli spaces

Let $X$ be a finite type DM stack over a finite field $k$ with a coarse moduli space $X_c$. (We only assume $X_c$ is an algebraic space and $X$ might have infinite inertia stack.) Under which ...
3
votes
1answer
243 views

What if the base change of an algebraic space is representable

Let $k\subset L$ be an extension of fields of characteristic zero. Suppose that $X/k$ is an algebraic space such that $X\otimes_k L$ is representable by a finite type $L$-scheme. I am sure there are ...
5
votes
1answer
426 views

On the local structure of Deligne-Mumford stacks

Is it true that for any DM stack $\mathcal{X}$ (quasicompact, separated, of finite type over a field) there is a Zariski covering $\mathcal{U}_i \to \mathcal{X}$ by open substacks, such that for all ...
4
votes
1answer
245 views

Torsors in the analytic topology versus torsors in the etale topology

Let $S= \mathbb A^1_{\mathbb C}$ be the affine line, and let $G$ be a smooth connected reductive group over $S$, e.g., $G = \mathbb G_m, \mathrm{SL}_n$ or $SO_n$. Is every analytic $G$-torsor over ...
5
votes
1answer
247 views

Is there a Riemann existence theorem for orbifolds?

For smooth algebraic varieties $X$ over $\mathbb{C}$, the Riemann existence theorem establishes an equivalence of categories between the category of finite etale covers of $X$ and finite unramified ...
7
votes
1answer
520 views

Relation between $BG$ in topology and in algebraic geometry

This could as well have been asked in the comments to this question, but I prefer to open a new one for the sake of clarity. Say $G$ is a reductive group over the complex numbers, with compact real ...
3
votes
0answers
133 views

Properties of finite quotients of quasi-projective varieties

Let $G$ be a finite group acting on a (smooth) quasi-projective variety over $\mathbb C$. One can consider the stacky quotient $[X/G]$ or the "classical" quotient $X/G$. In general, $[X/G]$ is not a ...
37
votes
12answers
5k views

Good introductory references on algebraic stacks?

Are there any good introductory texts on algebraic stacks? I have found some readable half-finsished texts on the net, but the authors always seem to give up before they are finished. I have also ...
7
votes
3answers
438 views

Gerbes and Stacks

The definition of a gerbe on a smooth manifold that I know is that - after fixing an open cover $U_i$, a gerbe consists of the data of line bundles $L_{ij}$ on two-fold-intersections $U_{ij}$, ...
12
votes
0answers
274 views

Coarse moduli spaces of stacks for which every atlas is a scheme

Let $X = [P/G]$ be a smooth finite type separated DM-stack over $\mathbb C$ given as the quotient of a smooth projective scheme $P$ by the action of a smooth (finite type separated) reductive group ...
4
votes
0answers
111 views

Nowhere vanishing section of vector bundles over varieties as connectivity of morphism of stacks

The following is, amongst others, a Hartshorne exercise: Let $V$ be a $k$-variety of dimension $n$ and $\mathcal{E}$ a vector bundle of rank greater than $n$, then, generically, a generating section ...
8
votes
2answers
461 views

Finite etale atlas for Deligne-Mumford stacks

Let $X$ be a smooth finite type separated connected Deligne-Mumford stack over $\mathbb C$. Does there exist a finite etale morphism $Y\to X$ with $Y$ a scheme? What if $X$ is an algebraic space ...
4
votes
2answers
175 views

Map between stacks and automorphism groups

I know that the Torelli morphism $t_g:\mathcal{M}_g\rightarrow \mathcal{A}_g$ between the stacks of smooth curves of genus $g$ and principally polarized abelian varieties of dimension $g$ is of order ...
2
votes
0answers
88 views

quotient a scheme by a stratified vector bundle

Let $k$ be a field. Let $X$ be a $k$-scheme of finite type, normal and integral. We consider $f,g:R\rightarrow X$ an equivalence relation, surjective and such that it is a stratified vector bundle, ...
2
votes
0answers
237 views

Is the upper half plane an algebraic stack?

Here by algebraic stack I mean an algebraic stack over the etale site $\textbf{Sch}/\mathbb{C}$. So I've read from various nonrigorous sources that the upper half plane $\mathcal{H}$ is a fine moduli ...
5
votes
0answers
107 views

Can the hyperbolic core of a curve over $\mathbb Q$ be defined over $\mathbb Q$ as an algebraic stack

Here is a question I've been wondering about for a while. Currently it is mere curiosity and I do not have any direct applications in mind. Let $X$ be a smooth quasi-projective geometrically ...
11
votes
2answers
625 views

The quotient stack $[\mathbb{A}^n / \mathrm{GL}_n]$

Consider the affine space $\mathbb{A}^n$ (over some base scheme) with the usual $\mathrm{GL}_n$-action. What does the quotient stack $[\mathbb{A}^n / \mathrm{GL}_n]$ classify? If $n=1$, then we get ...
4
votes
1answer
138 views

Question regarding 2-mathematics: Can you stackify a 2-functor without prestackifying it first?

Let $C$ be a site and $CAT$ the 2-category of categories. Given a contravariant 2-functor $A:C\rightarrow CAT$, we can of course consider the associated stack. This is done by first considering the ...
0
votes
0answers
89 views

How can information about morphisms between modular curves be read from the morphism of their corresponding stacks?

$\newcommand{\yY}{\mathcal{Y}}$ $\newcommand{\PSL}{\text{PSL}}$ $\newcommand{\SL}{\text{SL}}$ $\newcommand{\ZZ}{\mathbb{Z}}$ $\newcommand{\CC}{\mathbb{C}}$ $\newcommand{\mM}{\mathcal{M}}$ Let $Y(1)$ ...
1
vote
0answers
114 views

Why $D^b(S)\cong D^b_{\text{Car}}(\text{cosq}(X\rightarrow S))$?

Let $p: X\rightarrow S$ be a map between topological spaces and we can construct the simplicial space $\text{cosq}(X\rightarrow S)$ where $X_0=X$, $X_1=X\times_S X$ and $$ X_n=\underbrace{ X\times_S ...
4
votes
1answer
216 views

Pulling back quasi-coherent sheaves from a quotient stack

In a problem I am trying to solve, the following situation occurs. $X$ is a smooth variety and $G$ is a reductive group acting transitively on $X$. We have the stack $X/G$ and a morphism $\pi : X \to ...
2
votes
0answers
96 views

Is the cotangent complexes of groupoids bounded above by degree $1$?

Let $\mathcal{X}$ be a stack given by a groupoid $X_1\rightrightarrows X_0$, where $X_0$ and $X_1$ are smooth $k$-varieties. Let $\mathbb{L}_{\mathcal{X}/k}$ be the cotangent complex of ...
9
votes
0answers
255 views

Other examples of the algebro-geometric Ran space

First off, sorry if this seems vague. Let's recall some definition. Let $X$ be a curve over a field $k$ and $G$ an algebraic group, then the space $Ran_G(X)$ as defined by Lurie in his Tamagawa ...
6
votes
0answers
181 views

Derived Functors, Projection Formula and Base Change in the Derived $\infty$-Category

Let $X$ be a smooth stack and $\mathcal O_X$ the ring of smooth functions on $X$, i.e. for any smooth $M \to X$, $\mathcal O_X(M \to X) = C^\infty(M)$. In HigherAlgebra, the derived category ...
0
votes
1answer
106 views

étalé space of sheaves on a differentiable stack

If $F$ is a sheaf on a topological space $X$, the well-known étalé space contruction gives rise to a bundle $\Gamma F$ on $X$ such that $F$ is isomorphic to the sheaf of sections of $\Gamma F$. On ...