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### 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 ...

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**1**answer

173 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: ...

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211 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. ...

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133 views

### Finitness 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 ...

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**1**answer

298 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 ...

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105 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 ...

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191 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 ...

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144 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 ...

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**1**answer

219 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 ...

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**1**answer

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 $BG=[*/G]$ ?

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104 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 ...

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101 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 ...

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437 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 ...

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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 ...

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138 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 ...

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555 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 ...

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115 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 ...

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**1**answer

240 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 ...

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**1**answer

244 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 ...

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246 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 ...

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131 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 ...

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### 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 ...

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436 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 $U_{ij}$, ...

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269 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 ...

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### 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 ...

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456 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 ...

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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 ...

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### 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, ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 $Y(1)$ ...

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### 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 ...

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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 ...

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### 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 ...

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253 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 ...

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### 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 ...

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### Embedding dimension: local finiteness & intuition for more general spaces

Can every complex analytic space be covered by Stein spaces of finite embedding dimension?
I am almost sure that ought to be true. Here the definition of embedding dimension I have in mind is
$$
...

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### Properties of the induced map between inertia stacks

Let $\mathcal X$ and $\mathcal Y$ be (separated) Deligne-Mumford stacks. A morphism of stacks $f:\mathcal X \to \mathcal Y$ induces a morphism between inertia stacks $\tilde f:I\mathcal X \to ...

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### Moduli of curves in characteristic zero

Let $K$ be a field of characteristic zero, and let $\overline{K}$ be its algebraic closure. Let $\overline{M}_{g,n}(K)$ and $\overline{M}_{g,n}(\overline{K})$ be the coarse moduli spaces parametrizing ...

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### Is a stack that is finite etale over an algebraic stack also algebraic?

Here by "algebraic stack" I mean a stack in groupoids over $(\textbf{Sch}/S)_\text{etale}$ (for some scheme $S$) whose diagonal morphism is representable (by schemes), and which is covered by a ...

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### Representable torsors on geometric groupoid

Let $(C,\tau,\mathbb P)$ be a geometric context, as defined by Toen and Vezzosi. Let $(X_1\rightrightarrows X_0)$ be a groupoid object in $C$ such that the source and target morphisms are in $\mathbb ...

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### 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 ...

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### Deforming curves to other curves over the field of rational numbers

Let $X$ and $Y$ be smooth projective geometrically connected curves over $k$ of genus $g$ at least two.
If $k$ is an algebraically closed field of characteristic zero, there exists a connected ...

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### Is a Deligne-Mumford curve defined over Qbar if and only if its coarse moduli space is

Let $\mathcal X$ be a smooth proper finite type Deligne-Mumford stack over $\mathbb C$ that is generically a scheme. Let $X$ be its coarse moduli space.
If $\mathcal X$ can be defined over ...

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### Associating a principal bundle to a torsor

in arXiv:math/0212266, Moerdijk defines a torsor to be a sheaf $\mathcal{S}$ on $X$ with a freely transitive left-action of a sheaf of groups $\mathcal{G}$, such that $X =\bigcup \{ U \in ...

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### Quasi-coherent sheaves on classifying stacks

Let $G$ be a smooth group scheme over some base $S$. Then we have the $S$-stack $BG$ whose $T$-points are the $G$-torsors on $T$. Under which conditions do we have $\mathsf{Qcoh}(BG) \simeq ...

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### Quasi-finite morphisms of stacks

Let $f:X\to Y$ be a morphism of ``nice" stacks over $\mathbf C$ such that the induced morphism on coarse moduli spaces is quasi-finite. Is $f$ quasi-finite?
By a "nice" stack I mean a smooth finite ...

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### Cohomology of BG, algebraically

Let $k$ be a field (algebraically closed if you will) and $G$ be a connected reductive group over $k$. I would like to know a purely algebraic computation of the cohomology of $BG$, as the quotient ...