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58 views

### Is this map representable? what is the fiber?

Consider the following map of stacks:
Let $S$ be the stack whose $A$ points are diagrams of the form
\begin{array}{ccccc}
{} & {} & U & {} \\
{} & {} & \downarrow & \searrow \\...

**0**

votes

**0**answers

66 views

### On orders of stabilisers of group actions and stacks

Let $X$ be a finite type irreducible separated DM-stack over $\mathbb C$. Let $x$ be an object of $X(\mathbb C)$ with stabilizer $G_x$. Let $y$ be an object of $X(\mathbb C)$ with stabilizer $G_y$.
...

**0**

votes

**2**answers

323 views

### The non-existence of the fine moduli scheme of vector bundles. Why?

The reference I am using is this one. The question is about the moduli space of vector bundles. I am trying to understand why the fine moduli scheme does not exist. Let $C$ a projective curve. Let $S$ ...

**1**

vote

**0**answers

106 views

### Is there a reference for boundedness of smooth canonically polarized varieties over Z (No…)

In Kollár's paper Quotient spaces modulo algebraic groups, Kollár mentions right above Theorem 1.8 that the stack $\mathcal M_P$ of smooth canonically polarized varieties over Spec $\mathbb Z$ with ...

**2**

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**0**answers

132 views

### Are these moduli problems of curves “well-behaved”?

Let X be a smooth projective surface over $\mathbb C$, and let $d\geq 3$ be an integer. Suppose that all smooth hypersurfaces of degree $d$ are of genus $g\geq 2$.
Let $H_{X,d}$ be the Hilbert scheme ...

**3**

votes

**0**answers

60 views

### Psi-classes on moduli spaces of weighted curves

Let $\overline{\mathcal{M}}_{g,A[n]}$ be the stack of weighted genus $g$ curves with weights $A[n]=(a_1,...,a_n)$, and let $\pi:\mathcal{C}\rightarrow \overline{\mathcal{M}}_{g,A[n]}$ be the universal ...

**2**

votes

**0**answers

94 views

### Mayer-Vietoris sequence for orbifolds

Is there a version of the Mayer-Vietoris long exact sequence for orbifolds? I am interested in orbifold homology as opposed to the homology of the underlying topological space.

**1**

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151 views

### $\mathbb E$-descent maps in topological spaces in terms of different sites?

The paper Facets of Descent I by Janelidze and Tholen defines $\mathbb E$-descent maps as those for which $\Phi^p:\mathbb EB\longrightarrow \mathsf{Des}_\mathbb{E}(p)$ is an equivalence of categories.
...

**1**

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80 views

### Pushforward for differentiable stacks/ Lie groupoids

Let $X$ be a differentiable stacks, and let $(G_{0}, G_{1}, s,t)$ be a Lie groupoid representing $X$. Let $NG_{\bullet}$ be the nerve of the above groupoid. The De rham complex of $X$ can be defined ...

**3**

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102 views

### groupoids representing mapping stacks

1)Let $X$ be a differentiable stack ((2,1) sheaf over the category of smooth manifolds $Man$) and that is geometric. Let $N\in Man$, then
$$
Map(y(N),X)
$$
is again a differentiable stack ($y$ is the ...

**2**

votes

**0**answers

91 views

### representability of some mapping stack

Let $S$ be an Artin stack of finite type.
We assume that it contains a point as an open dense.
Is it always true that the mapping stack:
$Hom^{0}(\mathbb{P}^{1},S)$
which consists of sections ...

**4**

votes

**1**answer

192 views

### Essential dimension and the moduli space of abelian varieties

The following problem is listed here: http://www-personal.umich.edu/~erman/Papers/Questions2.pdf and attributed to Vistoli:
Let $\mathcal A_g$ denote the moduli stack of principally polarized abelian ...

**4**

votes

**1**answer

175 views

### Smooth algebraic stacks with precisely two $\mathbb C$-objects

In my quest of "understanding" stacks, I recently tried to figure out the structure of a smooth algebraic stack of finite type $\mathcal X$ over $\mathbb C$ with affine diagonal and precisely one $\...

**6**

votes

**1**answer

219 views

### Are Picard stacks group objects in the category of algebraic stacks

I've been wondering about what a "group algebraic stack" should be, and ran into the notion of a Picard stack.
I'm slightly confused by the terminology here.
Given an algebraic stack $\mathcal X$ ...

**10**

votes

**1**answer

260 views

### Counting isomorphism classes in open subsets of Bun_G

Let $G$ be a split semisimple algebraic group and let $C$ be a curve of genus $g$ over $\mathbb F_q$. Assume $g \geq 2$.
The number of $\mathbb F_q$-points of $\# \operatorname{Bun}_G(C)$, where each ...

**15**

votes

**1**answer

609 views

### Algebraic spaces as locally ringed spaces

Let $S$ be a scheme (although I am more than happy to have $S=\text{Spec}(k)$ for a field $k$) and $\mathsf{AlgSp}/S$ the category of algebraic spaces over $S$.
Does there exist an embedding $\...

**7**

votes

**1**answer

220 views

### Fiberwise criterion for a stack to be a gerbe

Let $f:X\to Y$ be a morphism of algebraic stacks.
If the geometric fibres of $f$ are algebraic spaces, then $f$ is representable by algebraic spaces.
I'm wondering about analogues of this fiberwise ...

**3**

votes

**1**answer

171 views

### Are there any non-trivial $G$-gerbes over the analytic space $\mathbb C$

Does there exist a finite (abstract) group $G$ and a non-trivial $G$-gerbe $\mathcal X\to \mathbb C$, where we work in the category of analytic stacks.
My guess is that $G$-gerbes for $G$ an abelian ...

**4**

votes

**1**answer

384 views

### Representable map of Deligne-Mumford stacks

Let $\mathscr{M}\to\mathscr{N}$ be a map of (Deligne-Mumford) stacks. Recall that it is said to be representable by affine schemes if for all affine maps $\operatorname{Spec}R\to \mathscr{N}$, the ...

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vote

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130 views

### Gromov-Witten invariants for arithmetic surfaces counting sections passing through points

Suppose we are given an arithmetic surface, $X\to \text{Spec}\mathbb{Z}[1/N]$ smooth and quasi-projective, and a finite set of closed points all in different vertical fibers.
Can we count the number ...

**6**

votes

**0**answers

112 views

### How to realize the descent data of Qcoh as a (pseudo)-limit in Cat?

It is well-know that $Qcoh$ is a fibered category on $Sch$. In more details let $\mathcal{C}$ be the category $(Sch/S)$ of schemes over a fixed base scheme S. For each scheme $U$ we define $Qcoh(U)$ ...

**11**

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**1**answer

233 views

### Are coarse spaces of 1-dimensional smooth proper Artin stacks smooth?

Let $\mathcal{X}$ be a regular proper 1-dimensional Artin stack with finite diagonal, with coarse space morphism $\mathcal{X} \to X$.
Question: Is $X$ regular?
Some comments:
I'm happy to assume ...

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108 views

### Representablility of fiber product over stacks

Let $(\mathcal{C},\mathcal{T})$ be a site and $\mathcal{X}$ a stack over it. Suppose $(\mathcal{C},\mathcal{T})$ subcanonical and hence for every object $U$ in $\mathcal{C}$ consider the associated ...

**4**

votes

**1**answer

159 views

### Are automorphism groups of polarized varieties of finite type

It is "well-known" that the stack of polarized varieties is an algebraic stack with quasi-compact and separated diagonal.
In particular, if $(X,L)$ and $(Y,M)$ are polarized schemes over a scheme $S$,...

**6**

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**0**answers

218 views

### Is the stack of varieties with a big line bundle algebraic

In Starr's paper https://www.math.stonybrook.edu/~jstarr/papers/moduli4.pdf the folk result that the fibred category of pairs $(X\to S, L)$, where $S$ is an affine scheme, $X\to S$ is flat proper ...

**1**

vote

**1**answer

139 views

### Stacks with representable morphisms to algebraic stacks

If $Y$ is an algebraic stack over a scheme $S$ and $X$ is a stack such that there exists an $S$-morphism $X\to Y$ representable by algebraic spaces, then is $X$ an algebraic stack (in the sense that ...

**4**

votes

**1**answer

213 views

### The stack of group algebraic spaces

The fibred category $\mathcal A$ of algebraic spaces over a scheme $S$ is a stack (over the category of affine schemes with the etale topology). This is proved in Laumon and Moret-Bailly's book (see (...

**4**

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**1**answer

200 views

### Stacks with a small coarse moduli space

Let $k$ be a field of characteristic zero.
Let $X$ be a finite type algebraic stack over $k$ with a coarse (or good) moduli space $M$.
Suppose that $M$ is isomorphic to a point, i.e., $M = Spec k$.
...

**3**

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119 views

### Is there a difference between the inertia stack and the universal automorphism group

Let $\mathcal M$ be a stack representing some moduli problem. Let $\mathcal X\to \mathcal M$ be the corresponding universal family.
What is the difference between the inertia stack $I\to \mathcal M$ ...

**5**

votes

**1**answer

295 views

### Algebraic spaces which are automatically schemes

Let $S$ be a scheme, and let $f:X\to S$ be a morphism of algebraic spaces.
If $f$ is smooth proper curve of genus at least two, then $X$ is a scheme. (Here I mean that $f$ is a smooth proper morphism ...

**1**

vote

**1**answer

164 views

### On functors which are generically representable

Let $F$ be a set-valued (contravariant) functor on the category of schemes. Let $F_{\mathbb Q}$ be the associated functor on the category of schemes over $\mathbb Q$.
Suppose that $F_{\mathbb Q}$ is ...

**4**

votes

**1**answer

124 views

### Singularities of the moduli stack of polarized hyperkahler varieties

Inspired by the recent question on singularities of the moduli stack of Calabi-Yau threefolds (Singularities of the moduli stack of Calabi-Yau threefolds) I'd like to ask the following question.
Is ...

**4**

votes

**1**answer

144 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 ...

**8**

votes

**1**answer

310 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 ...

**6**

votes

**0**answers

182 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

**1**answer

205 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

**2**answers

336 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. Math....

**3**

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221 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 ...

**4**

votes

**1**answer

348 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**

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130 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 ...

**6**

votes

**1**answer

310 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 ...

**6**

votes

**1**answer

171 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

**1**answer

300 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 ...

**3**

votes

**1**answer

177 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**

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**1**answer

236 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**

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107 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 ...

**4**

votes

**2**answers

483 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 $-...

**3**

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**1**answer

134 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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183 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 ...

**13**

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**1**answer

738 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 away ...