The algebraic-stacks tag has no usage guidance.

**19**

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**1**answer

825 views

### What is $Aut(Ell)$?

Consider the stack $Ell$ (of groupoids) of elliptic curves. I'm interested in the autoequivalence 2-group of $Ell$, the objects of which consists of transformations $Ell \Rightarrow Ell: Ring \to Gpd$ ...

**13**

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**0**answers

351 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 ...

**12**

votes

**1**answer

351 views

### Verifying that $\epsilon^!$ is indeed the right adjoint of $\epsilon_*$ in the context of algebraic stacks

The question is about the last paragraph of Remark 12.4.3 in the book on algebraic stacks by Laumon and Moret-Bailly.
Let $S$ be a (quasi-separated) scheme and let $\mathscr{X}$ be an algebraic stack ...

**11**

votes

**4**answers

1k views

### Soft question: beginners reference to moduli spaces

What is a geometrically intuitive yet reasonably general first introduction to the theory of Moduli spaces?
(Possibly introducing stacks also)?
I'm looking for something which really gets the pictures ...

**11**

votes

**1**answer

642 views

### Several simple questions on the geometry of higher stacks

I'm trying to understand definition/work out some examples. So, there are some simple questions about higher stacks.
For the simplicity assume that we are working with higher DM (Deligne-Mumford) ...

**11**

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**1**answer

672 views

### coarse moduli space and $\pi_0$

I've been reading this really nice paper by Alper http://math.columbia.edu/~jarod/good_moduli_spaces.pdf, and there's a question that doesn't seem to be answered (perhaps it's not relevant).
Any ...

**11**

votes

**1**answer

232 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 ...

**10**

votes

**1**answer

464 views

### Ramification of the map from the stack of elliptic curves to the $j$-line

Let $\mathcal{M}_{1, 1}$ be the stack of elliptic curves. Its coarse moduli space is $\mathbb{A}^1_{\mathbb{Z}}$ with the map $\mathcal{M}_{1, 1} \rightarrow \mathbb{A}^1_{\mathbb{Z}}$ given by the $j$...

**9**

votes

**1**answer

362 views

### Is $\mathcal{D} \bigl( \mathrm{QCoh}(\mathfrak{X}) \bigr)$ compactly generated?

An object $E$ in a triangulated category $\mathcal{T}$ with (small) coproducts is called compact if the functor $\mathrm{Hom}_{\mathcal{T}}(E,-)$ commutes with arbitrary coproducts or, equivalently, ...

**8**

votes

**3**answers

518 views

### Definition of étale (etc) for non-representable morphisms of algebraic stacks?

I've stumbled upon the statement that the morphism $\pi$ from a root stack of the form $\sqrt[r]{\mathscr{L}/\mathscr{Y}}$ (i.e. the "generic" version, not the one concentrated along a divisor) to its ...

**7**

votes

**2**answers

444 views

### 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 $\overline{...

**7**

votes

**1**answer

805 views

### Is every module the colimit of its finitely generated submodules? (for algebraic spaces or stacks)

For (quasi-compact and quasi-separated) schemes there is a categorical way to characterise quasi-coherent sheaves of finite type using purely the abelian category $\operatorname{QCoh}(X)$. In an ...

**7**

votes

**1**answer

217 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 ...

**7**

votes

**1**answer

344 views

### Degree formalism for line bundles on Deligne-Mumford stacks

Let $k$ be an algebraically closed field and let $\mathcal{C}$ be proper, Cohen-Macaulay, purely $1$-dimensional Deligne-Mumford stack over $k$. From looking at section 4.3 on page 135 of the paper "...

**6**

votes

**1**answer

395 views

### The singularity of the algebraic stack and the singularity of the coarse moduli space

It is possible that an algebraic stack is smooth while the coarse moduli space is not smooth. I want to know what is relationship between the singularity of the algebraic stack and that of its coarse ...

**6**

votes

**1**answer

536 views

### Is “stackiness” transitive? (and a couple other basic questions about stacks)

Say, $B$ is a category fibered in groupoids over some category $C$, and $A$ is a category fibered in groupoids over $B$.
Suppose $A$ is a stack (over whatever site) over $B$, and $B$ is a stack over $...

**6**

votes

**1**answer

416 views

### If $X$ is a smooth and proper stack, does it admit a smooth and proper atlas?

Fix a ground scheme $S$ (a field say).
By atlas for an algebraic stack I mean a smooth and surjective morphism $Y \to X$ from a scheme (or algebraic space or affine scheme) $Y$.
If the stack $X$ is ...

**6**

votes

**1**answer

311 views

### universal property of blow up for stacks?

I will use as a reference Hartshorne Prop. II.7.14, the universal property of blow-up. $\tilde{X}$ is the blow up of $X$ along a sheaf of ideals. $Z\to X$ is the morphism that is to be lifted to $\...

**6**

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**1**answer

209 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 $\mathcal X$ ...

**6**

votes

**1**answer

304 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**

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**1**answer

353 views

### on a Deformation long exact sequence of moduli space of stable maps

I am reading the book "mirror symmetry" by Hori,Katz,Klemm,etc. And I want to understand the following Deformation long exact sequence
\begin{align}
0 & \to Aut(Σ, p_1, . . . , p_n, f)\to Aut(Σ, ...

**6**

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**0**answers

217 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 ...

**5**

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**1**answer

814 views

### What is the intuition inertia orbifold (or stack)?

I am studying orbifolds with view towards Chen-Ruan cohomology. I have been struggling with inertia orbifolds but have no intuition about them at this point. I would appreciate your motivating me by ...

**5**

votes

**1**answer

207 views

### Stabilizer Action on vector bundle on a stack

Suppose you have a Deligne Mumford stack $X$ and a geometric point $x:Spec{k}\rightarrow X$ with stabilizer group $Stab(x)$.Let $F$ be a locally free sheaf on $X$.
How is the action of $Stab(x)$ on $...

**5**

votes

**1**answer

455 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 $...

**5**

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**0**answers

118 views

### Testing the vanishing of cohomology fiberwise for a proper morphism from an Artin stack

Let $S$ be a Noetherian scheme, let $f\colon\mathscr{X} \rightarrow S$ be a proper morphism with $\mathscr{X}$ an algebraic stack, and let $\mathscr{F}$ be an $S$-flat coherent sheaf on $\mathscr{X}$. ...

**5**

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**0**answers

219 views

### Homology of stack points (from math.stackexchange)

(This question is duplicated here)
This is a very basic question about how definitions in homology carry over to the easiest example of stacks. Let $G$ be a finite cyclic group. Consider the ...

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**0**answers

346 views

### Coarse moduli spaces of quotient stacks

Suppose you have a separated Deligne Mumford quotient stack $[V/G]$ over a field of characteristic $0$, where $V$ is a quasiprojective variety and $G$ is an algebraic group that does not necessarily ...

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133 views

### Normalization of quotient stacks

Suppose you have a Deligne Mumford stack which is a quotient $[X/G]$ of a scheme $X$ by an algebraic group $G$ .
What is the normalization of that? Is it true that its normalization is a quotient ...

**4**

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**1**answer

294 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 ...

**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 $S$,...

**4**

votes

**2**answers

432 views

### Explicit description of the stack associated to a groupoid

Let $\{X_1 \rightrightarrows X_0\}$ be a smooth groupoid object in the category of affine schemes ($X_0 \to X_1$, $X_1 \to X_1$ and $X_1 \times_{X_0} X_1 \to X_1$ also belong to the datum). ...

**4**

votes

**1**answer

212 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

195 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., $M = Spec k$.
...

**4**

votes

**1**answer

457 views

### Algebraicity of the stack of coherent sheaves

I am trying to understand the proof of Theorem 4.6.2.1 in the book on algebraic stacks by Laumon and Moret-Bailly. The setting is this: $S$ is a Noetherian scheme, $f\colon X \rightarrow S$ is a ...

**4**

votes

**1**answer

413 views

### What does it mean for a Deligne-Mumford stack to have trivial generic stabilizers?

I have stumbled upon some literature on Deligne-Mumford stacks, and it seems to me, at least superficially, that there is a strong link between DM-stacks which have "trivial generic stabilizers" and "...

**4**

votes

**1**answer

262 views

### Do Deligne-Mumford curves also have rational functions

If $X$ is a curve over a field of characteristic zero, then $X$ has a rational function, i.e., a finite morphism to the projective line.
Question. Suppose that $X$ is a Deligne-Mumford (or just ...

**4**

votes

**1**answer

167 views

### A question on $Isom(p_1^*E,p_2^*E) \rightrightarrows X$

I am learning the moduli stacks of vector bundles and have trouble understanding some definitions. Let $E$ be a rank $n$ vector bundle over the scheme $X$. We denote by $p_i$ the $i$th projection $p_i:...

**4**

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**0**answers

162 views

### Vanishing theorems on toric DM stacks

In chapter 9 of the book Toric varieties by Cox-Little-Schenck several cohomology vanishing theorems for toric varieties are proved or mentioned.
In this question I am interested in references for ...

**4**

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**0**answers

196 views

### Stack theoretic image?

If $X \xrightarrow{f} Y$ is a morphism of schemes then the scheme theoretic image of $f$ is the smallest closed subscheme $Z \subset Y$ through which $f$ factors through.
Is this notion defined for ...

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**0**answers

330 views

### A question on an intuitive way to look at stacks

I am reading the chapter "Introducing Algebraic Stacks" in The Stacks Projects to get a feeling for them. There is a small point that throws me off. They denote $\mathcal{M}_{1, 1}$ the moduli stack ...

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215 views

### Cohomology of line bundles on smooth projective toric Deligne-Mumford stacks

Let $\mathcal{X}$ be a smooth projective toric Deligne-Mumford stack with coarse moduli scheme $\pi\colon \mathcal{X}\rightarrow X.$
Let $[D]$ be a Cartier divisor on $\mathcal{X}$ such that $[D']=\...

**3**

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**2**answers

275 views

### Heuristics for 2-morphisms of (algebraic) stacks

For topological spaces and simplicial sets one can consider each pair of parallel morphisms $f,g:X\rightrightarrows Y$ as equipped with a set of 2-morphisms given by homotopies $H:f\simeq g$ (let's ...

**3**

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**2**answers

218 views

### Sheaf of isogenies representable?

It is well-known that "the" stack of elliptic curves (allow me to be vague as to singular curves, compactifications etc) has a presentation by a groupoid in schemes. One of the things that needs to be ...

**3**

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**2**answers

449 views

### Differentials for algebraic stacks

Let $S$ be a base scheme. For which algebraic stacks $X$ over $S$ can we define a sheaf of differentials $\Omega^1_{X/S}$ (classifying derivations)? Probably it works when $X$ is Deligne Mumford ...

**3**

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**1**answer

142 views

### The line bundle of the divisor at infinity of the moduli stack of stable curves of genus $g \ge 2$

Let $\overline{\mathscr{M}}_g$ be the $\mathbb{Z}$-algebraic stack of stable curves of genus $g \ge 2$, as constructed in the paper of Deligne and Mumford. The degeneracy locus of the universal stable ...

**3**

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**1**answer

118 views

### Do equivariant morphisms induce representable maps of quotient stacks?

Let $f: X \to Y$ be a $G$-equivariant map between schemes $X$, $Y$ with action of a flat group scheme $G$. Then why is the induced map of algebraic stacks $[X/G] \to [Y/G]$ representable?

**3**

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**1**answer

148 views

### Derived pullback of the coarse moduli morphism

Let $f: \mathcal{X}\to X$ be a morphism from a smooth DM-stack $\mathcal{X}$ to its coarse moduli space $X$. Assume that $X$ is also smooth. Is it true that $Lf^*$ is fully faithful and induces an ...

**3**

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**1**answer

166 views

### Question about canonical DM stacks

Let $\mathcal{X}$ be a canonical stack (edit: I forgot to say I also want $\mathcal{X}$ smooth), and $\pi : \mathcal{X}\to X$ its coarse moduli space morphism. Let also $D$ be a prime divisor (i.e. ...

**3**

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**1**answer

430 views

### The fibre product of two quotient stacks

My question is to know whether the fibre product of $[X/G]$ by $[Y/H]$ over a base scheme is $S$ is $[X \times_S Y/G \times H]$? And if yes, do you have any reference for it?
Thank you.