The algebraic-stacks tag has no usage guidance.

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### What is the conditional probability of the following expression? [closed]

Conditional probability of the following expression

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**1**answer

59 views

### representing base changes of the unit section

Let $S$ be a scheme and $G$ be a sheaf in groups on the big étale site over $S$. Let $e:S\rightarrow G$ be the unit section. Is it true that given an algebraic space in groups $H$, étale over $S$, and ...

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315 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 $S$ ...

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82 views

### representability of a certain extension of group algebraic spaces

Let S be a scheme. Suppose we have sheaves in abelian groups $A,B,C$ over the big étale site of $S$. Suppose that $A$ and $C$ are representable by algebraic spaces in groups locally of finite type ...

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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 $j$...

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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?

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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 "...

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**1**answer

211 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 $\mathcal X$ ...

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**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 ...

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**3**answers

519 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 ...

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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 $\mathscr{X}$. ...

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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. ...

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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 ...

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**1**answer

118 views

### On an example by Romagny about fixed point stack not commuting with coarse moduli space

This is to understand better Example 3.9 on page 221 of Group actions on stacks and applications by M.Romagny.
For an action of an algebraic group (scheme) $G$ on an algebraic stack $\mathcal{M}$, ...

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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 ...

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168 views

### Is the residual gerbe really independent of the choice of a representative?

My question is about the passage (11.1) in the book of Laumon and Moret-Bailly on algebraic stacks. There we have a scheme $S$, an algebraic stack $\mathscr{X}$ over $S$, and a point $\xi$ of $\...

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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 ...

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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 ...

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152 views

### Help for reference of moduli stack of fake elliptic curve

I see everywhere says the following:
Let $B$ be an indefinite quaternion algebra over $\mathbb{Q}$ of discriminant $D$, $\mathcal{O}_B$ be a maximal order, $N$ be an positive integer coprime to $D$. ...

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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 $S$,...

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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 ...

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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 ...

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### 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 (...

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### 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., $M = Spec k$.
...

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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 ...

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372 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 ...

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219 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 ...

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### 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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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 ...

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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 $BG=[*/G]$ ?

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### 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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### 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 ...

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### 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 ...

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114 views

### Stack of curves and universal deformations

I've just started studying algebraic stacks and I have a very basic question.
I've learned the notion of Deligne Mumford stack and I've seen as the stack of stable curves $\overline{\mathcal{M}_g}$ ...

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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 $...

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### pushing out families of curves

Let $f:X\rightarrow Y$ be a morphism of schemes with smooth curves as fibers. Let $g:X\rightarrow Z$ be a family of smooth or nodal curves with $Z$ a regular scheme. Does the push-out $Z\coprod_X Y$ ...

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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 ...

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### 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 ...

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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 ...

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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, ...

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### 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 ...

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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 $\overline{...

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### Grothendieck duality for stacks

Let $\mathcal{X}$ be a smooth, proper and separated Deligne-Mumford stack and let $\pi:\mathcal{X}\rightarrow X$ be its coarse moduli space. Does Grothendieck duality hold for the morphism $\pi$ ?
In ...

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### 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 ...

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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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### 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) ...

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### 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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### Finite-type Artin Stack over $\mathbb C$

Suppose I have an Artin stack $\mathfrak M$ locally of finite-type over $\mathbb C$ with presentation $M\rightarrow \mathfrak M$. Suppose further that $\mathfrak M$ "represents" (in the stack sense) ...

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### 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 $...

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### 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.