Questions tagged [algebraic-stacks]
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169
questions
2
votes
0answers
70 views
Operation on algebraic stack
Let $X$ be a smooth algebraic stack and $D$ be a normal crossing divisor. Let $\pi: \tilde{D}\rightarrow D$ be the normalization of $D$. Let $\sigma: \tilde{D}\rightarrow \tilde{D}$ be an involution. ...
18
votes
6answers
2k views
What are the occurrences of stacks outside algebraic geometry, differential geometry, and general topology?
What are the occurrences of the notion of a stack outside algebraic geometry, differential geometry, and general topology?
In most of the references, the introduction of the notion of a stack takes ...
1
vote
1answer
63 views
Determine whether a rational function on the codomain of a surjective morphism is regular
Let $X$ be a smooth affine algebraic variety with a (not necessarily free) action by an algebraic torus $T$. Let $Y$ be the quotient stack $X/T$ and let $p:X\rightarrow Y$ be the quotient map. Suppose ...
1
vote
0answers
129 views
Is there a stacky definition of irreducible symplectic manifold?
I am now interested in studying symplectic structures in the field of stacks.
In particular, is there a stacky definition of irreducible symplectic manifold ?
I'm also interested in similar things in ...
1
vote
0answers
91 views
Relationship between Quasicoherent sheaves and $\mathbb A^1$-fpqc modules over an fpqc stack
In what follows, assume several universes for simplicity.
Let $X$ be a stack in groupoids on the fpqc site of small affine schemes $\mathbf{Aff}_{\text{fpqc}}$. We can define $\mathbf{QCoh}(X)$ ...
3
votes
0answers
68 views
How big is the complement of stable locus $\operatorname{Bun}G$
Let $\Sigma$ be a smooth projective curve, and $G$ a reductive group. Let $\operatorname{Bun}G$ be the stack of principal $G$ bundles on $\Sigma$ (with a fixed topological type).
What is the ...
2
votes
1answer
144 views
Representaility of morphism of stacks for schemes
I have seen two definitions of representability of a morphism of stacks, which should be at least compatible with the definition of a morphism of categories fibered in groupoids.
(Representable ...
2
votes
0answers
157 views
Why not consider categorical quotient in stacks?
Let's say we have a $k$ group scheme $G$ acting on a $k$ scheme $X$, we can consider its quotient in the category of stacks, the usual definition of it would be the quotient stack $[X/G]$ defined by
...
1
vote
1answer
121 views
Are morphisms from affine schemes to Artin stacks affine morphisms?
It is explained in this MO discussion that if one has a morphism $f:X\rightarrow Y$ of schemes such that $X$ is an affine scheme, then $f$ need not be an affine morphism. However, if $Y$ is separated, ...
1
vote
0answers
145 views
A quotient stack $[X/G]$ and irreducibility of $X$
Let $k$ be an algebraically closed field.
Let $X$ be a variety over $k$ and $G$ be a algebraic group over $k$ acting on $X$.
Question
If the quotient stack $[X/G]$ is irreducible, then is $X$ also ...
4
votes
2answers
306 views
Serre's theorem on global generations on stacks
Let $X$ be a quasi-projective scheme, the followings are quite useful.
Every coherent sheaf is globally generated after tensoring with a suitable line bundle.
Every coherent sheaf has trivial ...
3
votes
0answers
141 views
Reference request: Derived structure on the moduli stack of Higgs bundles
I am reading arXiv:1708.08124. When talking about the moduli stack of Higgs bundles on a projective curve $X$. It is said on page 59, first paragraph that
It is often better to put
derived ...
7
votes
0answers
261 views
Stacky proof of no elliptic curves over Z
It is a well known result that there are no Elliptic curves over the integers with every where good reduction. In fact this is even true for abelian varieties (and hence higher genus curves) but let ...
4
votes
0answers
162 views
connection on principal bundles over algebraic/geometric stacks
Is there a notion of connection on a principal bundle over an algebraic or geometric stack?
By a geometric stack, I mean a stack over category of manifolds that is representable by a Lie groupoid; ...
8
votes
1answer
186 views
Representable diagonal map $\Delta: \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ for DM-Stacks/algebraic spaces
Following the standard definitions of a algebraic space or Deligne–Mumford stack one imposed condition is that the diagonal morphism $\Delta: \mathcal{X} \to \mathcal{X} \times \mathcal{X}$
has to be ...
1
vote
0answers
38 views
Definition of Morphisms of algebraic stacks smooth of relative dimension n
There is a notion of smooth morphism of algebraic stacks e.g. Tag 075U and a notion of
relative dimension of a locally of finite type morphism $T\to \mathcal{X}$ from an algebraic space into an ...
3
votes
0answers
66 views
Smoothness of the stack of Shtukas without modifications
It is a well-known fact, that the stack of shtukas $\mathop{\mathcal{S}\!\mathit{ht}_{\pm 1}^1}$ with two legs and elementary modifications (i.e. those of type $(1,0,\dotsc,0)$ or $(0,\dotsc,0,-1)$) ...
4
votes
1answer
421 views
What is the local structure of a general Artin stack?
Let $X$ be an Artin stack over the complex numbers. What can one say about the local structure of $X$, i.e. what is the simplest class of stacks by which were can always find a cover of $X$ by open ...
3
votes
1answer
332 views
Stacks as local quotients or via atlases
If one looks up the definition of a Deligne--Mumford stack or an Artin stack, one usually finds something like:
A DM (resp. Artin) stack is a stack $X$ satisfying [insert condition in the diagonal ...
1
vote
0answers
117 views
Perfect complexes on affine schemes
I'm reading a paper on algebraic stacks and in some part is stated the following:
Let $X$ be an algebraic stack and let
$P\in D_{qc}(X)$ be a perfect complex. Then, for every $x\in |X|$, there ...
2
votes
0answers
118 views
Reference for calculating the dimension of algebraic stacks
I am interested in the dimension of algebraic stacks.But, I’m in trouble because I can’t calculate it.Are there any good reference or tools for calculating it?
I use the “Champs algébriques” by ...
1
vote
0answers
104 views
Deformation theory about filtered sheaves
I am reading the paper “ Components of the stack of torsion-free sheaves of rank 2 on ruled surfaces” by C.Walter.
In the proof of Lemma 4.1(a) ,in $Filtcoh(X)$ and an object $F_1 \subset F$ the ...
2
votes
0answers
85 views
The morphism between the moduli stack of filtered sheaves and of coherent sheaves
I am thinking about the morphism from the moduli stack of filtered coherent sheaf on $X$ to the moduli stack of coherent sheaves defined by forgetting filtrations,i.e.$Filtcoh(X)$ $\rightarrow$ $Coh(X)...
1
vote
0answers
114 views
About morphisms of Artin stacks
I read the following question .
(Morphism of Artin stacks)
In Mr.skirvin’s answer,I found the statement “ Given surjective, representable $\phi:\mathcal{F} \to \mathcal{G}$, we obtain a surjective map ...
3
votes
1answer
227 views
Can being schematic be checked on an atlas?
Let $X\to Y$ be a map between algebraic stacks, and $U\to Y$ a smooth atlas of $Y$. Suppose we know that $X\times_Y U$ is a scheme, can we show that $X\to Y$ is schematic?
5
votes
1answer
286 views
K-theory for a (geometric) stack
There is a notion of $K$-theory for a manifold $M$.
Is there a notion of $K$-theory for a stack $\mathcal{D}\rightarrow \text{Man}$ that is representable by a Lie groupoid $\mathcal{G}$; that is $...
2
votes
1answer
174 views
Stack associated to Groupoid object in category $\text{Sch}/S$
Consider the category of manifolds $\text{Man}$.
A groupoid object in the category of manifolds is called a Lie groupoid, denoted by $\mathcal{G}$. There is a way to associate a stack (over the ...
3
votes
0answers
338 views
Prerequisites for understanding algebraic geometry of “algebraic gerbes”
I am trying to learn about algebraic geometry of gerbes.
I am familiar with set up of gerbes in the case of differential geometry. Though there is some similarity between differentiable gerbes and ...
2
votes
0answers
84 views
Calculating intermediate extension on the stack of coherent sheaves of rank $1$
Let $L$ be a line bundle of degree $d$ on a curve $X$ and let $x$ be a point of $X$. I want to describe the intermediate extension of the constant sheaf from the stack of line bundles of degree $d$ to ...
5
votes
1answer
256 views
Does the compactified Torelli map extend to a proper map of stacks?
Let $M_g^{ct}$ denote the moduli stack of compact type genus $g$ stable curves and $A_g$ the moduli stack of principally polarized $g$-dimensional abelian varieties.
Can someone provide a reference ...
3
votes
0answers
104 views
Theta bundles on moduli space of principal G-bundles
Let $G$ be a simply connected, semi-simple affine algebraic group and $C$ be a smooth projective curve with $g \geq 3$.
Let $\mathcal{M}_G$ be a moduli stack of principal G-bundles on the curve $C$ ...
2
votes
1answer
222 views
Local question and descent category for a quasi-coherent sheaf on $\mathbb{G}_m$-gerbe
Update: I removed what I thought was unecessary and tried to be more straightforward in the hope to get an answer.
Context:
Suppose I have a $\mathbb{G}_m$-gerbe $\mathcal{G}$ over a scheme $X$ with ...
2
votes
0answers
207 views
Notions of algebraic/differential geometry of scheme/manifolds extended to algebraic/differential stacks
Given a manifold, one can associate a stack over the category of manifolds, which is a differential geometric stack. This gives a functor $\text{Man}\rightarrow \text{D.Stacks}$. This is an embedding....
7
votes
0answers
1k views
Visualization of an algebraic stack
As the visuallization of an algebraic stack is virtually impossible I warn about this is a soft question.
I am interested in thinking visually about algebraic stacks (also higher and derived stacks, ...
3
votes
0answers
205 views
Stacks in moduli spaces of sheaves research
I want to get some practice and build more appreciation for the use of stacks in the context of classical moduli spaces of sheaves. Here by classical I vaguely mean hands-on description of the ...
4
votes
1answer
723 views
Generalized Behrend version for Grothendieck-Lefschetz trace formula
[MOVED HERE FROM MSE.]
The statement of the Grothendieck-Lefschetz fixed point theorem is well-known. For a proper algebraic variety $X$ over $\mathbb F_q$,
$$\#X(\mathbb F_q) =\sum_i (−1)^i Tr(Fr_X, ...
5
votes
1answer
174 views
Topological realisation of a stack (explicit description)
Let $X$ be an Artin stack over $k=\mathbf{C}$. I've heard that $X$ has a topological realisation, but I've not been able to find an explicit decription.
My first guess would be: take a smooth cover $...
4
votes
0answers
253 views
MSRI Workshop videos and lecture notes
I am referring to MSRI workshop "Introductory Workshop on Algebraic Stacks, Intersection Theory, and Non-Abelian Hodge Theory".
There are videos and lecture notes available which can be seen on ...
18
votes
3answers
947 views
Are there prominent examples of operads in schemes?
There is an abundance of examples of operads in topological spaces, chain complexes, and simplicial sets. However, there are very few (if any) examples of operads in algebraic geometric objects, even ...
4
votes
1answer
287 views
Stack being represented by a scheme/manifold
On page $10$ of the survey article Algebraic stacks, by T. Gomez (arXiv:math/9911199), we have following result
If a stack has an object with an automorphism other than the identity, then the ...
3
votes
0answers
392 views
English translation of G.Laumon, L.Moret-Bailly book Champs algébriques
Is there an English translation of G.Laumon, L.Moret-Bailly book Champs algébriques.
Most questions on this site on stacks received this book as reference in comments/answers.
So, I want to ask if ...
2
votes
0answers
123 views
Is this construction with stacks a blow-up?
Let $X$ be the stack of rank $1$ degree $b$ coherent sheaves $E$ with torsion of length at most 1 on an elliptic curve $C$. Let $Y$ be the stack of pairs $E^{'} \subset E$ such that $E \in X$ and $E/E^...
2
votes
0answers
111 views
Atlas for a stack of sheaves of rank 1 with torsion
I would like to construct an atlas for the stack of sheaves E of rank 1 and degree b on an elliptic curve C such that E has torsion of length at most 1. Am I allowed to fix both the determinant L of ...
2
votes
1answer
170 views
Is the stack of stable curves with no rational component algebraic?
Let $g\geq 2$ be an integer and let $\overline{\mathcal{M}}_g$ be the (smooth proper Deligne-Mumford) algebraic stack of stable curves of genus $g$.
Let $\mathcal{M}_g^{nr}$ be the substack of ...
3
votes
0answers
222 views
from functor of points to stacks (or almost)
I want to have a better "working knowledge" of stacks. However every time that I approach the topic, I finish hitting a wall of technical details. They are essential, but I don't need them for now.
...
3
votes
0answers
151 views
A naive question about representations of group stacks
For an algebraic group $G$ over a field $k$, the abelian category $ Rep_k(G)$ of $k$-linear locally finite representations of $G$ can be identified $QCoh(BG_{lis-et})$. Suppose now I replace $G$ with ...
1
vote
0answers
159 views
On presentations of algebraic stacks
Every algebraic stack $X$ admits a presentation as quotient stack $[U/R]$ of a groupoid in algebraic spaces
($U$ is an algebraic space of objects, $R$ is an algebraic space of arrows, and there are ...
7
votes
2answers
406 views
Understand the difference between two stacks
Let us work over $\mathbb{C}$. Let $G$ be a finite group, acting on $\mathbb{A}^1$ via a character, and let $H$ be the kernel of the action.
Assume that $\mathbb{A}^1$ is the coarse moduli space of ...
2
votes
0answers
335 views
Tangent and cotangent bundle of a smooth algebraic stack
Are there any good notion of tangent and cotangent bundle (and stacks) of a smooth algebraic stack, similar to the notion of tangent and cotangent bundles (and spaces) of smooth schemes?
I am ...
0
votes
1answer
131 views
Properties inherited by the coarse moduli space
Let $\mathcal{X}$ be a regular, seperated Deligne-Mumford stack and $X$ be the coarse moduli scheme associated with $\mathcal{X}$. Then, is $X$ regular? I guess this is a basic fact but am not able to ...
