Questions tagged [stacks]
In mathematics a stack or 2-sheaf is a sheaf that takes values in categories rather than sets.
493
questions
4
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1
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Stacks for the extensive topology?
Recall that any extensive category can be canonically endowed the structure of a site via the extensive topology, which is the Grothendieck topology whose covering morphisms are the coproduct ...
2
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0
answers
190
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Is there a formal local criterion of finiteness?
Vague question: Is there a criterion to deduce that a morphism between algebraic stacks is finite based on the local deformation functors?
For sure this is not enough, so let me be more specific.
...
6
votes
0
answers
507
views
How does one define the complete local ring of an algebraic stack at a (geometric) point?
Question
How does one define the complete local ring of an algebraic stack at a (geometric) point?
Including what the right definition might be, this is all I'm asking.
I can do this for schemes ...
7
votes
1
answer
476
views
What is the motivic class of a quotient stack?
The Grothendieck ring of complex varieties $K(Var_\mathbb C)$ is the free abelian group generated by isomorphism classes $[X]$ of $\mathbb C$-varieties, modulo the scissor relation $[X]=[Z]+[X\...
6
votes
0
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276
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When stacks don't help
Say I have a map between two coarse moduli spaces $f:M\to M'$, which is very clearly described (by performing some operation that works well for families). I want to prove that $f$ is formally ...
2
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0
answers
97
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concretely checking descent data condition in moduli stack of elliptic curves (simple question)
I am trying to understand in a very down to earth way how the definition of stack (really, prestack, i.e. Homs form a sheaf) allows for the presence of twists; specifically in the example of the ...
2
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0
answers
268
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Einstein's field equation on orbifolds
I was wondering if there is some kind of Einstein's field equation for orbifolds (say semi-Riemannian of Lorentz signature if this make sense).
Here, by an orbifold I mean the "stacky" quotient of, ...
1
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0
answers
273
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How to think about the quotient field of an integral stack?
This is the definition given in Vistoli's paper.
Let $F$ be an integral stack. A rational function of $F$ is a morphism $G \rightarrow A^1_S$ defined on a nonempty open substack $G$ of $F$.
...
28
votes
2
answers
2k
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morphisms representable by algebraic spaces vs morphisms representable by schemes
So I've been working with moduli stacks in algebraic geometry for a while now, with no formal training in the technicalities of the theory of algebraic stacks (ie, I've read a few articles and I learn ...
5
votes
1
answer
552
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Perfect chain complexes
In Thomason-Trobaugh in Remark 2.4.4 it is written: "On a general scheme, the perfect complexes are locally finitely presented objects in the "homotopy stack" of derived categories."
I was wondering ...
14
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1
answer
741
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Hodge to de Rham spectral sequence for stacks
For some work I'm doing, I need a version of the Hodge to de Rham spectral sequence for stacks. I am not at all an expert on stacks, so please excuse me if I make minor technical mistakes in stating ...
2
votes
2
answers
800
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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
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0
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129
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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
159
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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 ...
2
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0
answers
266
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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 ...
3
votes
0
answers
216
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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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179
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$\mathbb E$-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.
...
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0
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179
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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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0
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154
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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
111
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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 ...
5
votes
1
answer
496
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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
247
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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
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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 $\mathcal X$ ...
11
votes
1
answer
345
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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 ...
19
votes
1
answer
2k
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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
368
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
266
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
593
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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 ...
2
votes
0
answers
196
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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
165
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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 $Qcoh(U)$ ...
12
votes
1
answer
330
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 ...
1
vote
0
answers
248
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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
271
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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
votes
0
answers
252
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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
182
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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 ...
5
votes
1
answer
299
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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 (...
4
votes
1
answer
299
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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., $M = Spec k$.
...
3
votes
0
answers
179
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
745
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
214
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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
187
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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
210
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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 ...
9
votes
1
answer
567
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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 ...
7
votes
0
answers
291
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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 non-...
2
votes
1
answer
239
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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: \...
2
votes
2
answers
845
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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
votes
0
answers
301
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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 ...
6
votes
1
answer
943
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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
votes
0
answers
196
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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 ...
7
votes
1
answer
934
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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 ...