Questions tagged [algebraic-stacks]
The algebraic-stacks tag has no usage guidance.
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What about stacks of categories in algebraic geometry?
Stacks qua moduli spaces were introduced to keep track of nontrivial automorphisms of the objects they parameterize. In essence they are groupoids of objects with some form geometric cohesion. The ...
11
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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 ...
28
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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 ...
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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 ...
13
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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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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
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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.
7
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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 ...
6
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Ferrand pushouts for algebraic stacks
Given algebraic spaces $X$, $Y$, $Z$ with a finite morphism $Y \rightarrow X$ and a closed immersion $Y \hookrightarrow Z$, the pushout $P \cong X \amalg_Y Z$ exists as an algebraic space (cf. Temkin ...
4
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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
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0
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"standard limit arguments" involved in showing that roughly every DM stack is locally a quotient stack
I am trying to understand proposition 3.6 of this paper (perhaps I am in over my head):
https://arxiv.org/pdf/math/0703310.pdf
If we denote the stack $\mathcal{M}$ and its coarse moduli space as $M$ ...
3
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1
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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
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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 ...
2
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0
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Dimension of artin stacks
I was reading the article of Laumon 1988: Un analogue global du cone nilpotent (I am sorry but I could not find an available link to share).
He fixes a curve $X$ (over $\mathbb{C}$) and considers ...
2
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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 ...
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Semistable curves of genus $g\geq 2$ form an Artin algebraic stack in the etale topology?
I need the reference to a detailed proof the following fact.
Let $g\geq 2$ be an integer. Let $\mathcal M^{ss}_g: Sch\rightarrow Groupoids$ be the prestack which sends a scheme $T$ to the groupoid of ...