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8
votes
0answers
168 views

Is an étale morphism of algebraic stacks locally quasi-finite?

An étale morphism of schemes is unramified, and an unramified morphism is locally quasi-finite. Does the same hold for étale morphisms of algebraic stacks? Let us recall the definitions, following ...
12
votes
0answers
122 views

Sheaves of categories

I recently read an answer on this MO post explaining that one reason people are interested in higher category theory is to make reasonable sense of something like a "sheaf of categories" on a ...
3
votes
0answers
43 views

How to describe the points of a quotient stack?

Let $G$ be a finite algebraic group acting on a projective complex variety $X$. Then a quotient $Y=X/G$ exists as a scheme and, if $G$ acts freely, $Y$ is an orbit space and the natural map ...
5
votes
1answer
88 views

Is my algebraic space a scheme?

Consider $\mathcal{M}_{1,1}$ over $\bar{\mathbb{Q}}$. I have an algebraic stack $\mathcal{M}$ finite etale over $\mathcal{M}_{1,1}$ I can prove that it is an algebraic space (essentially because ...
1
vote
1answer
89 views

Why the existence of automorphism of varieties makes a functor not being a fine moduli space?

Let $F: (Sch) \to (Sets)$ be a functor sends schemes to sets (for example, $F$ sends a scheme $S$ to families of K3 surfaces over $S$ with some fixed polarization). Then it is known that because of ...
1
vote
1answer
190 views

The moduli space of stable maps

I am reading Katz' book Enumerative Geometry and String Theory. I have a few questions regarding the moduli space of degree $d$ genus $0$ stable maps into $\mathbb P^n$, denoted ...
3
votes
1answer
68 views

Showing that the moduli stack of triangles is equivalent to the quotient stack $[\tilde{T}/S_3]$

$\require{AMScd}$ A good reference for this question is found in the first chapter in the unfinished book Algebraic Stacks by Behrend, Conrad, Edidin, Fulton, Fantechi, Göttsche, and Kresch. ...
4
votes
0answers
51 views

What are Mumford's 'moduli topologies'?

I've been reading Mumford's Paper 'Picard Groups of Moduli Problems'. Stated in modern language, the most famous result from the paper is that the moduli stack of elliptic curves has Picard group ...
1
vote
0answers
66 views

Does faithfully flat descent work using restriction of scalars rather than extension?

Vistoli's notes on fibred categories and descent - http://homepage.sns.it/vistoli/descent.pdf - introduce (section 4.2.1) descent on modules over a commutative ring. The idea is as follows: ...
3
votes
1answer
107 views

Understanding the stack $B\mathbb{Z}$

Here, let $\mathbb{Z}$ be the group scheme whose functor of points is the constant functor which takes a connected affine scheme to the group $\mathbb{Z}$. I'm having a bit of trouble understanding ...
4
votes
1answer
75 views

Examples of algebro-geometric moduli problems without a “natural” choice of pullback?

When I try to learn about stacks, something that is mentioned all over the place (as a reason not to define moduli problems as functors from schemes to groupoids) is the fact that the pullback of a ...
8
votes
1answer
103 views

Three meanings of étale sheaf on X

When I am studying stacky stuffs, I am always confused by the notion of étale abelian sheaves on $X$, because conceivably there might be three different meanings of that: Take the global étale site ...
3
votes
1answer
104 views

How does a section of a stack give a sheaf?

At nLab in the article constant stack and a few other related articles, a pattern is mentioned where a section of a constant sheaf is a locally constant function, a section of a constant stack is a ...
4
votes
1answer
119 views

Stacks versus sheaves with values in categories

A (small) category is a perfectly valid algebraic structure like Groups, Rings, vector spaces, groupoids etc. So on a topological space or more generally on a site, it makes perfectly sense to ...
1
vote
0answers
46 views

Stack on commutative ring spectra?

One approach to stacks to call a stack a "sheaf of groupoids" which means a functor $$ \mathcal{C}^{\text{op}} \rightarrow \mathcal{G} $$ from a category $\mathcal{C}$ with a Grothendieck topology to ...
7
votes
1answer
104 views

Homology of stack points

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 classifying stack $\mathcal{B}G$. This has a ...
3
votes
1answer
228 views

Reference for Deligne-Mumford

What is a good reference for someone new to the theory of Deligne-Mumford stacks, other than the original Deligne-Mumford paper itself? The paper itself seems readable with some effort; but the fear ...