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### Reference request on Moduli stacks

I am starting now my self-studies on moduli stacks, and saw some material on internet, but unlike the moduli spaces, I feel that they lack on geometrical meaning. So I would apreciate good references ...
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### The non-existense of the fine moduli scheme of vector bundles. Why?

The reference I am using is this enter link description here. 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 ...
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### 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 ...
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### 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 ...
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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 $$\eta:[X/... 1answer 95 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 ... 1answer 95 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 ... 1answer 207 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 \overline{M}(\mathbb{... 1answer 77 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. (http://... 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 \... 0answers 73 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: ... 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 ... 1answer 77 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 ... 1answer 108 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 (... 1answer 106 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 ... 1answer 124 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 ... 0answers 48 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 ...
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 ...