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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 ...
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1answer
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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 ...
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39 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. ...
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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 ...
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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: ...
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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 ...
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1answer
63 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 ...
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81 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 ...
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87 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 ...
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1answer
59 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 ...
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40 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 ...
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93 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 ...
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193 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 ...