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4
votes
0answers
324 views

A question on an intuitive way to look at stacks

I am reading the chapter "Introducing Algebraic Stacks" in The Stacks Projects to get a feeling for them. There is a small point that throws me off. They denote $\mathcal{M}_{1, 1}$ the moduli stack ...
5
votes
1answer
359 views

If $X$ is a smooth and proper stack, does it admit a smooth and proper atlas?

Fix a ground scheme $S$ (a field say). By atlas for an algebraic stack I mean a smooth and surjective morphism $Y \to X$ from a scheme (or algebraic space or affine scheme) $Y$. If the stack $X$ is ...
2
votes
1answer
287 views

Representability of the diagonal morphism of stacks

Corollaire 3.13 in "Champs algebriques" says that the diagonal 1-morphism of stacks $\Delta:\mathcal{X} \to \mathcal{X} \times_S \mathcal{X}$ is representable if and only if the sheaf ...
3
votes
0answers
202 views

Cohomology of line bundles on smooth projective toric Deligne-Mumford stacks

Let $\mathcal{X}$ be a smooth projective toric Deligne-Mumford stack with coarse moduli scheme $\pi\colon \mathcal{X}\rightarrow X.$ Let $[D]$ be a Cartier divisor on $\mathcal{X}$ such that ...
1
vote
0answers
251 views

Fourier-Mukai transforms on stacks

I have "generic" questions about Fourier-Mukai transforms. Question 1: Does there exist a well-defined notion of Fourier-Mukai transform on (Deligne-Mumford) stacks? Question 2: Do there exist ...
11
votes
1answer
613 views

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 ...