7
votes
2answers
371 views

Is a Deligne-Mumford curve defined over Qbar if and only if its coarse moduli space is

Let $\mathcal X$ be a smooth proper finite type Deligne-Mumford stack over $\mathbb C$ that is generically a scheme. Let $X$ be its coarse moduli space. If $\mathcal X$ can be defined over ...
2
votes
0answers
66 views

Quasi-finite morphisms of stacks

Let $f:X\to Y$ be a morphism of ``nice" stacks over $\mathbf C$ such that the induced morphism on coarse moduli spaces is quasi-finite. Is $f$ quasi-finite? By a "nice" stack I mean a smooth finite ...
11
votes
1answer
528 views

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) ...
2
votes
1answer
183 views

Grothendieck duality for stacks

Let $\mathcal{X}$ be a smooth, proper and separated Deligne-Mumford stack and let $\pi:\mathcal{X}\rightarrow X$ be its coarse moduli space. Does Grothendieck duality hold for the morphism $\pi$ ? In ...
2
votes
1answer
169 views

Algebraic stacks: limit preserving versus locally of finite presentation

I'm wondering what the precise relationship is between an algebraic stack being locally of finite presentation and being limit preserving. Under some mild hypotheses on the diagonal (in force ...
6
votes
1answer
467 views

Is “stackiness” transitive? (and a couple other basic questions about stacks)

Say, $B$ is a category fibered in groupoids over some category $C$, and $A$ is a category fibered in groupoids over $B$. Suppose $A$ is a stack (over whatever site) over $B$, and $B$ is a stack over ...
3
votes
1answer
280 views

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.
4
votes
1answer
237 views

Do Deligne-Mumford curves also have rational functions

If $X$ is a curve over a field of characteristic zero, then $X$ has a rational function, i.e., a finite morphism to the projective line. Question. Suppose that $X$ is a Deligne-Mumford (or just ...
3
votes
0answers
242 views

On the local structure of Deligne-Mumford stacks

Is it true that for any DM stack $\mathcal{X}$ (quasicompact, separated, of finite type over a field) there is a Zariski covering $\mathcal{U}_i \to \mathcal{X}$ by open substacks, such that for all ...
1
vote
0answers
142 views

Zariski's Main Theorem for stacks

Let $X$ be a separated Deligne Mumford stack of finite type over a base field $k$ and $Y$ be a proper Deligne Mumford stack over $k$. Assume there is a quasifinite, representable, surjective and ...
4
votes
1answer
165 views

Stabilizer Action on vector bundle on a stack

Suppose you have a Deligne Mumford stack $X$ and a geometric point $x:Spec{k}\rightarrow X$ with stabilizer group $Stab(x)$.Let $F$ be a locally free sheaf on $X$. How is the action of $Stab(x)$ on ...
3
votes
0answers
146 views

Chow ring of a $\mu_2$-gerbe

Suppose that $X$ is a stack, and $Y \to X$ is a $\mu_2$-gerbe. Is there any relationship between the integral Chow rings (in the sense of Edidin and Graham) of $X$ and $Y$? (I assume they become ...
1
vote
1answer
221 views

Finite-type Artin Stack over $\mathbb C$

Suppose I have an Artin stack $\mathfrak M$ locally of finite-type over $\mathbb C$ with presentation $M\rightarrow \mathfrak M$. Suppose further that $\mathfrak M$ "represents" (in the stack sense) ...
5
votes
1answer
247 views

universal property of blow up for stacks?

I will use as a reference Hartshorne Prop. II.7.14, the universal property of blow-up. $\tilde{X}$ is the blow up of $X$ along a sheaf of ideals. $Z\to X$ is the morphism that is to be lifted to ...
4
votes
1answer
149 views

A question on $Isom(p_1^*E,p_2^*E) \rightrightarrows X$

I am learning the moduli stacks of vector bundles and have trouble understanding some definitions. Let $E$ be a rank $n$ vector bundle over the scheme $X$. We denote by $p_i$ the $i$th projection ...
5
votes
1answer
322 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 ...
3
votes
0answers
194 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 ...