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### 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 ...

**2**

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186 views

### Lang isogeny for group stacks

Let $G$ be a commutative algebraic group stack over $\mathbb{F}_q$ (I don't really care about the precise definition: I'm secretly thinking about the Picard stack of a projective curve). To what ...

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236 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**

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609 views

### What is the intuition inertia orbifold (or stack)?

I am studying orbifolds with view towards Chen-Ruan cohomology. I have been struggling with inertia orbifolds but have no intuition about them at this point. I would appreciate your motivating me by ...

**5**

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279 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 ...

**3**

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59 views

### Are two “nice” transformation groupoids with the same coarse moduli and isomorphic inertia isomorphic?

Hi!
I am stuck with the following question: suppose we have a semisimple connected algebraic group acting on a quasi-affine variety X by closed orbits, and suppose that inertia is flat.
Suppose we ...

**4**

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**1**answer

157 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 ...

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**1**answer

138 views

### Some bounded theorem of algebraic stack of coherent sheaves

Let $X$ be a connected projective scheme. Let $U$ be a finite type open substack of the algebraic stack of coherent sheaves on $X$ with a fixed Hilbert polynomial. Can one take $p>0$ such that ...

**6**

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**1**answer

312 views

### on a Deformation long exact sequence of moduli space of stable maps

I am reading the book "mirror symmetry" by Hori,Katz,Klemm,etc. And I want to understand the following Deformation long exact sequence
\begin{align}
0 & \to Aut(Σ, p_1, . . . , p_n, f)\to Aut(Σ, ...

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327 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 ...

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380 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 ...

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**1**answer

296 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 ...

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204 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 ...

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257 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 ...

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629 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 ...