The algebraic-stacks tag has no usage guidance.

**10**

votes

**1**answer

412 views

### Ramification of the map from the stack of elliptic curves to the $j$-line

Let $\mathcal{M}_{1, 1}$ be the stack of elliptic curves. Its coarse moduli space is $\mathbb{A}^1_{\mathbb{Z}}$ with the map $\mathcal{M}_{1, 1} \rightarrow \mathbb{A}^1_{\mathbb{Z}}$ given by the ...

**13**

votes

**0**answers

346 views

### Coarse moduli spaces of stacks for which every atlas is a scheme

Let $X = [P/G]$ be a smooth finite type separated DM-stack over $\mathbb C$ given as the quotient of a smooth projective scheme $P$ by the action of a smooth (finite type separated) reductive group ...

**6**

votes

**0**answers

217 views

### Is the stack of varieties with a big line bundle algebraic

In Starr's paper https://www.math.stonybrook.edu/~jstarr/papers/moduli4.pdf the folk result that the fibred category of pairs $(X\to S, L)$, where $S$ is an affine scheme, $X\to S$ is flat proper ...

**5**

votes

**0**answers

116 views

### Testing the vanishing of cohomology fiberwise for a proper morphism from an Artin stack

Let $S$ be a Noetherian scheme, let $f\colon\mathscr{X} \rightarrow S$ be a proper morphism with $\mathscr{X}$ an algebraic stack, and let $\mathscr{F}$ be an $S$-flat coherent sheaf on $\mathscr{X}$. ...

**5**

votes

**0**answers

219 views

### Homology of stack points (from math.stackexchange)

(This question is duplicated here)
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 ...

**5**

votes

**0**answers

336 views

### Coarse moduli spaces of quotient stacks

Suppose you have a separated Deligne Mumford quotient stack $[V/G]$ over a field of characteristic $0$, where $V$ is a quasiprojective variety and $G$ is an algebraic group that does not necessarily ...

**5**

votes

**0**answers

132 views

### Normalization of quotient stacks

Suppose you have a Deligne Mumford stack which is a quotient $[X/G]$ of a scheme $X$ by an algebraic group $G$ .
What is the normalization of that? Is it true that its normalization is a quotient ...

**4**

votes

**0**answers

160 views

### Vanishing theorems on toric DM stacks

In chapter 9 of the book Toric varieties by Cox-Little-Schenck several cohomology vanishing theorems for toric varieties are proved or mentioned.
In this question I am interested in references for ...

**4**

votes

**0**answers

196 views

### Stack theoretic image?

If $X \xrightarrow{f} Y$ is a morphism of schemes then the scheme theoretic image of $f$ is the smallest closed subscheme $Z \subset Y$ through which $f$ factors through.
Is this notion defined for ...

**4**

votes

**0**answers

330 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 ...

**3**

votes

**0**answers

217 views

### Finiteness of the connected components of a stack

Let $X$ be an algebaic stack over a scheme $S$, for any $S$-scheme $Y$ we can consider the groupoid $X(Y)$ of $Y$-points. Denote by $\pi_0(X(Y))$ the set of isomorphism classes of the groupoid.
Are ...

**3**

votes

**0**answers

137 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 ...

**3**

votes

**0**answers

146 views

### Are there any results on stable maps to Artin stacks with infinite stabilizers?

The Abramovich-Vistoli/Chen-Ruan theory of twisted stable maps into Deligne-Mumford stacks is extremely useful, as is the generalization to tame Artin stacks in positive characteristic. I am ...

**3**

votes

**0**answers

164 views

### Subgroups of a group algebraic space

I found in the literature many references on the representability of quotients of group schemes but almost nothing about subgroups. For this reason I hope that my question is a silly one and that what ...

**3**

votes

**0**answers

169 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 ...

**3**

votes

**0**answers

60 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 ...

**3**

votes

**0**answers

212 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 ...

**2**

votes

**0**answers

152 views

### Help for reference of moduli stack of fake elliptic curve

I see everywhere says the following:
Let $B$ be an indefinite quaternion algebra over $\mathbb{Q}$ of discriminant $D$, $\mathcal{O}_B$ be a maximal order, $N$ be an positive integer coprime to $D$. ...

**2**

votes

**0**answers

114 views

### Stack of curves and universal deformations

I've just started studying algebraic stacks and I have a very basic question.
I've learned the notion of Deligne Mumford stack and I've seen as the stack of stable curves $\overline{\mathcal{M}_g}$ ...

**2**

votes

**0**answers

221 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 ...

**2**

votes

**0**answers

190 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 ...

**1**

vote

**0**answers

273 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 ...

**0**

votes

**0**answers

95 views

### pushing out families of curves

Let $f:X\rightarrow Y$ be a morphism of schemes with smooth curves as fibers. Let $g:X\rightarrow Z$ be a family of smooth or nodal curves with $Z$ a regular scheme. Does the push-out $Z\coprod_X Y$ ...