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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 ...

**9**

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**1**answer

428 views

### Why is there no stack of $\ell$-adic sheaves on a curve?

One of the main players in the categorical geometric langlands correspondence is the moduli stack of rank n integrable connections on a complex curve. The reason for considering such objects is that ...

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196 views

### Smooth map to the stack of G-bundles

Let $G$ a semisimple group and $B$ a Borel subgroup.
We denote by $Bun_{G}$ the stack of G-bundles.
Is it true that a certain open subset $Bun_{B,r}$ maps smoothly to $Bun_{G}$?
My question comes ...

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97 views

### KK-witnesses of Gysin maps between differentiable stacks

In 1982 Alain Connes gave the construction of a KK-element $f! \in KK(C(X), C(Y))$ that "witnesses" the fiber integration/Gysin/Umkehr/wrong-way map on topological $K$-theory along a K-orientable map ...

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104 views

### Co-completeness of differential stacks?

I once heard a rumour that various nice categories of stacks were co-complete. Gepner and Henriques, working from the groupoids point of view, give a construction [link] of 2-colimits of topological ...

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227 views

### Rigidification and good moduli space (morphism) in the sense of Alper

Let $\mathcal{X}$ be an Artin stack. In "Abramovich, Graber, Vistoli - Twisted bundles and admissible covers", the authors describe a procedure to rigidify $\mathcal{X}$ by a central subgroup $H$ of ...

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176 views

### Uniqueness of the canonical etale coverings

This is a construction [Definition 6.1] given in the paper D-equivalence and K-equivalence by Kawamata.
Let $X$ be a normal quasiprojective variety such that the canonical divisor $K_X$ is a ...

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484 views

### are moduli stacks deligne-mumford stacks in general

Let M be your favorite moduli stack over the field of complex numbers.
Is it reasonable to expect M to be a Deligne-Mumford stack?
I know this is true for the moduli space of curves of genus g, ...

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82 views

### How does the machinery of left-exact comonads generalize from sheaves to stacks?

Suppose that we have two Grothendieck sites, their associated sheaves $\mathcal{E}=\rm{Sh}(\bf{C},J)$ and $\mathcal{F}=\rm{Sh}(\bf{D},K)$ and a geometric surjection $f:\mathcal{E}\to\mathcal{F}$. This ...

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**1**answer

182 views

### universal families and maps to quotient stacks

Suppose I have a certain (contravariant) moduli functor $M:Schemes \to Groupoids$ that is represented by a quotient stack $[X//G]$ where $X$ is a scheme and $G$ a linearly reductive group. Roughly ...

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222 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) ...

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557 views

### Geometric description of the Deligne-Mumford stacks

It is well known that a one-dimensional smooth Deligne-Mumford stack (over $\mathbb{C}$) could be described as a collection of its "stacky" points (finitely many) on its coarse moduli space with the ...

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432 views

### Passage from the moduli functor to the functor of points of the coarse moduli space

Let $F: (Sch)^{o}\to (Set)$ be a functor that admits a coarse moduli $Y$ (a scheme). We can consider $Y$ as a representable functor $h_{Y}: (Sch)^{o}\to (Set)$. Is there a direct way to produce $h_Y$ ...

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196 views

### What is the structure of the stack of complexes supported in dimension less than r?

Let $X$ be something. (smooth and projective variety over C are my assumptions)
The stack $M$ parameterising coherent sheaves on $X$ splits as a disjoint union of open and closed substacks $M_\alpha$, ...

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199 views

### Definition of derived category of a stack

In their book, Bernstein an Lunts define the equivariant derived category in several ways. One can be expressed as follows:
Let $X$ be a say complex variety with an action by an algebraic group $G$.
...

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**1**answer

167 views

### Twisting an object P by an H-Torsor I

I am reading Brylinski's Loop Spaces, Characteristic Classes, and Geometric Quantization.
The Statement
Let $C$ be a gerbe on a space $X$ with "abelian" band $H$, $f: Y \to X$ a local homeomorphism ...

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266 views

### The Grothendieck Ring of Higher Stacks

The Grothendieck ring of varieties is defined to be the free abelian group spanned by isomorphism classes of varieties modulo the cut & paste (or scissor) relations, which say that $[X] = [U] + ...

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317 views

### How to specify a finite group up to inner automorphism?

I want some finite set of data to which I can canoically associate a "group up to inner automorphism", and which can be constructed canoically from a "group up to inner automorphism". I have a few ...

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442 views

### examples of moduli functors for which coarse moduli space does not exists

Well, the title almost says it all. I would like to list as many examples as possible of moduli functors, for which a coarse moduli space does not exist (and maybe explain why). So, examples such as ...

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258 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 ...

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783 views

### When does sheaf cohomology commute with arbitrary direct sums?

It is well known and more or less proven in Hartshorne's 'Algebraic Geometry' (p. 209) that for every noetherian scheme $X$ and every collection of abelian sheaves $\mathcal{F}_i$ the canonical map
...

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409 views

### On the local structure of stacks

1) Is it true that any Deligne-Mumford stack is locally a quotient stack $[X/G]$ with a finite group $G$?
2) Is it true that any Deligne-Mumford stack can be globally presented as a quotient stack ...

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489 views

### On the coarse moduli space of a stack

Consider a stack $\mathcal{X}$ over $\mathbb{C}$ as a category fibred in groupoids over the category of schemes. Let $\mathcal{X}^s$ be the $\pi_0$ of this category, i.e. objects of $\mathcal{X}^s$ ...

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174 views

### Enumerativity of Gromov-Witten invariants of orbifolds

For smooth Deligne-Mumford stacks, there is a well-defined Gromov-Witten theory, see http://arxiv.org/pdf/math/0103156.pdf and http://arxiv.org/pdf/math/0603151.pdf.
Is there some sense, or some ...

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578 views

### Teaching stacks to differential geometry students

Does anyone have any experience teaching stacks over the category of manifolds to students whose background is, say, a semester-long course on manifolds? Does anyone know of any publicly available ...

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150 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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501 views

### How does descent theory imply a sheaf is a scheme?

I've noticed that often authors will comment that "descent theory" shows that some sheaf in the étale topology is actually a scheme. I was wondering what result in descent theory actually implies ...

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### Cohomologically trivial stacks

The following theorem of Serre is well-known:
A noetherian scheme $X$ is affine if and only if $H^i(X; \mathcal{F}) = 0$ for all quasi-coherent sheaves $\mathcal{F}$ on $X$ and all $i>0$. ...

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123 views

### quotient by a proper equivalence relation

Let X be a scheme and R be a proper equivalence relation on X. What can be said about the geometric structure of the quotient X/R?
Is it representable by a stack, for example?

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249 views

### family of gerbes over smooth and proper algebraic varieties

Let $X$ be a smooth and proper variety over $\mathbb{C}$. Let $F$ be an $\mathbb{A}^1$ family of $\mathbb{G}_m$ gerbes over $X$. Suppose the fibers over every point away from 0 in $\mathbb{A}^1$ are ...

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461 views

### Recovering classical Tannaka duality from Lurie's version for geometric stacks

In Lurie's paper Tannaka Duality for Geometric Stacks, it is essentially shown that specifying a morphism of geometric objects
$$ f \colon X \to Y$$
is equivalent to giving a corresponding pullback ...

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280 views

### fpqc sheafification and localisation

I am slightly confused about sheafification at the moment.
I first learned sheaves defined as a subcategory of presheaves, then I was told that sheaves are also a localisation of presheaves, then I ...

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117 views

### Can stabilizer groups in an orbifold have global twisting?

Can stabilizer groups in an orbifold have global twisting?
For example, consider the two groups $\mathbb Z/3\times\mathbb Z$ and $\mathbb Z/3\rtimes\mathbb Z$ (where $\mathbb ...

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260 views

### Are quotients of stacks flat?

Let $\cal X$ be a DM stack of finite type over a field (if necessary, I will assume that $k=\mathbb{C}$ and $\cal X$ is a scheme, or even a variety) and $G$ be a finite group. Then we have a quotient ...

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278 views

### Divisorial contraction: when is the image an algebraic space or a stack?

Let $X$ be a smooth projective surface (in the category of varieties, or schemes), and let $C\subset X$ be a curve (a priori not irreducible, but the irreducible case in itself is already ...

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697 views

### Homotopy theory of topological stacks/orbifolds

Motivation $\newcommand{\T}{\mathscr{T}}$
I have many times found myself saying some variant of the following. Let $\T_g$ be the Teichmüller space of a surface of genus $g$, and $\Gamma_g$ its ...

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533 views

### what exactly is the moduli functor for classifying elliptic curves with (full) level N structure?

So, when people say, "the moduli problem of classifying elliptic curves over $\mathbb{C}$ with level $N$ structure", there are usually two associated functors I've seen:
$P_N : ...

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329 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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146 views

### Irreducibility of monodromy of eigenspaces of families of cyclic coverings

In the article "La conjecture de Weil", Deligne proves that for the primitive cohomology of a universal family $f:X \rightarrow S$ for $M_{d,n}$ the moduli stack of hypersurfaces of degree $d$ in ...

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311 views

### Chern classes of vector bundles on a stack

Is there literature about chern classes of vector bundles on DM-stacks? I had a look at a lot of different papers about intersection theory on stacks and related stuff and this seems to be known, but ...

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### stack quotient question

Hi,
I have the following question:
let $k$ a field with $char(k)= p>0$, which we can assume to be perfect, $W(k)$ the ring of Witt vector, and $a,b$ positive integers.
Consider the ring ...

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232 views

### Representability of Hom-sheaves of various moduli spaces

(May be a poor title, happy to update)
Recall that for a stack $\mathcal{X} \to Sch$ on schemes (e.g. fppf site) and a pair of morphisms $x,y\colon U\to \mathcal{X}$ ($U$ a representable stack) there ...

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### Stacks in modern number theory/arithmetic geometry

Stacks, of varying kinds, appear in algebraic geometry whenever we have moduli problems, most famously the stacks of (marked) curves. But these seem to be to be very geometric in motivation, so I was ...

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### 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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288 views

### how to construct a $C^\infty$ stack from a holomorphic stack

Given a complex manifold, you can `weaken' its structure to give a smooth manifold. Is there an analogous construction that constructs a stack over the category of smooth manifolds from a stack over ...

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### Does every stack with a connection admit an atlas with a connection?

Dear all,
Let $S$ be a scheme in characteristic $0$,
and let $\mathscr{X}/S$ mean a crystal in Artin stacks over $S$ in the sense of this handout, page 4, Definition 0.5, where we replace the scheme ...

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### Reference request: an algebraic stack whose closed points have no automorphisms is an algebraic space

The question is stated in the title. I think BCnrd states in a comment here
Is every (Artin/DM) algebraic stack fibered in sets an algebraic space?
that while the answer is not found in Laumon & ...

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### Why are sheaves not preserved in this case?

Suppose that $C$ is a Grothendieck site, and $\mathscr{X}$ is a stack over $C$ (which is NOT equivalent to a sheaf). Let $$\pi_{\mathscr{X}}:\int_{C} \mathscr{X}\to C$$ denote the associated fibered ...

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### Quasi-coherent sheaves on $M_{FG}$ and the exact functor theorem

I'm struggling with these notes, and one of the things I don't really understand is the following. The notes consider the stack $M_{FG}$ of formal groups; this is the stack associated to the prestack ...

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### Separation condition for higher Deligne-Mumford stacks

Let $X$ be a stack of $n$-groupoids on the site of affine schemes over a fixed base, with the etale topology. If $n=1$ then for $X$ to be Deligne-Mumford, aside from having an etale atlas from an ...