Questions tagged [algebraic-stacks]
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260
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etale fundamental group of global quotient algebraic stacks
I am reading about fundamental group of algebraic stacks, and in my opinion, the class of quotient stacks is an important one in algebraic stacks. However, I am not clear about how to compute ...
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Cartesian square in the category of Algebraic stacks
Suppose we have a commutative diagram of Artin stacks
$ \newcommand{\ra}[1]{\kern-1.5ex\xrightarrow{\ \ #1\ \ }\phantom{}\kern-1.5ex} \newcommand{\ras}[1]{\kern-1.5ex\xrightarrow{\ \ \smash{#1}\ \ }\...
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Topological property of an algebraic stack and its presentation
I started to learn algebraic stacks this January. I found there are several properties of algebraic stacks which are defined in terms of their underlying topological spaces, for example, connectedness,...
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Moduli of morphisms between varieties
Let $\mathcal{M}, \mathcal{N}$ be two "well-behaved" (i.e. representable by an algebraic stack parametrising flat, proper, surjective, finitely presented morphisms with geometric fibres ...
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Closed embedding into weighted projective stack/space
Let $ Z $ be a proper or even, projective scheme. I have two related questions. (everything is over complex numbers)
(1) What is a criteria for a map $ \phi: Z \rightarrow \left[ \mathbb{A}^{n+1} - (0)...
- 754
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Defining log prestacks (and their structures)
It's possible to define log schemes, and Olsson's thesis defines log stacks.
Is there a definition of log prestack in the literature? Is it easy to extend Gaitsgory-Rozenblyum type results (...
- 5,112
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Residual gerbes and coarse moduli space of a DM stack
Let $X$ be a nice DM stack (Noetherian, separated), and $X_0$ its coarse
moduli space which exist by the Keel-Mori theorem.
(I like the exposition in D. Rydh. “Existence and properties of geometric ...
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Local structure of smooth morphisms of stacks
Let $\varphi:X \to Y$ be a smooth morphism of schemes.
There is a well-known local structure theorem: Zariski locally $\varphi$ is given by the composition of an etale morphism and an affine space (...
- 19.4k
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Explicit computation of inertia stacks
I am learning algebraic stacks myself with some reference recommended by my friends. I know from here Section 6.1 that an explicit description of the fibre product of stacks $\mathfrak{X}\times_\...
- 363
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How the automorphism group of an elliptic curve acts at the localization of the stack $\mathcal{M}_{1, 1, k}$ at the corresponding point
I am studying the enlightening article "The Picard Group of $\mathcal{M}_{1, 1, S}$", written by Fulton and Olsson, but I have some problems with a proof.
Setting
Let $\mathcal{M}_{1, 1, k}$ ...
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Smallest atlas for algebraic stack
Let $X$ be an algebraic stack of finite type over a field.
Is there an intrinsic way to calculate the minimum of the dimensions of all atlases of $X$?
By intrinsic here I mean using constructions such ...
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Is $\operatorname{Rep}(G,\operatorname{SL}_2)$ representable by an algebraic space?
Let $G$ be a finite group. Consider the category of rigid analytic spaces over $\operatorname{Spf}\mathbb{Q}_p$, and let $\operatorname{Rep}(G, \operatorname{SL}_2)$ be the fibred category above it, ...
- 675
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a connected geometrically unibranch algebraic stack of finite type over a field is irreducible
Let $f:X\to \mathfrak{X}$ be a smooth presentation of geometrically unibranch connected algebraic stack by a scheme, which is geometrically unibranch since being geom. unibranch is local in smooth ...
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Semistable curves of genus $g\geq 2$ form an Artin algebraic stack in the etale topology?
I need the reference to a detailed proof the following fact.
Let $g\geq 2$ be an integer. Let $\mathcal M^{ss}_g: Sch\rightarrow Groupoids$ be the prestack which sends a scheme $T$ to the groupoid of ...
- 482
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Quotient stack is an algebraic space when $G$ is finite and acts freely
I have been following Jarod Alper's lecture series on YouTube on Stacks https://youtube.com/playlist?list=PLhFI5R_xInjdhtWuhgYlA8NZGXO-unnl4
From what I understand -
If a smooth affine group scheme $...
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Decomposition of vector bundles on the inertia stack of a DM stack
Let $X$ be a tame DM stack over $\mathbb{C}.$ Let $IX$ denote the inertia stack of $X.$ Let $K(IX)$ denote the Grothendieck group of vector bundles on $IX.$ By the discussion on page 20 of Toen's ...
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Connected components of inertia stacks
Let $k$ be a field. Let $X$ be a connected tame DM stack over $k.$ Let $IX$ be the inertia stack of $X.$ Then $IX$ is a disjoint union of connected components.
Is this always a finite union? If not, ...
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Smoothness of inertia stacks
Let $k$ be a field of characteristic zero. Let $X$ be a smooth DM stack over $k.$ Is the inertia stack $IX$ always smooth over $k$?
I believe this is true, but cannot find a proof in the literature. I ...
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When is a quotient stack of finite type?
Let $k$ be a field. Let $X$ be a scheme over $k.$ Let $G$ be an affine smooth group scheme over $k$ acting on $X.$ Suppose $X$ is of finite type over $k.$ Does this guarantee that the quotient stack $[...
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Birational morphisms from DM stacks to their coarse moduli spaces
Let $X$ be an integral scheme over a field. Let $G$ be a finite group acting on $X$ faithfully. Assume the quotient stack $[X/G]$ is separated (e.g., when $G$ acts on $X$ properly). Then $[X/G]$ is a ...
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Questions about root stacks
Let $\cal X$ be a DM stack and ${\cal D}\hookrightarrow{\cal X}$ an effective Cartier divisor on it. Suppose that $n$ is a positive integer invertible in ${\cal X}$.
Let $\sqrt[n]{{\cal D}}\to{\cal X}$...
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Cohomology of coherent sheaves on Deligne Mumford stacks
Suppose that $\cal X$ is tame Deligne Mumford stack with generic trivial inertia. Let $X$ be its muduli space and $f:{\cal X}\to X$ the projection.
Let $\cal F$ be a coherent sheaf on $\cal X$.
Is it ...
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Moduli stacks of algebraic surfaces—obstructions to existence?
The moduli stack $\mathcal{M}_g$ of genus $g$ curves is one of the deepest objects in mathematics, so of course you wonder to what extent you can construct an (Artin?) stack parametrising algebraic ...
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2-shifted 2-form on the classifying stack 𝐵𝐺
Let $G$ be a reductive group. A $2$-shifted $2$-form on the classifying stack $BG$ is by definition a a morphism of quasi-coherent complexes
\begin{equation}
\mathcal O_{BG}\rightarrow (\wedge^2 \...
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$\mathbb G_{\mathrm{m}}$-gerbes are to (derived) Azumaya algebras as $G$-gerbes are to …?
Let $X$ be a quasicompact quasiseparated scheme over a field $k$. The connection between Azumaya algebras over $X$ and $\mathbb G_{\mathrm{m}}$-gerbes over $X$ is well-known: there exists an injection ...
- 435
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Base change and quotient stacks
Let $X$ be an algebraic space over a scheme $S$, let $G$ be a smooth group scheme over $S$, acting on $X$, and let $T\to S$ be a morphism of schemes. Is it true that there is an isomorphism
$$[X/G]\...
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In what sense is an orbifold a DM stack?
My advisor mentioned in passing that orbifolds are Deligne-Mumford stacks, and I'd like to know in which sense this is true. The only reference I can find is this article (https://arxiv.org/abs/0806....
- 181
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Representability of blow-up of stacks
Suppose $\mathcal{Z}\subseteq\mathcal{X}$ is a closed immersion of stacks. Is the blow-up morphism $p:Bl_{\mathcal{Z}}(\mathcal{X})\to\mathcal{X}$ representable? More precisely, if $C\to\mathcal{X}$ ...
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Quotient of a quotient stack: interesting examples?
Let $X$ be a scheme acted on by an algebraic group $G$. Also, let $H$ be an algebraic group acting on the quotient stack $X/G$, for the definition of "act", see Romagny - Group Actions on ...
- 5,112
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Geometric quotients of DM stacks by group actions
Let $G$ be a finite group acting on a DM stack $X$ and $Y$ the quotient stack. I.e., $Y \to BG$ has fiber $X$. Is there a geometric quotient $X/G$ in some sense? I want automorphism groups of points ...
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Is there a degeneration formula for Gromov-Witten K-theoretic invariants?
By Gromov-Witten K-theoretic invariants (call them KGW) I mean the invariants defined by Givental and Lee.
I expect the formula expresses the KGW of the generic fiber of a given degeneration in terms ...
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Square root of a line bundle up to a finite surjective morphism
Given a projective variety $X$ over a field of any characteristic, consider a line bundle $\mathcal{L}$ over $X$.
The existence of a line bundle $\mathcal{L}^\prime$ with an isomorphism ${\mathcal{L}^...
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Cohomology of quotient stack
Let $X$ be an algebraic variety over $\mathbb{C}$ (the ground field is not important but this makes things easier I think) and $G$ an algebraic group acting over it. Let's say we know that there's a ...
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Moduli stack of quiver representations
Let $Q$ be a finite quiver. As far as I know there's a great amount of work concerning the so-called quiver varieties one can associate to it. Loosely speaking, these are obtained by taking GIT ...
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Residue field of point on an algebraic stack
$\DeclareMathOperator{\Spec}{Spec}$ Let $X$ be an algebraic stack.
Is there is a well-defined notion of the residue field of a point $x \in |X|$?
Attempts:
Recall that a point on a stack is an ...
- 19.4k
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Character stack and character variety
Let $\Sigma$ be a Riemann surface of genus $g$. We can consider two different type of objects associated to it parametrising representations of its fundamental groups. On one side we have the ...
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Artin's "Versal Deformations and Algebraic stacks": Question concerning proof of Theorem 3.3
I have been reading Artin's paper titled "Versal deformations and algebraic stacks" and am a bit confused about a statement he makes in the proof of Theorem 3.3, in the first few lines of pg....
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Generic stabiliser of moduli space of vector bundles on a curve
Let $C$ be a curve of genus $g >1$. Fix a line bundle $L$ on $C$. Let $Bun_{n,L}$ be the moduli stack of vector bundles of rank $n$ on $C$ with determinant isomorphic to $L$. I have a few questions ...
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Criterion for relative Deligne-Mumfordness?
A quotient stack $X/G$ of a scheme by a smooth algebraic group is Deligne-Mumford if and only if (approximately) the stabiliser groups of points are finite (for the precise statement see corollary 8.4....
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Finiteness result for higher direct image of $\ell$-adic sheaves
Let $f:X\to Y$ be a representable map of finite type (or is finite dimensional enough?) Artin stacks, whose fibres (which are schemes) have dimension at most $n$. Then is it true that $R^qf_*\mathbf{...
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Chevalley complex and $\text{BG}$
For a long time I've been under the impression that the Chevalley complex $\text{CE}(\mathfrak{g})$ of a semisimple (maybe can weaken this) Lie algebra $\mathfrak{g}$ can be extracted from the ...
- 5,112
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Measuring non-separatedness of algebraic stacks
Let $X$ be an algebraic stack of finite type over a field $k$, and let $U\subset X$ be an open dense separated sub-scheme of $X$. Let $D=\text{Spec}~ k[[t]], D^*=\text{Spec}~ k((t))$.
Fix a map $f:D^*\...
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Irreducible components of an algebraic stack
Let $\mathcal{X}$ be an algebraic stack of finite type over a (separably closed) field $ k$. Let's say that $\mathcal{X}$ has finite dimension $d \in \mathbb{Z}$. Is it still true that the number of ...
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Stacky points detect nilpotent cohomology
Take a finite type scheme $X/\mathbf{C}$ acted on by a reductive group $G$. Then $X/G$ is an Artin stack. Let $\alpha\in H^*(X/G)$ be a rational cohomology class on it.
Question: Is it true that $\...
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Explicit natural correspondence between cusps of $X(N)$ and isomorphism classes of level $N$ structures on Tate($q^N$)
In Katz' paper Antwerp III, section 1.4 (Ka-14) one reads (we assume $n \geq 3$ integer):
''The scheme $\overline{M}_n - M_n$" over $\mathbb{Z}[1/n]$ is finite and étale, and over $\mathbb{Z}[1/n,...
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Relation between stacky curves and "M-curves"
A tame stacky curve over a field $k$ is a geometrically connected proper smooth DM stack of dimension 1 which has a dense open substack which is a scheme, and whose automorphism group of each ...
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The classifying stack of $\operatorname{PGL}(2)$ and the moduli space of genus zero curves
$\DeclareMathOperator\PGL{PGL}$The classifying stack of $\PGL(2)$ is the stack quotient $[\operatorname{Spec} k/\PGL(2)]$ where $\PGL(2)$ acts trivially on $\operatorname{Spec} k$.
Since $[\...
- 709
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Automorphism of a stack morphism
Let $X$ be an algebraic stack and let $f: S \to X$ be a smooth covering of $X$ by a scheme $S$.
Motivation: Forgetting about stacks for a moment and going back to covering spaces: Given a covering map ...
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Does inclusion from n-stacks into (n+1)-stacks preserve the sheaf condition?
I'm going to describe two situations that seem to contradict each other, and I'm interested to know precisely what's wrong with this reasoning.
Let $M$ be a manifold, and consider the presheaf $C^*(-,...
- 14.7k
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Generalizations of Artin–Verdier duality?
Constructible étale abelian sheafs on $Spec\ O_\mathbb K$, for number fields $\mathbb K$, satisfy Artin-Verdier duality. Are there known any algebraic schemes or algebraic stacks, other than $Spec\ O_\...
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