Questions tagged [algebraic-stacks]

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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 ...
jimmy's user avatar
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5 votes
1 answer
404 views

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}^...
user158892's user avatar
1 vote
1 answer
727 views

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 ...
Tommaso Scognamiglio's user avatar
3 votes
2 answers
453 views

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 ...
Tommaso Scognamiglio's user avatar
10 votes
1 answer
449 views

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 ...
Daniel Loughran's user avatar
7 votes
0 answers
325 views

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 ...
Tommaso Scognamiglio's user avatar
3 votes
0 answers
227 views

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....
user's user avatar
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1 vote
0 answers
97 views

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 ...
iron feliks's user avatar
2 votes
0 answers
107 views

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....
Pulcinella's user avatar
  • 5,506
2 votes
1 answer
376 views

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{...
Pulcinella's user avatar
  • 5,506
2 votes
1 answer
386 views

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 ...
Pulcinella's user avatar
  • 5,506
4 votes
0 answers
143 views

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^*\...
Alexander Braverman's user avatar
6 votes
1 answer
453 views

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 ...
Tommaso Scognamiglio's user avatar
5 votes
0 answers
158 views

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 $\...
Pulcinella's user avatar
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4 votes
1 answer
373 views

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,...
FelixBB's user avatar
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4 votes
1 answer
512 views

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 ...
k.j.'s user avatar
  • 1,352
3 votes
0 answers
198 views

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 $[\...
user's user avatar
  • 719
3 votes
0 answers
187 views

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 ...
user's user avatar
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9 votes
2 answers
726 views

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^*(-,...
David Corwin's user avatar
  • 15.1k
2 votes
0 answers
264 views

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_\...
Adam's user avatar
  • 2,370
7 votes
0 answers
276 views

Moduli stacks and representability of diagonal by schemes

The answer to my question might very well be standard, but I have had trouble finding the right keywords to search for it, so I apologize if this is something well-known to the experts. I am learning ...
Wojowu's user avatar
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4 votes
0 answers
87 views

Classifying twists for a general moduli problem

Suppose $\mathfrak X$ is a (Deligne-Mumford/Artin/...) stack and denote by $|\mathfrak X(k)|$ the set of isomorphism classes of it's $k$-valued point. Can we say something in general about the fibers ...
Asvin's user avatar
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5 votes
1 answer
468 views

About an argument in Olsson's book

The following picture is from Algebraic spaces and stacks (p.54) by Martin Olsson. I don't understand how to conclude that $\alpha$ is induced by a nonzero class in the end. It seems that there might ...
Lao-tzu's user avatar
  • 1,856
6 votes
2 answers
307 views

Relation between finite dimensional representations of an affine group scheme and quasicoherent sheaves on the classifying stack

Let $G$ be an affine group scheme over a field $k$ of characteristic zero. I understand that when $G$ is algebraic there is an equivalence between the category $\text{Rep}(G)$ of (rational) ...
Patrick Elliott's user avatar
2 votes
0 answers
239 views

Are universal geometric equivalences of DM stacks affine?

Let $f:X\to Y$ be a map of Deligne-Mumford stacks. Let's say that the map $f$ is a geometric equivalence if the induced map on small étale topoi is a geometric equivalence. Moreover, let's say that ...
Harry Gindi's user avatar
  • 19.4k
4 votes
1 answer
349 views

A de Rham space for meromorphic connections?

To any space $X$ you can associate its de Rham space $X_{dR}$. Vector bundles on $X_{dR}$ are the same thing as vector bundles on $X$ with a flat connection. Can anything like this be said for ...
Pulcinella's user avatar
  • 5,506
1 vote
0 answers
286 views

Is there an intrinsic Gauss map?

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Gr{Gr}$This may all be well known, too vague, or stupid; my apologies. The Gauss map is defined by embedding your $k$-dimensional smooth scheme/...
Leo Herr's user avatar
  • 1,084
3 votes
0 answers
280 views

Sheaf $\operatorname{Isom}(x,y)$ isomorphic to fibered product $U \times_{(x,y),X \times X, \Delta} X$

$\DeclareMathOperator\Mor{Mor}\DeclareMathOperator\Isom{Isom}$Let $S$ be a scheme and $C$ be the category $(Sch/S)$. Let moreover $p:X \to C$ be an algebraic stack over $C$. Consider an arbitrary ...
user267839's user avatar
  • 5,948
2 votes
0 answers
86 views

Map from the stack of coherent sheaves on a curve to the Grothendieck group

Let $X$ be a smooth, projective curve. We let $Coh(X)$ be the stack of coherent sheaves on $X$. Its Grothendieck group is $Pic(X)\times\mathbf{Z}$. Is the map $$ Coh(X)\rightarrow Pic(X)\times \mathbf{...
hennlu's user avatar
  • 323
5 votes
1 answer
423 views

Do quotient stacks help classify the orbits of group actions on varieties?

I am trying to understand algebraic stacks and I have a newbie question. Let $X$ be an affine variety over an algebraically closed field to keep things simple and let $G$ be a reductive group acting ...
Adam's user avatar
  • 2,370
5 votes
1 answer
453 views

Construction of an atlas for the moduli stack $\mathcal{Bun}_X^{n,d}$ in F. Neumann's 'Algebraic Stacks and Moduli of Vector Bundles'

I'm reading Frank Neumann's "Algebraic Stacks and Moduli of Vector Bundles" and have some problems to understand a construction from the proof of: Theorem 2.67. (page 81) The moduli stack $...
user267839's user avatar
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14 votes
0 answers
322 views

Do connected algebraic stacks have a smooth cover by a connected scheme?

An algebraic stack $X$ has an induced topological space $|X|$ given by equivalence classes of fields mapping to $X$ as outlined in the stacks project. If $|X|$ is connected, does that imply there ...
Leo Herr's user avatar
  • 1,084
4 votes
1 answer
450 views

Are universally submersive morphisms of stacks universal descent morphisms for relative étale stacks?

Recall that a map of schemes, algebraic spaces, stacks, etc is called submersive if the associated map on underlying topological spaces is a quotient map. Recall moreover that a map is called ...
Harry Gindi's user avatar
  • 19.4k
3 votes
0 answers
194 views

$2$-vector spaces and algebraic $2$-stacks

I am thinking about higher Artin stacks in the sense of Simpson, concretely I would like to calculate the dimension and compare these two cases: $\mathfrak{X}_{1}=$ Higher linear stack classifying (...
Martin Hurtado's user avatar
8 votes
0 answers
188 views

Closed immersion → Pro-open immersion factorization for residual gerbes

Let $X$ be a quasi-separated algebraic stack. Then it is a theorem of Rydh that every point $x$ in $X$ admits a residual gerbe. More or less, the construction proceeds by first taking the closure ...
Harry Gindi's user avatar
  • 19.4k
5 votes
1 answer
518 views

When quotient stacks (for nonsmooth group) are algebraic and related questions

Let $k$ be a field. Consider a group $k$-scheme $G$ and let $X$ be a $k$-scheme equipped with an action of $G$. Then one can define the quotient stack $[X/G]$. Objects of $[X/G]$ over $k$-scheme $T$ ...
Slup's user avatar
  • 532
5 votes
0 answers
297 views

Čech nerve $C(U)$ corresponds to $BG$ in same manner as a hypercover $\mathcal{H}(U)$ to

We can via the bar construction canonically associate to a monoid $A$ the nerve $N(B A)$, a simplicial set with $N(\mathbf{B}A)_k := \times^{k+1} A $ and canonical face maps and degeneracy maps ...
user267839's user avatar
  • 5,948
3 votes
1 answer
377 views

K/G-theory of affine bundles

Setting: $f : C \to D$ is a morphism of Artin stacks over $X$ which is a torsor for a vector bundle $T \to X$: étale-locally in $X$, we have $C \simeq D \times_X T$. I want to conclude that $f^*: G(D)...
Leo Herr's user avatar
  • 1,084
6 votes
1 answer
373 views

Zariski's main theorem for non-representable morphisms?

Let $f:\mathcal{X}\to \mathcal{Y}$ be a separated quasi-finite map of qcqs Deligne-Mumford stacks. Is there a version of Zariski's main theorem that makes sense in this context? Rydh proved a ...
Harry Gindi's user avatar
  • 19.4k
2 votes
0 answers
410 views

Dimension of the moduli stack of vector bundles over a curve

Let $Vect_{n}(C)$ the moduli stack of vector bundles $V$ of rank $n$ over a smooth curve $C$ of genus $g$. It is well known that $Vect_{n}(C)$ is a smooth stack of dimension $\dim(H^{0}(C,End(V)))-\...
Martin Hurtado's user avatar
4 votes
0 answers
177 views

Gysin map and $B\mathbf{G}_m$, confusion

Write $\text{Sh}(X)$ for the triangulated/stable $\infty$ category of $\ell$-adic sheaves on $X$, and $k\in\text{Sh}(X)$ fo the unit object. In playing around with $\text{Sh}(B\mathbf{G}_m)$ I've ...
Pulcinella's user avatar
  • 5,506
9 votes
0 answers
472 views

Geometric stacks, groupoids and étendues

If $(C, \tau)$ is a site with pullbacks and $\tau$ subcanonical, it is well known that these things are essentially equivalent: Groupoids $s,t: U_1 \to U_0$ where $s,t$ are covering for the $\tau$-...
Damien Robert's user avatar
4 votes
0 answers
224 views

Dimension of derived Artin stacks and perfect complexes

I am interested in the concept of dimension of derived and $n$-Artin stacks. Take for example the definition 4.10 of From HAG to DAG: derived moduli stacks. in which they define the dimension of a ...
Martin Hurtado's user avatar
8 votes
1 answer
667 views

Milnor excision for algebraic stacks

Recall that a commutative square of commutative rings $$\begin{matrix} A&\to&B\\ \downarrow &&\downarrow\\ A^\prime&\to&B^\prime\end{matrix}$$ is called a Milnor square if the ...
Harry Gindi's user avatar
  • 19.4k
12 votes
0 answers
278 views

birational geometry of moduli spaces: why work on the coarse space?

In studying the birational geometry of $\overline{\mathcal{M}}_g$, it seems standard to work with the coarse space $\overline{M}_g$ rather than the smooth stack $\overline{\mathcal{M}}_g$. Why is this?...
Hans Sachs's user avatar
6 votes
1 answer
348 views

Ferrand pushouts for algebraic stacks

Given algebraic spaces $X$, $Y$, $Z$ with a finite morphism $Y \rightarrow X$ and a closed immersion $Y \hookrightarrow Z$, the pushout $P \cong X \amalg_Y Z$ exists as an algebraic space (cf. Temkin ...
some1random's user avatar
1 vote
0 answers
215 views

Two definitions of cotangent complex

I have reading a paper by Professor Pridham(https://arxiv.org/abs/0905.4044v4). Page 47-48 contains a comparison of the two definitions of the cotangent complex, but there is a part I don't understand....
Walter field's user avatar
3 votes
0 answers
145 views

Perfect complexes on stacks are strict?

As mentioned in the comments of this question, on a quasiprojective scheme over a field, every perfect complex is globally a complex of vector bundles. I have some question about the extension of this ...
Pulcinella's user avatar
  • 5,506
3 votes
1 answer
212 views

Algebraic spaces in the étale topology (proof from Stacks project)

I have a question about the proof of Lemma 78.12.1 from Stacks Project. The aim of the last paragraph of the proof is to verify that the map of sheaves in the étale topology $F \to U/R$ is an ...
user267839's user avatar
  • 5,948
1 vote
1 answer
569 views

Stabilizer $G_x$ of a $k$-valued point of an algebraic Stack

An algebraic stack or Artin stack is a stack in groupoids $\mathcal{X}$ over the étale site such that the diagonal map of $\mathcal{X}$ is representable and there exists a smooth surjection from (the ...
user267839's user avatar
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