# Tagged Questions

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### Is this map representable? what is the fiber?

Consider the following map of stacks: Let $S$ be the stack whose $A$ points are diagrams of the form \begin{array}{ccccc} {} & {} & U & {} \\ {} & {} & \downarrow & \searrow \\...
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### On orders of stabilisers of group actions and stacks

Let $X$ be a finite type irreducible separated DM-stack over $\mathbb C$. Let $x$ be an object of $X(\mathbb C)$ with stabilizer $G_x$. Let $y$ be an object of $X(\mathbb C)$ with stabilizer $G_y$. ...
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### The non-existence of the fine moduli scheme of vector bundles. Why?

The reference I am using is this one. The question is about the moduli space of vector bundles. I am trying to understand why the fine moduli scheme does not exist. Let $C$ a projective curve. Let $S$ ...
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### Is there a reference for boundedness of smooth canonically polarized varieties over Z (No…)

In Kollár's paper Quotient spaces modulo algebraic groups, Kollár mentions right above Theorem 1.8 that the stack $\mathcal M_P$ of smooth canonically polarized varieties over Spec $\mathbb Z$ with ...
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### Are these moduli problems of curves “well-behaved”?

Let X be a smooth projective surface over $\mathbb C$, and let $d\geq 3$ be an integer. Suppose that all smooth hypersurfaces of degree $d$ are of genus $g\geq 2$. Let $H_{X,d}$ be the Hilbert scheme ...
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### Psi-classes on moduli spaces of weighted curves

Let $\overline{\mathcal{M}}_{g,A[n]}$ be the stack of weighted genus $g$ curves with weights $A[n]=(a_1,...,a_n)$, and let $\pi:\mathcal{C}\rightarrow \overline{\mathcal{M}}_{g,A[n]}$ be the universal ...
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### Mayer-Vietoris sequence for orbifolds

Is there a version of the Mayer-Vietoris long exact sequence for orbifolds? I am interested in orbifold homology as opposed to the homology of the underlying topological space.
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### $\mathbb E$-descent maps in topological spaces in terms of different sites?

The paper Facets of Descent I by Janelidze and Tholen defines $\mathbb E$-descent maps as those for which $\Phi^p:\mathbb EB\longrightarrow \mathsf{Des}_\mathbb{E}(p)$ is an equivalence of categories. ...
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### Pushforward for differentiable stacks/ Lie groupoids

Let $X$ be a differentiable stacks, and let $(G_{0}, G_{1}, s,t)$ be a Lie groupoid representing $X$. Let $NG_{\bullet}$ be the nerve of the above groupoid. The De rham complex of $X$ can be defined ...
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### groupoids representing mapping stacks

1)Let $X$ be a differentiable stack ((2,1) sheaf over the category of smooth manifolds $Man$) and that is geometric. Let $N\in Man$, then $$Map(y(N),X)$$ is again a differentiable stack ($y$ is the ...
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### representability of some mapping stack

Let $S$ be an Artin stack of finite type. We assume that it contains a point as an open dense. Is it always true that the mapping stack: $Hom^{0}(\mathbb{P}^{1},S)$ which consists of sections ...
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### Essential dimension and the moduli space of abelian varieties

The following problem is listed here: http://www-personal.umich.edu/~erman/Papers/Questions2.pdf and attributed to Vistoli: Let $\mathcal A_g$ denote the moduli stack of principally polarized abelian ...
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### Fiberwise criterion for a stack to be a gerbe

Let $f:X\to Y$ be a morphism of algebraic stacks. If the geometric fibres of $f$ are algebraic spaces, then $f$ is representable by algebraic spaces. I'm wondering about analogues of this fiberwise ...
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### Are there any non-trivial $G$-gerbes over the analytic space $\mathbb C$

Does there exist a finite (abstract) group $G$ and a non-trivial $G$-gerbe $\mathcal X\to \mathbb C$, where we work in the category of analytic stacks. My guess is that $G$-gerbes for $G$ an abelian ...
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### Representable map of Deligne-Mumford stacks

Let $\mathscr{M}\to\mathscr{N}$ be a map of (Deligne-Mumford) stacks. Recall that it is said to be representable by affine schemes if for all affine maps $\operatorname{Spec}R\to \mathscr{N}$, the ...
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### Gromov-Witten invariants for arithmetic surfaces counting sections passing through points

Suppose we are given an arithmetic surface, $X\to \text{Spec}\mathbb{Z}[1/N]$ smooth and quasi-projective, and a finite set of closed points all in different vertical fibers. Can we count the number ...
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### How to realize the descent data of Qcoh as a (pseudo)-limit in Cat?

It is well-know that $Qcoh$ is a fibered category on $Sch$. In more details let $\mathcal{C}$ be the category $(Sch/S)$ of schemes over a fixed base scheme S. For each scheme $U$ we define $Qcoh(U)$ ...
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### Are coarse spaces of 1-dimensional smooth proper Artin stacks smooth?

Let $\mathcal{X}$ be a regular proper 1-dimensional Artin stack with finite diagonal, with coarse space morphism $\mathcal{X} \to X$. Question: Is $X$ regular? Some comments: I'm happy to assume ...
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### Representablility of fiber product over stacks

Let $(\mathcal{C},\mathcal{T})$ be a site and $\mathcal{X}$ a stack over it. Suppose $(\mathcal{C},\mathcal{T})$ subcanonical and hence for every object $U$ in $\mathcal{C}$ consider the associated ...
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### Are automorphism groups of polarized varieties of finite type

It is "well-known" that the stack of polarized varieties is an algebraic stack with quasi-compact and separated diagonal. In particular, if $(X,L)$ and $(Y,M)$ are polarized schemes over a scheme $S$,...
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### 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 ...
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### Stacks with representable morphisms to algebraic stacks

If $Y$ is an algebraic stack over a scheme $S$ and $X$ is a stack such that there exists an $S$-morphism $X\to Y$ representable by algebraic spaces, then is $X$ an algebraic stack (in the sense that ...
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### The stack of group algebraic spaces

The fibred category $\mathcal A$ of algebraic spaces over a scheme $S$ is a stack (over the category of affine schemes with the etale topology). This is proved in Laumon and Moret-Bailly's book (see (...
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### Stacks with a small coarse moduli space

Let $k$ be a field of characteristic zero. Let $X$ be a finite type algebraic stack over $k$ with a coarse (or good) moduli space $M$. Suppose that $M$ is isomorphic to a point, i.e., $M = Spec k$. ...
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### Is there a difference between the inertia stack and the universal automorphism group

Let $\mathcal M$ be a stack representing some moduli problem. Let $\mathcal X\to \mathcal M$ be the corresponding universal family. What is the difference between the inertia stack $I\to \mathcal M$ ...
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### Algebraic spaces which are automatically schemes

Let $S$ be a scheme, and let $f:X\to S$ be a morphism of algebraic spaces. If $f$ is smooth proper curve of genus at least two, then $X$ is a scheme. (Here I mean that $f$ is a smooth proper morphism ...
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### On functors which are generically representable

Let $F$ be a set-valued (contravariant) functor on the category of schemes. Let $F_{\mathbb Q}$ be the associated functor on the category of schemes over $\mathbb Q$. Suppose that $F_{\mathbb Q}$ is ...
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### Singularities of the moduli stack of polarized hyperkahler varieties

Inspired by the recent question on singularities of the moduli stack of Calabi-Yau threefolds (Singularities of the moduli stack of Calabi-Yau threefolds) I'd like to ask the following question. Is ...
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### Descending a monomorphism of stacks

The question is about Proposition 3.8.1 in Laumon and Moret-Bailly book on algebraic stacks. Let $S$ be a scheme and let $F: \mathscr{X} \rightarrow \mathscr{Y}$ be a morphism of $S$-stacks (for the ...
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### Singularities of the moduli stack of Calabi-Yau threefolds

Let $M$ be the moduli of polarized Calabi-Yau threefolds over $\mathbb C$ with fixed Euler characteristic. The coarse moduli space is singular (as usual), but what about the stack? In many cases I ...
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### Is there a more general obstruction to the existence of moduli spaces than the existence of automorphisms?

We are taught that, in general: A type of objects that has nontrivial automorphisms cannot have a fine moduli space. The proof generally goes along the lines of: Take an object $X$ with a ...
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### reference for “curves over S are locally the base change of a curve over S' which is finite type over R”

So recently I heard someone claiming that if $X\rightarrow S$ is a smooth curve (not necessarily proper?) and $S$ is an arbitrary scheme over $\text{Spec }R$ (for $R$ sufficiently nice), then there is ...
Let $X$ be a scheme (I'm happy to assume that $X$ is of finite type, separated, and over $\mathbb{C}$) and let $E$ be an elliptic curve which acts freely on $X$. Does the quotient stack $[X/E]$ have ...
Suppose a scheme $X$ has tame quotient singularities. Does there exist a smooth DM stack $\mathcal X$ with coarse space $X$ so that the coarse space morphism $\mathcal X\to X$ is an isomorphism away ...