All Questions
Tagged with stacks moduli-spaces
57 questions
24
votes
2
answers
1k
views
Different interpretations of moduli stacks
I'm taking my first steps in the language of stacks, and would like something cleared up. The intuitive idea of moduli spaces is that each point corresponds to an object of what we're trying to ...
- 1,386
16
votes
2
answers
3k
views
Is the Torelli map an immersion?
The Torelli map $\tau\colon M_g \to A_g$ sends a curve C to its Jacobian (along with the canonical principal polarization associated to C); see this question for a description which works for families....
- 10.5k
14
votes
1
answer
648
views
Does a degeneration always have a larger-dimensional automorphism group?
Suppose $\newcommand{\X}{\mathcal{X}}\X$ is an algebraic stack over a field $k$, $\xi$ is a $k$-point which has another $k$-point $x$ in its closure ($x$ is an isotrivial degeneration of $\xi$). Must ...
12
votes
1
answer
2k
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 : \textbf{Ell}\...
- 10.7k
11
votes
1
answer
350
views
Counting isomorphism classes in open subsets of Bun_G
Let $G$ be a split semisimple algebraic group and let $C$ be a curve of genus $g$ over $\mathbb F_q$. Assume $g \geq 2$.
The number of $\mathbb F_q$-points of $\# \operatorname{Bun}_G(C)$, where each ...
- 148k
11
votes
0
answers
379
views
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 ...
- 5,701
10
votes
0
answers
410
views
What is the signficance of the existence of a moduli stack to a moduli problem?
This is a question in pretty unfamiliar territory for me, so if I have conceptual mistakes please correct me.
Let's say we begin with a naive moduli problem: we want a moduli space (whatever space ...
9
votes
1
answer
593
views
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 ...
- 93
9
votes
2
answers
839
views
$Pic$ of the stack of elliptic curves vs. $Pic$ of the coarse space
There's a natural map $f:\overline{\mathcal{M}}_{1,1}\to \overline{M}_{1,1}\cong \mathbb{P}^1$ from the stack of elliptic curves to the coarse space. Both spaces have $Pic=\mathbb{Z}$ hence $f^*:\...
- 3,779
9
votes
1
answer
2k
views
Kodaira-Spencer Theory and moduli of curves
I was looking at a paper of Farkas and the following confusing point came up.
Let $\mathscr{M}_g$ be the moduli stack of smooth genus $g$ curves and let $\pi: \mathscr{C} \to \mathscr{M}_g$ be the ...
- 10.5k
8
votes
2
answers
656
views
Moduli 'space' of stacks?
In algebraic geometry, we are frequently interested in parametrizing geometric objects. Formally, parametrization of geometric objects having some property can be viewed as a functor $F:Sch\rightarrow ...
user74900
8
votes
0
answers
333
views
Do automorphisms actually prevent the formation of fine moduli spaces?
I have found similar questions littered throughout this site and math.SE (for example [1], [2], [3],…), but I feel like like most of them usually just say that non-trivial automorphisms prevent the ...
- 821
7
votes
0
answers
312
views
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 non-...
- 907
6
votes
3
answers
834
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 $[...
- 3,779
6
votes
2
answers
651
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 $h_Y$ ...
- 128
6
votes
1
answer
432
views
Is there a Riemann existence theorem for orbifolds?
For smooth algebraic varieties $X$ over $\mathbb{C}$, the Riemann existence theorem establishes an equivalence of categories between the category of finite etale covers of $X$ and finite unramified ...
- 10.7k
6
votes
0
answers
277
views
When stacks don't help
Say I have a map between two coarse moduli spaces $f:M\to M'$, which is very clearly described (by performing some operation that works well for families). I want to prove that $f$ is formally ...
- 61
6
votes
0
answers
254
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 ...
- 171
5
votes
3
answers
1k
views
Proving that a map is a morphism
Example : Consider the (open, not compactified) moduli space of stable maps $ \mathcal M_g(1,d)$ of maps of smooth curves of genus $ g$ to $\mathbb P^1$. To each map
we associate its branch divisor, ...
- 238
5
votes
1
answer
524
views
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 ...
- 18.7k
5
votes
0
answers
272
views
Is the analytification of the coarse space equal to the coarse moduli space of the analytification?
If $X$ is a smooth finite type separated DM algebraic stack over $\mathbb C$ with coarse space $X^c$, then do we know whether the analytification of $X^c$ is the coarse space of the analytification of ...
- 113
5
votes
0
answers
308
views
Properties of the Zariski-Riemann topology on the set of valuations
One can classify all valuations on a function field $K$ of transcendence degree $2$ over $\mathbf{C}$. Let's consider the set $S_K$ of all valuations on $K$ endowed with the Zariski-Riemann topology.
...
- 51
4
votes
1
answer
705
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 ...
user17778
4
votes
1
answer
198
views
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 ...
- 41
4
votes
1
answer
562
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 ...
- 35.4k
4
votes
0
answers
154
views
$\mathcal{M}_{g,n}$ a scheme for $n \gg 0$? [duplicate]
I think that for $n \geq 3$, the Deligne-Mumford moduli stack $\mathcal{M}_{0,n}$ is a scheme. Is it more generally true that for every $g$, the Deligne-Mumford moduli stack $\mathcal{M}_{g,n}$ is a ...
user19475
3
votes
3
answers
1k
views
The non-existence of the fine moduli scheme of vector bundles. Why?
The reference I am using is Hoffmann - The moduli stack of vector bundles on a curve. The question is about the moduli space of vector bundles. I am trying to understand why the fine moduli scheme ...
- 587
3
votes
1
answer
280
views
Integral locus of Hitchin morphism
Let $\Sigma$ be a Riemann surface of genus $g$. To it, we can associated $M_{Dol}$ be the Higgs moduli space of rank $n$ and degree $d$. Fo simplicity let us take $(n,d)=1$. This quasiprojective ...
- 1,061
3
votes
1
answer
508
views
Grothendieck duality for stacks
Let $\mathcal{X}$ be a smooth, proper and separated Deligne-Mumford stack and let $\pi:\mathcal{X}\rightarrow X$ be its coarse moduli space. Does Grothendieck duality hold for the morphism $\pi$ ?
In ...
- 8,998
3
votes
0
answers
198
views
Interesting stacks with affine space as coarse moduli
I am looking for examples of Deligne-Mumford stacks whose coarse moduli space is $\mathbb{A}^n$ or at least an open subscheme of $\mathbb{A}^n$ whose complement has codimension $2$. (Thus the whole ...
- 12.1k
3
votes
0
answers
316
views
Reference request: Derived structure on the moduli stack of Higgs bundles
I am reading arXiv:1708.08124. When talking about the moduli stack of Higgs bundles on a projective curve $X$. It is said on page 59, first paragraph that
It is often better to put
derived ...
- 385
3
votes
0
answers
785
views
Road map for moduli space/moduli problem/moduli stack
I am familiar with (most of the) contents of Angelo Vistoli's notes on Descent theory (Stacks). I am also comfortable with basics of Schemes, their Cohomology (Cech), from Hartshorne's Algebraic ...
- 6,023
3
votes
0
answers
240
views
Stacks in moduli spaces of sheaves research
I want to get some practice and build more appreciation for the use of stacks in the context of classical moduli spaces of sheaves. Here by classical I vaguely mean hands-on description of the ...
- 1,190
3
votes
0
answers
291
views
Properties of finite quotients of quasi-projective varieties
Let $G$ be a finite group acting on a (smooth) quasi-projective variety over $\mathbb C$.
One can consider the stacky quotient $[X/G]$ or the "classical" quotient $X/G$. In general, $[X/G]$ is not a ...
- 31
2
votes
2
answers
1k
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, ppav'...
- 246
2
votes
2
answers
927
views
Universal curve of stacks of stable curve
Let $\overline{M}_{g,A}$ the moduli stack of pointed genus $g$ stable curves with weights $A = (a_1,...,a_n)$ introduced in
Brendan Hassett, Moduli spaces of weighted pointed stable curves, Adv. Math....
user58604
2
votes
1
answer
312
views
Sheaf of elliptic curves up to isogeny
For a scheme $X$, denote by $\mathcal{Ell}_X[\text{isog}^{-1}]$ the category of elliptic curves on $X$ localized at isogenies. Consider the functor
$$
\mathcal{Ell}^{isog}:Sch/S^{op}\rightarrow \text{...
- 4,386
2
votes
1
answer
363
views
Moduli of curves in characteristic zero
Let $K$ be a field of characteristic zero, and let $\overline{K}$ be its algebraic closure. Let $\overline{M}_{g,n}(K)$ and $\overline{M}_{g,n}(\overline{K})$ be the coarse moduli spaces parametrizing ...
user55899
2
votes
1
answer
258
views
picard group of moduli of elliptic r-prym curves
Let $\overline{\mathcal{M}}_{1,1}$ be the DM compactification of the moduli stack of elliptic curves. Its Picard group is $\mathbb{Z}$. Let us now consider stack of $r$-prym curves $\overline{\mathcal{...
- 3,779
2
votes
0
answers
103
views
Representability of stack of finite maps between curves
I am interested in the following moduli problem: The moduli functor $\mathcal{F}$ has $T$-points:
a nodal $n$-pointed curve $C/T$ of genus $g$.
a nodal $b$-pointed curve $D/T$ of genus $h$.
a finite ...
- 223
2
votes
0
answers
61
views
Stack of smooth fiber bundles with fiber $F$
I'd like to premise that while I know the definition of (differentiable) stack, I'm not really into the language of schemes so my understanding of what is a moduli stack is pretty concrete and ...
- 105
2
votes
0
answers
98
views
Tangent Space of Moduli of Log-Smooth Curves
We consider an algebraically closed field $\underline{k}$ and all constructions that we will consider are over this field. It is well known that for each relative nodal curve $\underline{f}: \...
- 223
2
votes
0
answers
123
views
Moduli stack of doubly periodic complexes?
Let $\mathcal{A}$ be an abelian category.
In HAG II Toen and Vessozi built a higher derived stack $X$ whose category of perfect complexes is $\text{Perf}(X)\simeq D^b(\mathcal{A})$. So $X$ is a good ...
- 5,701
2
votes
2
answers
443
views
What is the pull-back of a polarization of abelian schemes over different bases?
The following came up when reading the definition of the moduli stack of principally polarized abelian varieties in [1].
Let $\pi_1:A_1 \to S_1$ and $\pi_2: A_2 \to S_2$ be abelian schemes over $S_i$, ...
- 1,286
2
votes
0
answers
734
views
Tangent and cotangent bundle of a smooth algebraic stack
Are there any good notion of tangent and cotangent bundle (and stacks) of a smooth algebraic stack, similar to the notion of tangent and cotangent bundles (and spaces) of smooth schemes?
I am ...
- 523
2
votes
0
answers
132
views
Irreducible components of $\partial \bar{H}_{g,n}$
Let $\bar{H}_{g,n}$ be the moduli space (or moduli stack) of $n$-pointed stable hyperelliptic curves. I am interested in understanding its rational Picard group.
In the case $n=0$, the rational ...
- 21
2
votes
0
answers
159
views
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 ...
- 21
2
votes
0
answers
292
views
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 ...
user82886
1
vote
1
answer
254
views
Examples when algebraic 1-stack = derived enhancement?
Are there any examples where a usual algebraic 1-stack $X$ and the corresponding derived stack enhancement $\mathbb{R}X$ coincide?
Let me take an example from notes of Bertrand Toen, page 41 of https:/...
- 163
1
vote
0
answers
60
views
Distribution of the marked points on the components of a stable n-pointed curve of genus zero
Let $\overline{M}_{0,n}$ be the fine moduli space of stable n-pointed curves of genus $g=0$. Let $[(D_{0},p_1,...,p_n)] \in \overline{M}_{0,n}$. Suppose that each component of $D_0$ contains at least ...
- 560