Skip to main content

All Questions

Tagged with
Filter by
Sorted by
Tagged with
0 votes
0 answers
129 views

Two elliptic curves with the same j-invariants

This is an interesting observation of mine when exploring moduli of elliptic curves. Let's fix a base field $k$ and assume that we have two elliptic curves $E$, $E'$ which are isomorphic over $\bar k$...
fe mu's user avatar
  • 1
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 ...
Matthias's user avatar
  • 223
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 ...
Manoel's user avatar
  • 560
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 ...
Coherent Sheaf's user avatar
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 ...
Kandinskij's user avatar
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}: \...
Matthias's user avatar
  • 223
1 vote
0 answers
180 views

Moduli stack of l-adic sheaves?

Let us work over a field $k$. Then for any smooth affine group scheme $G$ over $k$, we can consider the stack quotient $BG := [\text{pt} / G]$ which classifies étale $G$-torsors. Let $\ell$ be a prime ...
user577413's user avatar
1 vote
0 answers
138 views

What is bad when stabilizers are non-reductive in moduli stacks?

Here is J. Alper's definition of good moduli spaces. Consider in characteristic zero. Then we see that the classifying stack of any non-reductive group $H$ does not have a good moduli space. In ...
Yikun Qiao's user avatar
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:/...
Robert Hanson's user avatar
1 vote
0 answers
156 views

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 ...
Matthias's user avatar
  • 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 ...
Pulcinella's user avatar
  • 5,701
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 ...
Pulcinella's user avatar
  • 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 $S_i$, ...
red_trumpet's user avatar
  • 1,286
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 ...
Tommaso Scognamiglio's user avatar
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 ...
Lennart Meier's user avatar
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{...
curious math guy's user avatar
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 ...
Chan Ki Fung's user avatar
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 ...
Praphulla Koushik's user avatar
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 ...
Bananeen's user avatar
  • 1,190
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 ...
user avatar
2 votes
0 answers
733 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 ...
user124771's user avatar
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 ...
GeraldV's user avatar
  • 21
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 ...
George's user avatar
  • 113
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 ...
Wizzig's user avatar
  • 61
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 ...
Marion's user avatar
  • 587
1 vote
0 answers
136 views

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 ...
Canu's user avatar
  • 11
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 ...
Adnon's user avatar
  • 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 ...
user avatar
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 ...
John Pardon's user avatar
  • 18.7k
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 ...
Will Sawin's user avatar
  • 148k
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 ...
Pancho's user avatar
  • 171
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 ...
user2345897's user avatar
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 ...
El Nino's user avatar
  • 93
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-...
John Gowers's user avatar
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....
user avatar
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 ...
user avatar
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 ...
Will Chen's user avatar
  • 10.7k
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 ...
Rafen J.'s user avatar
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 ...
user avatar
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 ...
Puzzled's user avatar
  • 8,998
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^*:\...
IMeasy's user avatar
  • 3,779
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{...
IMeasy's user avatar
  • 3,779
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 ...
user avatar
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'...
Jonathan 's user avatar
0 votes
1 answer
651 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 ...
IMeasy's user avatar
  • 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 $h_Y$ ...
user32511's user avatar
  • 128
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 $[...
IMeasy's user avatar
  • 3,779
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}\...
Will Chen's user avatar
  • 10.7k
1 vote
0 answers
204 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 $\...
Jack's user avatar
  • 637
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 ...
David Roberts's user avatar
  • 35.4k