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8 votes
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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
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
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
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
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. ...
Saahmri's user avatar
  • 51
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
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
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
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
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
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
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
2 votes
0 answers
735 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
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
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
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
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
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
1 vote
0 answers
205 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