12
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Understand the difference between two stacks
Often you hear the informal statement that a stack is like a scheme, except with a stabilizer group attached to each point. Your question shows why the intuition from this statement can be misleading. ...
- 40.2k
11
votes
Accepted
Fundamental group of $M_g^\circ$
Except in a few trivial cases, the locus of curves which have an extra automorphism will have codimension greater than one in $\mathcal M_g$. When that happens, the fundamental group of $\mathcal M_g$ ...
- 148k
11
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Fundamental group of $M_g^\circ$
Since $\mathcal{M}_g=[\mathcal{T}_g/\text{Mod}_g]$, and $\mathcal{T}_g$ Teichmüller space is contractible, (1) and (3) are going to be isomorphic.
This was mentioned previously on MathOverflow here, ...
- 2,372
11
votes
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A "comprehensive" family of abelian varieties
Welcome new contributor. The idea of such "comprehensive" families goes back very far. These were studied by Amitsur under the name "generic splitting varieties", primarily in ...
8
votes
Accepted
$\mathcal{M}_g$ and $\mathcal{A}_g$ have natural structures as quasi-projective varieties
The GIT proof gives very nice compactifications of these spaces (and is the "right" way to do this), but they were known to be quasiprojective varieties long before GIT was developed.
The classical ...
- 44.8k
8
votes
Understand the difference between two stacks
One can see the difference by writing down the functor of points explicitly. For a test scheme $T$,
\begin{align}
(\mathbb G_m\times BH)(T) & =\{(f,p)\mid f:T\to \mathbb G_m, p:T'\to T\text{ $H$-...
- 81
7
votes
Intermediate moduli spaces of stable maps
I am just adding some details to my comments above. Let $k$ be a field. Let $G$ be a simply connected, semisimple algebraic $k$-group; for simplicity, assume that $G$ is split. Let $X$ be a ...
7
votes
Accepted
Questions about root stacks
One way to construct the root stack is to consider the universal situation of $\Theta := [\mathbb{A}^1/\mathbb{G}_m]$. Here let us work over $\mathbb{Z}[\frac{1}{n}]$ since we assume $n$ is invertible....
- 3,290
7
votes
Accepted
Is the set of hyperelliptic curves with a K-point closed?
In the "more sophisticated" direction, we can ask a similar question about the moduli stack $\mathscr{M}_g$ of hyperelliptic curves of genus $g$. If $K$ is a topological field, there is a ...
- 19.6k
4
votes
Accepted
When is the coarse moduli space of genus $g$ stable curves smooth?
A more detailed description of the singular locus of $\mathrm{M}_g$ is as follows.
Theorem. Let $\mathrm{C}$ be a smooth curve of genus $g$.
If $g=2$, then $[\mathrm{C}]$ is a singular point of $\...
- 2,808
3
votes
Accepted
Tangent space to spaces of maps
I think this is not true, at least if $k\geq 6$. The Euler exact sequence pulled back to $\mathbb{P}^1$ is
$$0\rightarrow \mathscr{O}_{\mathbb{P}^1}\rightarrow \mathscr{O}_{\mathbb{P}^1}(d)^3\...
- 38k
3
votes
Accepted
"Generalized" clutching maps between moduli spaces of curves
You should read Arbarello-Cornalba more carefully! Let's consider the gluing map
$$ h:\overline M_{g,n+1} \times \overline M_{g',n'+1} \to \overline M_{g+g',n+n'}.$$
As you say we have $H^2(\overline ...
- 40.2k
3
votes
Accepted
Blowing-up projective spaces of parametrized rational curves
One modular interpretation of $X_1$ mimics the modular interpretation of the space of complete collineations. Let $V$ be a $k$-vector space of finite dimension.
Definition 1. For every $k$-scheme $...
3
votes
Accepted
$G$-invariant morphism and coarse moduli spaces
This is not true without the assumption that $G$ is reduced. Here is a counterexample.
Fix a prime number $p$ and any field $k$ of characteristic $p$. We define the nonreduced algebraic subgroup $\...
- 921
1
vote
The weight of a weighted filtration is given (for large $m$) by a polynomial
I'm aware this is an old question, but I'm answering it for the benefit of anyone who comes across this question in the future.
There are not one but two proofs of this result in the paper Uniform $K$-...
- 11
1
vote
Family over the coarse moduli space of curves
To close up loose ends and for everyone finding this questions: Such a family does not exist in general. An argument for elliptic curves can be found in Robin Hartshorne Deformation Theory in Remark ...
- 445
1
vote
Accepted
When is the moduli of generalized parabolic bundles with fixed determinant smooth?
It seems to me that the definition of generalized parabolic structure given in the question differs from the one given in the following paper (a GPB should fix a flag in the global sections of a twist ...
- 17.4k
1
vote
Accepted
Compactifications of spaces of morphisms
I started writing an answer about the generators and relations for the $\mathbb{Q}$-Picard group of the stack of stable maps of genus $0$ curves to an arbitrary projective homogeneous variety, but it ...
1
vote
Accepted
Linear systems on moduli spaces of stable maps
I am posting this as an answer because the comment thread is too long. A proper morphism of integral, Noetherian schemes, $f:X\to Y$, is a contraction if the sheaf homomorphism $f^\#:\mathcal{O}_Y\to ...
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