All Questions
15 questions
8
votes
0
answers
205
views
What are the Newton groupoids from Drinfeld's paper on the Grinberg-Kazhdan theorem?
The paper the Grinberg-Kazhdan formal arc theorem and the Newton groupoids by Drinfeld seems to contain many interesting things which are beyond me. For now, I am trying to get some intuition for the ...
- 10.5k
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
1
answer
1k
views
Diagonal is representable then any morphism is representable
Ariyan Javanpeykar said here in comments that,
If the diagonal is representable, then isn't any morphism $S\rightarrow \mathcal{X}$ with $S$ a scheme representable?
I could not find the statement (...
- 6,023
2
votes
0
answers
117
views
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 ...
- 3,472
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
1
answer
307
views
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 (...
4
votes
1
answer
301
views
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., $M = Spec k$.
...
- 41
3
votes
0
answers
180
views
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 $I\to \mathcal M$ ...
- 31
1
vote
1
answer
217
views
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 ...
- 11
3
votes
0
answers
303
views
Finiteness of the connected components of a stack
Let $X$ be an algebaic stack over a scheme $S$, for any $S$-scheme $Y$ we can consider the groupoid $X(Y)$ of $Y$-points. Denote by $\pi_0(X(Y))$ the set of isomorphism classes of the groupoid.
Are ...
- 845
3
votes
1
answer
544
views
What if the base change of an algebraic space is representable
Let $k\subset L$ be an extension of fields of characteristic zero.
Suppose that $X/k$ is an algebraic space such that $X\otimes_k L$ is representable by a finite type $L$-scheme.
I am sure there are ...
- 33
3
votes
0
answers
410
views
Quasi-finite morphisms of stacks
Let $f:X\to Y$ be a morphism of ``nice" stacks over $\mathbf C$ such that the induced morphism on coarse moduli spaces is quasi-finite. Is $f$ quasi-finite?
By a "nice" stack I mean a smooth finite ...
- 31
6
votes
0
answers
400
views
Differential Geometry of (Non-Abelian) Gerbes in the language of Brylinski
Context
In an effort to have a definition for connection on a non-abelian gerbe, in the style of Brylinski, I am reading Breen and Messing's Differential Geometry of Gerbes [BM]. It seems that there ...
- 1,466
10
votes
1
answer
1k
views
How does descent theory imply a sheaf is a scheme?
I've noticed that often authors will comment that "descent theory" shows that some sheaf in the étale topology is actually a scheme. I was wondering what result in descent theory actually implies ...
- 2,108
31
votes
7
answers
4k
views
Categorical construction of the category of schemes?
The answer to the following question is probably well known or the question itself is well known not to have a reasonable answer. In the latter case could you please let me know what the "right" ...
- 23.5k