All Questions
Tagged with fundamental-group ag.algebraic-geometry
115 questions
6
votes
0
answers
428
views
generating the etale fundamental group by sections?
Let $X$ be a proper smooth scheme over a field $k$ of characteristic zero (well you can naturally weaken the assumption to normal integral scheme over some "nice" base like $\mathbb{Z}$, $\mathbb{F}_q$...
- 1,305
12
votes
1
answer
2k
views
What is the wild fundamental group?
In the abstract of
Singularités irrégulières Correspondance et documents
Pierre Deligne, Bernard Malgrange, Jean-Pierre Ramis
Documents mathématiques 5 (2007), xii+188 pages (link)
there is a ...
- 35.5k
11
votes
2
answers
1k
views
Finite vector bundles over punctured affine spaces
Let $X$ be a connected scheme. Recall that a vector bundle $V$ on $X$ is called finite if there are two different polynomials $f,g \in \mathbb N[T]$ such that $f(V) = g(V)$ inside the semiring of ...
- 30.6k
28
votes
2
answers
2k
views
Does the etale fundamental group of the projective line minus a finite number of points over a finite field depend on the points?
Clearly the etale fundamental group of $\mathbb{P}^1_{\mathbb{C}} \setminus \{a_1,...,a_r\}$ doesn't depend on the $a_i$'s, because it is the profinite completion of the topological fundamental group. ...
- 5,929
2
votes
2
answers
503
views
Lie Algebras and Simple Connectivity for general algebraic groups
In the representation theory of Lie groups (say, over $\mathbb{R}$ or $\mathbb{C}$), one can show that a Lie algebra homomorphism between the Lie algebras of two algebraic groups $G$ and $H$ always ...
- 15.4k
1
vote
0
answers
194
views
Inverting infinitely many points on an algebraic curve
This question is very naive, but that's why I'm asking it.
Say we begin with $\mathbb{A}^1_{\mathbb{C}}$. Let $U$ be the open disc around $0$ of radius $1$. Now invert all the $a$'s not in $U$: $Spec(...
- 6,348
23
votes
5
answers
7k
views
Grothendieck's Galois Theory today
I have recently become aware of, and started to study in my free time (abundant in these summer months) Grothendieck's Galois Theory (GGT), as formulated in SGA 1 and later by Grothendieck's ...
- 1,539
7
votes
1
answer
1k
views
étale fundamental group of projective space
What is the étale fundamental group of projective space over an algebraically closed field?
In char = 0 it is trivial (Lefschetz principle), as well as in dimension 1 (Riemann-Hurwitz).
- 227
20
votes
2
answers
2k
views
Unipotency in realisations of the motivic fundamental group
Deligne, in his 1987 paper on the fundamental group of $\mathbb{P}^1 \setminus \{0,1,\infty\}$ (in "Galois Groups over $\mathbb{Q}$"), defines a system of realisations for a motivic fundamental group. ...
- 2,976
9
votes
2
answers
2k
views
Functoriality of fundamental group via deck transformations
Problem
I'm trying to understand this with a view towards the etale fundamental group where we can't talk about loops. What I'm missing is how the fundamental group functor should work on morphisms, ...
- 5,929
36
votes
3
answers
3k
views
Tannaka formalism and the étale fundamental group
For quite a while, I have been wondering if there is a general principle/theory that has
both Tannaka fundamental groups and étale fundamental groups as a special case.
To elaborate: The theory of ...
- 4,450
8
votes
2
answers
981
views
Covers of the projective line over Z and arithmetic Grauert-Remmert
This question is the two-dimensional analogue of Etale coverings of certain open subschemes in Spec O_K
There I asked if one could characterize in a way certain covers of $\textrm{Spec} \ O_K$. As ...
- 9,497
32
votes
3
answers
4k
views
Fundamental groups of topoi
Just yesterday I heard of the notion of a fundamental group of a topos, so I looked it up on the nLab, where the following nice definition is given:
If $T$ is a Grothendieck topos arising as category ...
- 4,450
14
votes
4
answers
1k
views
Etale coverings of certain open subschemes in Spec O_K
Let $U$ be an open subscheme of $\textrm{Spec} \ \mathbf{Z}$. The complement of $U$ is a divisor $D$ of $\textrm{Spec} \ \mathbf{Z}$.
Q. Can we classify the etale coverings of $U$ of a given degree? ...
- 9,497
7
votes
4
answers
736
views
Simply connected quasi-projective varieties in positive characteristic
I am looking for examples of non-projective (quasi-projective) varieties $X$ defined over a field of positive characteristic, which have trivial étale fundamental group.
It is well known that the ...
- 4,450