All Questions
Tagged with ag.algebraic-geometry fundamental-group
115 questions
3
votes
1
answer
600
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Grothendieck's section conjecture and base change: restricting sections
Let $X$ be a smooth projective geometrically connected curve over $\mathbf{Q}$ of genus at least two. Fix an algebraic closure $\overline{\mathbf{Q}}$ of $\mathbf{Q}$ and let $G_{\mathbf{Q}}$ be the ...
- 149k
4
votes
2
answers
349
views
Number of connected components of the Hurwitz space $H_n^o$ and subgroups of the fundamental group
A cover (of $\mathbf{P}^1_{\overline{\mathbf{Q}}}$) is a finite morphism $X\to \mathbf{P}^1_{\overline{\mathbf{Q}}}$, where $X$ is a smooth projective connected curve over $\overline{\mathbf{Q}}$. ...
- 96.4k
4
votes
1
answer
604
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A question about the Tannakian etale fundamental group of a curve
Let $X$ be a smooth connected quasi-projective curve over $\mathbf{Q}$. Let $U$ be the pro-unipotent etale fundamental group of $X$ over $\mathbf{Q}_p$.
$U^1 = U$ and let $U^n =[U,U^{n-1}]$.
Let $n\...
- 1,838
14
votes
3
answers
1k
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Can we define homotopy groups using Tannakian categories
This is another vague question. Hope you guys don't mind.
Let $T$ be a Tannakian category. For any fibre functor $F$ on $T$ we define the fundamental group of $T$ at $F$, denoted by $\pi_1(T,F)$, to ...
- 8,972
11
votes
2
answers
1k
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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 ...
CommunityBot
- 1
6
votes
1
answer
393
views
finite quotients of fundamental groups in positive characteristic
For affine smooth curves over $k=\bar{k}$ of char. $p,$ Abhyankar's conjecture (proved by Raynaud and Harbater) tells us exactly which finite groups can be realized as quotients of their fundamental ...
CommunityBot
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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
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 ...
CommunityBot
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12
votes
1
answer
2k
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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 ...
- 22.6k
28
votes
2
answers
2k
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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. ...
- 4,450
2
votes
2
answers
503
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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 ...
- 3,246
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
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).
- 1,411
9
votes
2
answers
2k
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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, ...
CommunityBot
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7
votes
4
answers
736
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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 ...
CommunityBot
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