All Questions
Tagged with fundamental-group algebraic-curves
8 questions
2
votes
1
answer
270
views
Motivation of Zariski–Van Kampen theorem
The Zariski–Van Kampen theorem gives the presentation of the fundamental group of the complement of the plane curve of degree $d$. But what's the motivation of this theorem? More generally, why are ...
- 197
1
vote
0
answers
68
views
Glueing local systems over union of compact Riemann surfaces
Let $X,Y$ be two connected, non-singular compact Riemann surfaces such that $X$ intersects $Y$ transversely at two distinct points. Let $L$ be a $\mathbb{C}$-local system on $X$. Let $L'$ be the ...
- 1,593
5
votes
1
answer
585
views
Reconstruction of hyperbolic curves using the fundamental group
In the paper Curves and their Fundamental Groups written by Gerd Faltings, Mochizuki's proof of Grothendick's conjecture on anabelian curves is explained.
In the proof, he shows that for two ...
- 527
10
votes
0
answers
239
views
On tangent space to the fundamental group scheme
Let $X$ be a smooth, projective complex curve of genus at least $2$. If I understand correctly, after choosing a base point, one can associate to $X$, a fundamental group scheme $\pi$. I am trying to ...
- 2,126
3
votes
1
answer
294
views
Is the fundamental group of an open arithmetic Riemann surface contained in $\Gamma(2)$
Let $X$ be a non-compact Riemann surface with universal covering $\mathbb H$ and suppose that the fundamental group of $X$ is an arithmetic subgroup of $\mathrm{Aut}(\mathbb H) = \mathrm{PSL}_2(\...
- 31
3
votes
1
answer
600
views
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 ...
- 125
4
votes
1
answer
604
views
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\...
- 1,213
1
vote
1
answer
329
views
$P^1$ minus k points
For $k\geq 3$, and $k$ arbitrary points $S=( z_1,\cdots,z_k ) \in \mathbb{P}^1$, we can write
$$ P^1 \setminus S \cong \mathbb{H}/G $$
where $\mathbb{H}$ is the upper-half plane and $G\subset PSL(2,...
