Questions tagged [branched-covers]
The branched-covers tag has no usage guidance.
69
questions
2
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2
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Confusion about two statements about cohomology of curves with automorphisms
Let $\pi :C\rightarrow \mathbb{P}^{1}$ be a cyclic cover of degree $m$ of $\mathbb{P}^{1}$. So $C$ has an action of $\mathbb{Z}/m\mathbb{Z}$. Let $\xi$ be a primitive $m$-th root of unity. Consider ...
- 2,221
8
votes
2
answers
1k
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How to explicitly see the ramification over infinity
Take the equation $y^{d}=\Pi_{1}^{n}(x-t_{i})^{m_{i}}$ over $\mathbb{C}$. This affine equation gives a cyclic cover of $\mathbb{P}^{1}$. Now it is usually said without explanation that if the sum $\...
- 2,221
9
votes
3
answers
3k
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What prevents a cover to be Galois?
Let $f:X\rightarrow Y$ be a ramified cover of Riemann surfaces or algebraic curves over $\mathbb{C}$. My question is can one in terms of the ramification data of $f$, determine whether the cover is ...
- 2,221
4
votes
2
answers
184
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Dimension of the space of invariant quadratic differentials in Galois covers
Let $f: X \rightarrow Y $ be a Galois cover of with $X$ and $Y$ algebraic curves over $\mathbb{C}$. I want to compute the dimension of the subspace of $G$-invariants in $H^{0}(X,\omega^{\otimes2})$ (...
- 2,221
3
votes
1
answer
323
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Classification of fiber-preserving branched covers between Seifert fibered integer homology spheres
Is there an easy classification (and proof) of the possible branched covers between Seifert fibered integer homology spheres which are fiber-preserving and branched over fibers (or at least what the ...
- 215
3
votes
1
answer
415
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regularity of finite flat branched covers
Let $D$ and $S$ be two regular schemes and let $D$ be a divisor of $S$. Let $C \to S$ be a finite flat morphism, branched along $D$. Is $C$ regular as well?
- 3,737
2
votes
1
answer
689
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Trigonal curves of genus three: can their Galois closure be non-abelian
Let $X$ be a curve of genus three which is not hyperelliptic. Then $X$ is trigonal, i.e., there exists a finite morphism $X \to \mathbf P^1$ of degree $3$.
Let $Y\to X \to \mathbf P^1$ be a Galois ...
- 23
8
votes
1
answer
768
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Question about local description of the branch locus
Let $\pi:Y\to X$ be a dominant, finite morphism of nonsingular varieties over an algebraically closed field $\Bbbk$. Assume furthermore that for all $Q\in Y$, with $P=\pi(Q)$, we have
$$\mathcal O_{Y,...
- 4,862
4
votes
3
answers
374
views
Automorphism of finite groups and Hurwitz spaces
If $G$ is a finite group, embedded as a transitive subgroup of $S_n$ for some $n$, will every automorphism of $G$ extend to an inner automorphism of $S_n$?
I'm trying to connect the language that's ...
- 10k
4
votes
0
answers
394
views
Do regular noetherian schemes of dimension one only have finitely many etale covers of bounded degree
Let $X$ be a regular noetherian scheme of dimension one. Let $d$ be an integer.
Question. Are there only finitely many finite etale morphisms $Y\to X$ of degree $d$?
I want to exclude finite etale ...
- 381
3
votes
0
answers
462
views
Galois group decomposition of non-cyclic covers
If $\pi: C \rightarrow \mathbb{P}^{1}$ is a cyclic cover of $\mathbb{P}^{1}$ with Galois group $\mathbb{Z}/m \mathbb{Z}$ and thus with the (affine) formula
$y^{m}= (x_{1}-a_{1})^{t_{1}}....(x_{n}-a_{...
- 637
2
votes
3
answers
417
views
Equations for abelian coverings of $\mathbb{P^{1}}$
Cyclic coverings of $\mathbb{P^{1}}$ have a simple (affine) equation, namely the formula,
$y^{m}= (x_{1}-a_{1})^{t_{1}}....(x_{n}-a_{n})^{t_{n}}$. Is there such a nice equation for abelian non-cyclic ...
- 637
4
votes
0
answers
402
views
Every curve is a Hurwitz space in infinitely many ways
Diaz, Donagi and Harbater proved that every curve over $\overline{\mathbf{Q}}$ is a Hurwitz space.
A Hurwitz space is a connected component of the curve $H_n$. The curve $H_n$ is (the ...
- 163
1
vote
1
answer
838
views
Is this function field extension a Galois extension ?
Setting and question
Let $X$ be a variety over an algebraically closed field of null characteristic, and let $C$ be a (regular if you want) curve included in $X$.
Consider $X'$ the normalization of $...
- 1,044
4
votes
2
answers
342
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 $\overline{\mathbf{Q}}$. ...
- 163