# Questions tagged [branched-covers]

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48 questions
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### Pre-images of Seifert surfaces are incompressible?

Consider a knot $K \subset S^3$ and let $M_K$ be the associated double branched cover. The pre-image $S$ of a Seifert surface is a surface without boundary inside $M_K$. Can $S$ be incompressible? If ...
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### moduli stack of double covers of $\mathbb{P}^1$ with one marked point

I am trying to improve my moderate knowledge of moduli spaces/stacks by examining the moduli stack of stable double covers of $\mathbb{P}^1$ with one marked point. My idea is to ignore the stack ...
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### What is the preimage of a braid in a covering space branched over the braid?

For a knot $K\subset \mathbb{S}^3$, one can construct the covering space branched over that knot by assigning elements of the symmetric group $S_n$ to each arc of the knot. You can find the knot group ...
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### Monodromy representation of elementary simple covers

Let $X$, $Y$ be smooth, connected, compact manifolds (for instance, projective varieties) and $f \colon X \longrightarrow Y$ be a finite, branched cover of degree $n$, with branch locus $B \subset Y$. ...
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### Constructing ramified covers with prescribed multiplicities at ramification points

Let $Y$ be a smooth projective curve defined over number field $K$. Let $P_1,\dots ,P_m$ be some $K$-points to which we will associate "multiplicities" $m_i\in\{ 2,3,\dots \}$ for $i=1,\dots ,m$. The ...
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### Cohomology of ramified double cover of $\mathbb P^n$ (reference)

Let $X\rightarrow \mathbb P^n_{\mathbb C}$ be a double cover ramified over a smooth hypersurface $B$ of degre $2d$. In the case of hypersurfaces of $\mathbb P^n$ one can determine the integral ...
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### Is the number of ramified coverings of given degree of a curve with prescribed branch divisor finite?

Let Y be a smooth projective curve over C and prescribe a branch divisor B on Y. I want to know if the number of coverings of Y of fixed degree and branched along B is finite. If so, why? Or, where is ...
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### Two ways to look at a double cover of the projective line

Let $f:L\rightarrow \mathbb P^1_{\mathbb C}$ be the line bundle associated to the invertible sheaf $\mathcal O_{\mathbb P^1}(2)$, $\phi=(X_0-X_1)^3X_0\in H^0(\mathbb P^1, \mathcal O_{\mathbb P^1}(4))$ ...
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### Heegaard Floer Homology of double branched cover

The question is the following: Let $K\subset S^{3}$ be a knot, consider the double branched cover $Y$ of $S^{3}$ over $K$. We know $Y$ has a unique spin structure $\mathfrak{s}$, now the question is: ...
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### 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 ...
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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 ...
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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 $H^{0}(X,\omega^{\otimes2})$ (...
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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 ...
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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?
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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,...
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### 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 ...
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### 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 $\overline{\mathbf{Q}}$. ...
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### Fibre cardinality of an unramified morphism

Let $\varphi: X \to Y$ be a finite, dominant, unramified morphism of varieties over an algebraically closed field. If necessary, we can assume $X$ and $Y$ to be nonsingular. I am trying to prove that ...
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### degenerating surface II

In degenerating surface, Robert Bryant give us an example of a sequence of minimal immersions which converges (in $C^2$- topology) to $z\mapsto z^{2k+1}$ on the unit disc $\mathbb{D}$. My question is ...
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### Manin-Drinfeld and constructing a finite morphism with two given ramification points

Fix a smooth projective connected curve $X$ over $\overline{\mathbf{Q}}$ of genus $g\geq 1$ and distinct points $x,y \in X$ such that $x-y$ has infinite order in the Jacobian. Can we always find a ...
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Hi, i have a sequence of immersed disc $u_n: \mathbb{D} \rightarrow \mathbb{R}^3$ which converge to a singular cover of the disc: $z^k$ for $k\geq 2$, moreprecisely $u_n \rightarrow z^k$ in $C^2(\... 1answer 239 views ### Comparing heights of rational points on curves through covers Let$a$be a closed point in$\mathbf{P}^1_{\overline{\mathbf{Q}}}$. Let$Y \cong \mathbf{P}^1_{\overline{\mathbf{Q}}} $and let$\pi:Y\to \mathbf{P}^1_{\overline{\mathbf{Q}}}$be a finite morphism ... 2answers 2k views ### Finite, Étale Morphism Of Varieties I have a, probably very simple, question: My intuition tells me that the following statement should be true, but I couldn't find it anywhere and I wanted to make sure I am not missing something. Let$...
Let $\pi:Y\to X$ be a Galois cover, i.e. a finite morphism of nonsingular varieties over an algebraically closed field $\Bbbk$ such that $K(X)\hookrightarrow K(Y)$ is Galois. Let $H\subset X$ be the ...