The branched-covers tag has no usage guidance.

**7**

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### Conditions for underlying space of an orbifold $\Bbb T^n/\Gamma$ to be a sphere?

Given a $n$-dimensional torus, is it always possible to find a discrete action to produce an orbifold such that its underlying space is the $n$-dimensional sphere? Or does it only happens for specific ...

**3**

votes

**1**answer

155 views

### Definition and sigularity of Ramified covers

Let $X$ be a normal variety over $\mathbb{C}$.
In their book Birational geometry of algebraic varieties, Kollár and Mori define [Definition 2.50 and 2.51] a ramified m-th cyclic cover associate to a ...

**2**

votes

**1**answer

281 views

### Kummer Coverings

Let $L_1$, $\cdots$, $L_k$ be homogenous linear forms in three variables $z_0$, $z_1$, $z_2$ defining $k$ lines in $\mathbb{P}_{2}$. Consider the abelian extension $K
((L_2/L_1)^{1/n}, \cdots, ...

**3**

votes

**3**answers

382 views

### 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: ...

**7**

votes

**1**answer

279 views

### Finite morphisms to projective space

Let $X$ be a projective variety of dimension n. Then there exists a finite surjective morphism $X \to \mathbf P^n$. Let $d$ be the minimal degree of such a finite surjective morphism.
Let $d^\prime ...

**4**

votes

**1**answer

233 views

### To what extent does the branch locus determine the covering (Chisini's conjecture)?

Suppose that $X$ is a smooth projective surface over $\mathbb C$ and $f\colon X\to\mathbb P^2$ is a finite morphism branched over a curve $S\subset\mathbb P^2$. Assume in addition that all the ...

**5**

votes

**1**answer

293 views

### The cyclic branched covers of “simple” knots in $S^3$

Is there a convenient place in the literature where the geometric decompositions of cyclic branched covers of $S^3$ branched over "small" knots is recorded?
By small knots, I'm referring to things ...

**2**

votes

**0**answers

163 views

### branch locus of the discriminant map $\overline{\mathcal{H}}_{g',r} \to \overline{\mathcal{M}}_{g,n}$

Let $\overline{\mathcal{M}}_{g,n}$ be the moduli space of pointed, stable, genus $g$ curves. Let $\overline{\mathcal{H}}_{g',r}$ be the hurwitz space of cyclic covers of degree $r$ of genus $g$ curves ...

**3**

votes

**2**answers

269 views

### chern classes of push-pulled vector bundles

Let $f:X\to Y$ be a finite cover of smooth algebraic varieties, branched along a divisor $R\subset Y$. Let $E$ be a vector bundle on $Y$. What is the relation between the chern classes of $E$ and the ...

**7**

votes

**2**answers

380 views

### Orbifolds vs. branched covers

Forgive me if this is a basic question. I'm just learning about orbifolds, and covering spaces are my happy place for thinking about group actions.
If $M$ is a manifold and $G$ is a group acting ...

**2**

votes

**2**answers

204 views

### 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 ...

**3**

votes

**2**answers

417 views

### 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 ...

**8**

votes

**3**answers

760 views

### 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 ...

**4**

votes

**2**answers

140 views

### 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 $H^{0}(X,\omega^{\otimes2})$ ...

**3**

votes

**1**answer

205 views

### 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 ...

**2**

votes

**1**answer

198 views

### 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?

**1**

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**1**answer

218 views

### 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 ...

**8**

votes

**1**answer

572 views

### 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 ...

**4**

votes

**3**answers

281 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 ...

**4**

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**0**answers

196 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 ...

**2**

votes

**0**answers

295 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}= ...

**2**

votes

**3**answers

311 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 ...

**3**

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**0**answers

265 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 ...

**1**

vote

**1**answer

521 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 ...

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votes

**2**answers

232 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 $\overline{\mathbf{Q}}$. ...

**3**

votes

**2**answers

598 views

### 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
...

**5**

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**0**answers

277 views

### 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 ...

**1**

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**1**answer

252 views

### 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 ...

**5**

votes

**2**answers

667 views

### degenerating surface

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 ...

**2**

votes

**1**answer

224 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 ...

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**2**answers

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### 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 ...

**2**

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**0**answers

242 views

### cardinality of the fibre of a constantly branched, finite morphism over the branch locus

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 ...

**9**

votes

**4**answers

2k views

### Higher dimensional version of the Hurwitz formula?

In Hartshorne IV.2, notions related to ramification and branching are introduced, but only for curves. The main result is the Hurwitz formula.
Now if you have a finite surjective morphism between ...

**30**

votes

**4**answers

2k views

### Curves which are not covers of P^1 with four branch points

The following interesting question came up in a discussion I was having with Alex Wright.
Suppose given a branched cover C -> P^1 with four branch points. It's not hard to see that the field of ...