Questions tagged [branched-covers]

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11 votes
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What relationship is there between repeated roots of discriminants and orders of roots of the original polynomials?

Disclaimer: I asked this problem several days ago on MSE, I'm cross-posting it here. The title sounds like a high school problem, but (as a grad student not in algebra) it feels subtle/deep. ...
Harambe's user avatar
  • 225
16 votes
3 answers
889 views

Maximal degree of a map between orientable surfaces

Suppose that $M$ and $N$ are closed connected oriented surfaces. It is well-known that if $f \colon M \to N$ has degree $d > 0$, then $\chi(M) \le d \cdot \chi(N)$. What is an elementary proof of ...
Andrey Ryabichev's user avatar
1 vote
0 answers
99 views

Relation of branched covers and groups

I am self-studying covering spaces of topological spaces. The following question comes to my mind. In the case of topological covering spaces, we have a nice relation between the fundamental group of ...
KAK's user avatar
  • 321
3 votes
0 answers
83 views

(When) can you embed a closed map with finite discrete fibers into a (branched) cover?

Assume all spaces are topological manifolds. A branched cover is a continuous open map with discrete fibers. A finite branched cover is one with finite fibers. Questions. Given closed map $X\to S$ ...
Arrow's user avatar
  • 10.3k
4 votes
2 answers
305 views

Books for learning branched coverings

I am self-studying branched coverings. I read it from B. Maskit's Kleinian groups book. I want some more references for reading branched covers. In particular, I want to understand how to create ...
KAK's user avatar
  • 321
0 votes
0 answers
418 views

Irreducibility of plane algebraic curves

Given a plane algebraic curve $$ y^n + a_1(x)y^{n-1} + \dots +a_{n-1}(x) + a_n(x)y = 0, $$ with a branch point $P_0=(0, y_0)$ of order $n$. Can we prove that this curve is irreducible? What if the ...
minxin jia's user avatar
0 votes
1 answer
804 views

What is definition of branched covering?

What is definition of branched covering in the page 10 of following paper ? In Hatcher, Allen; Lochak, Pierre; Schneps, Leila, On the Teichmüller tower of mapping class groups, J. Reine Angew. Math. ...
Usa's user avatar
  • 119
11 votes
0 answers
513 views

A curious observation on the elliptic curve $y^2=x^3+1$

Here is a calculation regarding the $2$-torsion points of the elliptic curve $y^2=x^3+1$ which looks really miraculous to me (the motivation comes at the end). Take a point of $y^2=x^3+1$ and ...
KhashF's user avatar
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6 votes
3 answers
1k views

Graphs from the point of view of Riemann surfaces

I was listening to the lecture "Graphs from the point of view of Riemann surfaces" by Prof. Alexander Mednykh. I am looking for references for the basics of this topic. Any kind of ...
Lokenath Kundu's user avatar
6 votes
1 answer
848 views

Is there a, in depth, classification of branch points in complex analysis?

Disclaimer: This question was originally posted in math.stackexchange.com and, after 30 days with no answers, I followed the instructions of this topic. In complex analysis we have well known results ...
Diego Santos's user avatar
10 votes
1 answer
939 views

Visualizing genus-two Riemann surfaces: from the three-fold branched cover to the sphere with two handles

I am trying to visualize the genus-two Riemann surface given by the curve $$ y^3 = \frac{(x-x_1)(x-x_2)}{(x-x_3)(x-x_4)}. $$ We can regard this surface as a three-fold cover of the sphere with four ...
Holomaniac's user avatar
2 votes
1 answer
206 views

Recovering a family of rational functions from branch points

Let $Y$ be a compact Riemann surface and $B$ a finite subset of $Y$. It is a standard fact that isomorphism classes of holomorphic ramified covers $f:X\rightarrow Y$ of degree $d$ with branch points ...
KhashF's user avatar
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5 votes
1 answer
335 views

Thurston universe gates in knots: which invariant is it?

Today I discovered this nice video of a lecture by Thurston: https://youtu.be/daplYX6Oshc in which he explains how a knot can be turned into a "fabric for universes". For example, the unknot ...
Andrea Marino's user avatar
1 vote
0 answers
138 views

Monodromy Representation on $H_1$ of Elliptic Curve

I'm reading this post by Charles Siegel on Monodromy Representations and there is a construction in example a not unterstand. We look at the family $y^2z=x(x-z)(x-\lambda z)$ of projective elliptic ...
user267839's user avatar
  • 6,000
3 votes
0 answers
146 views

How to enumerate branched covers of $\mathbb{P}^1$ branched over $0,1$ and $\infty$?

Setup: Let $u:\Sigma \to X$ be a holomorphic map of closed Riemann surfaces with branch points $P \subset X$. For each branch point $p \in P$, we have a partition $\Gamma_p$ of $\text{deg}(u)$ given ...
Julian Chaidez's user avatar
1 vote
1 answer
569 views

Unibranch points (definition for varieties over arbitrary field)

In David Mumford's book Algebraic Geometry I, Complex Projective Varieties treating mainly complex varieties as objects of interest on page 43 he defines what is a topologically unibranch variety $X$ ...
user267839's user avatar
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3 votes
0 answers
260 views

Galois theory of ramified coverings vs classical Galois theory

That's an exact copy of my former MSE question I asked a couple of weeks ago and unfortunately not got the answer I was looking for. The question adresses reuns' answer in this thread: Algebraic ...
user267839's user avatar
  • 6,000
4 votes
0 answers
401 views

Is there a Galois theory for deformations of curves?

I have some general questions about the deformations of Galois covers of curves. Suppose we are given a $G$-Galois cover $k[[z]]/k[[x]]$, where $k$ is algebraically closed of characteristic $p>0$. ...
Huy Dang's user avatar
  • 245
4 votes
1 answer
216 views

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 ...
Vinicius Ambrosi's user avatar
4 votes
1 answer
270 views

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 ...
gmp's user avatar
  • 65
4 votes
1 answer
697 views

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 $B \subset Y$. ...
Francesco Polizzi's user avatar
3 votes
1 answer
540 views

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 ...
ralleee's user avatar
  • 131
17 votes
3 answers
2k views

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 ...
pi_1's user avatar
  • 1,433
10 votes
0 answers
241 views

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 ...
cduston's user avatar
  • 145
2 votes
1 answer
391 views

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 ...
enoch's user avatar
  • 23
5 votes
1 answer
1k views

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 $\mathcal O_{\mathbb P^1}(2)$, $\phi=(X_0-X_1)^3X_0\in H^0(\mathbb P^1, \mathcal O_{\mathbb P^1}(4))$ ...
pi_1's user avatar
  • 1,433
0 votes
1 answer
157 views

Uniformizing variable for branched covering of the Riemann sphere

Suppose I have a function $Q(z)$ of a complex variable $z\in\mathbb P^1$, possessing square root type branch points at the positions $\left\{z_i\right\}_{i=1}^{2M}$. I know that the Riemann surface $\...
Stefano's user avatar
  • 105
6 votes
1 answer
1k views

Smoothness of the branch divisor and ramification on surfaces

Let $f \colon X \longrightarrow Y$ be a finite morphism of degree $n \geq 2$ between smooth compact, complex surfaces. Let $B \subset Y$ be the branch divisor of $f$ and assume that the ...
Francesco Polizzi's user avatar
15 votes
1 answer
750 views

$S^3$ as cyclic branched cover of itself

In Chapter One of his notes (March 2002) Thurston says: If $K$ is the trivial knot the cyclic branched covers are $S^3$. It seems intuitively obvious (but it is not known) that this is the only ...
Antonio Alfieri's user avatar
9 votes
0 answers
331 views

Two transfers for ramified or branched covers

Let $\pi: X \rightarrow Y$ be a 2-fold branched cover of complex varieties. I know of (at least) two types of pushforwards associated to this situation: If I'm not mistaken, there is a pushforward ...
Dylan Wilson's user avatar
  • 13.2k
4 votes
1 answer
250 views

Flatness of Weil restriction

Let $X\rightarrow Y$ a ramified double cover of smooth projective curves, and let $$\mathcal G:=Res_{X/Y}(SL_n)$$ be the Weil restriction of the constant group scheme $SL_n$ over $X$. Question: Is ...
Z.A.Z.Z's user avatar
  • 1,871
6 votes
1 answer
404 views

Can you functorially "reconstruct" a branched cover of curves from its etale locus?

I'm sure this must be covered somewhere, but all the references I have only treat this in very special cases (mostly when working over fields). Suppose $f : X\rightarrow S$ is smooth of finite ...
Will Chen's user avatar
  • 10k
8 votes
4 answers
822 views

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 ...
arivero's user avatar
  • 437
5 votes
1 answer
1k 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 ...
Li Yutong's user avatar
  • 3,362
2 votes
1 answer
492 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, (L_k/...
jung y's user avatar
  • 21
5 votes
3 answers
1k 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: ...
user44651's user avatar
  • 1,049
9 votes
1 answer
1k 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 ...
Edgar JH's user avatar
4 votes
1 answer
610 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 ...
Serge Lvovski's user avatar
5 votes
1 answer
1k 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 ...
Ryan Budney's user avatar
  • 43.1k
2 votes
0 answers
378 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 ...
IMeasy's user avatar
  • 3,717
5 votes
2 answers
1k 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 ...
IMeasy's user avatar
  • 3,717
11 votes
2 answers
1k 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 ...
Taylor McNeill's user avatar
2 votes
2 answers
219 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 ...
Darius Math's user avatar
  • 2,181
7 votes
2 answers
1k 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 $\...
Darius Math's user avatar
  • 2,181
9 votes
3 answers
3k 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 ...
Darius Math's user avatar
  • 2,181
4 votes
2 answers
182 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 $H^{0}(X,\omega^{\otimes2})$ (...
Darius Math's user avatar
  • 2,181
3 votes
1 answer
323 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 ...
Tye Lidman's user avatar
3 votes
1 answer
402 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?
IMeasy's user avatar
  • 3,717
2 votes
1 answer
683 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 ...
Tridib's user avatar
  • 23
8 votes
1 answer
762 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 O_{Y,...
Jesko Hüttenhain's user avatar