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23 votes
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$S^3$ as cyclic branched cover of itself

The statement that for arbitrary K in $S^3$, if for some $n \ge 2$, the n-fold cyclic branched cover is $S^3$ (or in some versions, a homotopy 3-sphere) then K is the unknot, was known as the Smith ...
Danny Ruberman's user avatar
17 votes

Maximal degree of a map between orientable surfaces

I like the Gromov norm approach. Another approach uses the Milnor-Wood inequality (here I am assuming $\chi(M), \chi(N) < 0$ and $M, N$ are connected WOLG). A discrete faithful representation of $\...
Ian Agol's user avatar
  • 67.8k
13 votes

Cohomology of ramified double cover of $\mathbb P^n$ (reference)

If $\pi \colon X \longrightarrow Y$ is a cyclic cover of complex projective manifolds of dimension $n$ branched along an ample divisor $B \subset Y$, then by a variation of Lefschetz Hyperplane ...
Francesco Polizzi's user avatar
13 votes
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Visualizing genus-two Riemann surfaces: from the three-fold branched cover to the sphere with two handles

The picture (produced by Nick Schmitt) of the Lawson surface of genus 2 might help: It shows the genus 2 Riemann surface given by the algebraic equation $$y^3=\frac{z^2-1}{z^2+1}.$$ The lines show ...
Sebastian's user avatar
  • 6,745
12 votes
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Maximal degree of a map between orientable surfaces

The best, elementary, self-contained, and clarifying proof of Kneser's result, including the desired inequality, is due to Richard Skora. See Skora, Richard, The degree of a map between surfaces, Math....
Allan Edmonds's user avatar
12 votes

Maximal degree of a map between orientable surfaces

Allan Edmonds shows that such a map decomposes as the composition of a pinch (i.e., pinching off handles) and a branched cover, which immediately gives what you need. Edmonds, Allan L. Deformation of ...
Mikhail Katz's user avatar
  • 15.8k
10 votes

Graphs from the point of view of Riemann surfaces

It seems that the question is about the following lecture notes: http://math.nsc.ru/conference/g2/g2r2/files/pdf/Lecture-8.pdf At the start of these notes, five papers are mentioned. Based on the ...
J. van Dobben de Bruyn's user avatar
9 votes

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

(1) There is the following indirect explanation: For a generic curve neither of these phenomena happen - the discriminant has no repeated roots and the branch points all have ramification index two. ...
Will Sawin's user avatar
  • 141k
9 votes
Accepted

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

$\def\P{{\mathbb{P}}} \def\A{{\mathbb{A}}} \newcommand{\O}{\mathcal{O}} \DeclareMathOperator{\Disc}{Disc}$I think the story goes like this. The multiplicity of a zero of the discriminant counts ...
Jack Huizenga's user avatar
8 votes

Graphs from the point of view of Riemann surfaces

I believe one of the first paper was: Bacher, Roland; de la Harpe, Pierre; Nagnibeda, Tatiana The lattice of integral flows and the lattice of integral cuts on a finite graph, Bull. Soc. Math. Fr. 125,...
Roland Bacher's user avatar
8 votes
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Is there a, in depth, classification of branch points in complex analysis?

Yes, there is a classification. An isolated branch point can be algebraic or logarithmic. If the branch point is at 0, algebraic means that $f(z^n)$ has a pole or removable singularity at 0. It can ...
Alexandre Eremenko's user avatar
7 votes
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moduli stack of double covers of $\mathbb{P}^1$ with one marked point

It's very important to define things carefully the problem of interest before constructing the moduli space / stack. In the most general setting you want to carefully define the moduli functor, but ...
Will Sawin's user avatar
  • 141k
7 votes

Smoothness of the branch divisor and ramification on surfaces

It seems to me that the intersection is zero in general, i.e. the answer is YES. Let us prove that $f^{-1}(B)$ is a smooth curve in $X$. This clearly implies the desired result. The proof of ...
aglearner's user avatar
  • 14.1k
6 votes
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Thurston universe gates in knots: which invariant is it?

Here is a higher-quality video of the same material. My answer is a more algebraic version of Thurston's presentation, but I will tie this back to Thurston's "intention" at the end. ...
Sam Nead's user avatar
  • 27k
6 votes
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Is the number of ramified coverings of given degree of a curve with prescribed branch divisor finite?

This can be understood by looking at fundamental groups. Covers of $Y$ branched only along $B$ of degree $n$ correspond to homomorphisms $\pi_1(Y-B)\rightarrow S_n$ ($S_n$ is the symmetric group on $...
Will Chen's user avatar
  • 10k
6 votes

Two ways to look at a double cover of the projective line

The smooth points on a cuspidal cubic curve in $\mathbb P^2$ can be given the structure of an algebraic group, just as is done for smooth cubic curves. If the tangent directions of the cuspidal point ...
Joe Silverman's user avatar
6 votes
Accepted

Monodromy representation of elementary simple covers

I am just writing my comments as an answer. For every subgroup $H$ of the symmetric group $\mathfrak{S}_n$, define a relation on $\{1,\dots,n\}$ by $a\sim b$ if either $a$ equals $b$ or if the ...
5 votes

Cohomology of ramified double cover of $\mathbb P^n$ (reference)

Expanding a bit Jason Starr's comment: let $R_X$ (or $R_B$) the jacobian ring of $X$ (resp. $B$). Recall that if $X=V(F) \subset w \mathbb{P}(a_0, \ldots, a_n)$, the jacobian ring is defined as $$ ...
Enrico's user avatar
  • 776
5 votes
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Recovering a family of rational functions from branch points

The reason for this phenomenon is that you are afflicted with a serious mathematical condition, that being: Your monodromy has monodromy. To be less cryptic, the key thing is that the fundamental ...
Will Sawin's user avatar
  • 141k
5 votes

Graphs from the point of view of Riemann surfaces

The book Graphs on Surfaces and Their Applications by Sergei K. Lando and Alexander K. Zvonkin From Amazon page: Graphs drawn on two-dimensional surfaces have always attracted researchers by their ...
Tom Copeland's user avatar
  • 10.2k
4 votes
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Books for learning branched coverings

Montesinos wrote several papers defining the meaning of branched coverings and proving basic properties(not just between manifolds, but for general topological spaces): Montesinos-Amilibia, José María,...
Moishe Kohan's user avatar
  • 10.3k
4 votes
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Pre-images of Seifert surfaces are incompressible?

The incompressible Seifert surface $\Sigma$ will have incompressible preimage $S$ in the double branched cover if and only if the complement of a tubular neighborhood of $\Sigma$ in $S^3$ has ...
Ian Agol's user avatar
  • 67.8k
3 votes
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Flatness of Weil restriction

As in the commentry of of @nfdc23, $\mathcal G$ is even smooth. Ref: Néron Models, Prop. 5 of section 7.6.
Z.A.Z.Z's user avatar
  • 1,871
3 votes

Constructing ramified covers with prescribed multiplicities at ramification points

Theorem A of the following article of Pete Clark and Jon Voight seems to prove the existence of $X\to Y$ when $Y$ is $\mathbb{P}^1$ with three special points with multiplicities $(a,b,c)$. Pete L. ...
2 votes

How to determine the LS category of branched covers?

There's no relation, because every PL manifold is a branched covering of a sphere; see J. W. Alexander, Note on Riemann spaces, 1920, Bull. Amer. Math. Soc. 26.
Denis T's user avatar
  • 4,531
2 votes

Branched covering maps between Riemann surfaces

For infinite degree, the definition of "branched covering" can be somewhat ambiguous. But $$z\mapsto \cos z: \mathbb{C}\to \mathbb{C}$$ $$\wp: \mathbb{C}\to S$$ are a simple examples of ...
Alexandre Eremenko's user avatar
2 votes

Books for learning branched coverings

You can have a look at Makoto Namba's book "Branched coverings and algebraic functions".
Francesco Polizzi's user avatar
2 votes

What is definition of branched covering?

At the top of page ten of this paper the authors write "the standard two-sheeted branched covering of the sphere by the torus, branched over four points which become the four boundary circles of ...
Sam Nead's user avatar
  • 27k
1 vote
Accepted

Unibranch points (definition for varieties over arbitrary field)

For a scheme $X$, say that $X$ is topologically unibranch at $x$ if $\mathop{Spec} O_{X,x}$ is geometrically unibranch (meaning that $O_{X,y}$ is geometrically unibranch at all generisations $y$ of $x$...
Damien Robert's user avatar
1 vote

Cohomology of ramified double cover of $\mathbb P^n$ (reference)

This is not an area of expertise for me, so forgive me if I didn't understand the question properly and hence this answer isn't on point. I think the original reference for this might be Lazarsfeld's ...
meh's user avatar
  • 964

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