Questions tagged [branched-covers]
The branched-covers tag has no usage guidance.
69
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How to determine the LS category of branched covers?
Define the (normalized) Lusternik-Schnirelmann (LS) category of a space $X$, denoted $\mathsf{cat}(X)$ to be the least integer $n$ such that $X$ can be covered by $n+1$ number of open sets $U_i$ each ...
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Branched covering maps between Riemann surfaces
What is an example of a branched covering map between Riemann surfaces of infinite degree? i.e. something like a branched version of the exponential map $exp: \mathbb{C} \to \mathbb{C}^*$.
Thanks!
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Branched covers of real algebraic varieties
Let $X$ be a smooth complex algebraic variety and $L$ be an $n$-torsion line bundle on $X$, i.e., a line bundle $L$ such that $L^n=\mathcal{O}_X(B)$, where $B$ is a divisor $B$ on $X$. Such a bundle ...
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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.
...
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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 ...
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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 ...
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(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 $X\to S$ ...
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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 ...
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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 ...
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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. ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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