Questions tagged [branched-covers]
The branched-covers tag has no usage guidance.
69
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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 ...
- 19.1k
28
votes
5
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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 ...
- 4,862
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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 ...
- 1,443
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3
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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 ...
- 1,085
15
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1
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$S^3$ 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 ...
- 606
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2
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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 $...
- 4,862
11
votes
2
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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.
...
- 225
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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 ...
- 111
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0
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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 ...
- 3,554
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answer
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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 ...
- 173
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2
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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
...
- 4,862
10
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0
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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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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 ...
- 2,221
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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 ...
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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 ...
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