All Questions
Tagged with branched-covers ag.algebraic-geometry
41
questions
2
votes
0
answers
78
views
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 ...
- 63
11
votes
2
answers
840
views
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
0
votes
0
answers
462
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 ...
- 13
1
vote
1
answer
609
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 $X$ ...
- 5,780
4
votes
1
answer
280
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 ...
- 65
4
votes
1
answer
715
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 $B \subset Y$. ...
- 65.7k
3
votes
1
answer
583
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 ...
- 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 ...
- 1,443
2
votes
1
answer
412
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 ...
- 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 $\mathcal O_{\mathbb P^1}(2)$, $\phi=(X_0-X_1)^3X_0\in H^0(\mathbb P^1, \mathcal O_{\mathbb P^1}(4))$ ...
- 1,443
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 ...
- 65.7k
9
votes
0
answers
334
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 ...
- 13.3k
4
votes
1
answer
252
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 ...
- 1,871
6
votes
1
answer
416
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 ...
- 10k
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 ...
- 3,392