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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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### 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: ...
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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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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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 ...
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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 ...
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### Equations for abelian coverings of $\mathbb{P^{1}}$

Cyclic coverings of $\mathbb{P^{1}}$ have a simple (affine) equation, namely the formula, $y^{m}= (x_{1}-a_{1})^{t_{1}}....(x_{n}-a_{n})^{t_{n}}$. Is there such a nice equation for abelian non-cyclic ...
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### Every curve is a Hurwitz space in infinitely many ways

Diaz, Donagi and Harbater proved that every curve over $\overline{\mathbf{Q}}$ is a Hurwitz space. A Hurwitz space is a connected component of the curve $H_n$. The curve $H_n$ is (the ...
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Setting and question Let $X$ be a variety over an algebraically closed field of null characteristic, and let $C$ be a (regular if you want) curve included in $X$. Consider $X'$ the normalization of $... 2answers 740 views ### Fibre cardinality of an unramified morphism Let$\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$to be nonsingular. I am trying to prove that ... 0answers 280 views ### degenerating surface II In degenerating surface, Robert Bryant give us an example of a sequence of minimal immersions which converges (in$C^2$- topology) to$z\mapsto z^{2k+1}$on the unit disc$\mathbb{D}$. My question is ... 1answer 261 views ### Manin-Drinfeld and constructing a finite morphism with two given ramification points Fix a smooth projective connected curve$X$over$\overline{\mathbf{Q}}$of genus$g\geq 1$and distinct points$x,y \in X$such that$x-y$has infinite order in the Jacobian. Can we always find a ... 2answers 676 views ### degenerating surface Hi, i have a sequence of immersed disc$u_n: \mathbb{D} \rightarrow \mathbb{R}^3$which converge to a singular cover of the disc:$z^k$for$k\geq 2$, moreprecisely$u_n \rightarrow z^k$in$C^2(\...
Let $a$ be a closed point in $\mathbf{P}^1_{\overline{\mathbf{Q}}}$. Let $Y \cong \mathbf{P}^1_{\overline{\mathbf{Q}}}$ and let $\pi:Y\to \mathbf{P}^1_{\overline{\mathbf{Q}}}$ be a finite morphism ...