# Tagged Questions

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### Varieties with infinitely many topological covers of finite degree

Let $X$ be a smooth projective connected variety over $\mathbf C$ with infinitely many etale covers. If $\dim X =1$, this holds if and only if the genus of $X$ is positive. Do we have a similar ...
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### Pulling-back a divisor and reducing it

Let $f:C\to B$ be a finite morphism of curves. Let $D$ be a divisor on $B$. Does the equality of divisors $$(f^\ast D)_{red} = f^\ast (D_{red})$$ hold on $C$? (I'm asking for an equality of divisors, ...
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### Why is the rank of $f_\ast L$ the degree of $f$

Let $f:X\to Y$ be a finite morphism of curves. Let $L$ be a line bundle on $X$. Why is $f_\ast L$ a line bundle and is the degree of $f_\ast L$ equal to $\deg f$ or $\deg f+ \deg L$? Here is my ...
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Let $X$ and $Y$ be (smooth projective connected) varieties over $\mathbf{C}$. Let $\pi:X\to Y$ be a finite surjective flat morphism. Does this induce (by base change) a map $\mathrm{Aut}(Y) \to ... 1answer 78 views ### The universal cover of the multiplicative group over the field of algebraic numbers Let$X=\mathbf{A}^1_{\overline{\mathbf{Q}}}-\{0\} = \mathbf{G}_{m,\overline{\mathbf{Q}}}$be the multiplicative over the field of algebraic numbers. Each finite etale cover$Y\to X$(with$Y$... 0answers 53 views ### What are the branch points of$X(n)\to X(1)$Let$\Gamma \subset \mathrm{SL}_2(\mathbf{Z})$be a finite index subgroup. Let$X_\Gamma \to X(1)$be the corresponding morphism of compact connected Riemann surfaces (obtained by adding the cusps). ... 1answer 115 views ### Are these two notions of Galois morphism the same Let$f:X\to Y$be a finite morphism of integral schemes. Let$G$be the automorphism group of$X$over$Y$. Are the following two conditions equivalent? The function field extension$K(Y)\subset ...
One can associate a Hurwitz number to any rational function $f:X\to \mathbf{P}^1$ on a compact connected Riemann surface $X$ which ramifies over precisely FOUR points. Suppose that $X$ is an elliptic ...
Let $\pi:X\to E$ be a finite étale morphism, where $E$ is an elliptic curve over a number field $K$. Assume $X$ to be connected, and to be of genus 1. Edit: Assume $X$ and $E$ have semi-stable ...