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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 ...
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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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### 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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### cardinality of the fibre of a constantly branched, finite morphism over the branch locus

Let $\pi:Y\to X$ be a Galois cover, i.e. a finite morphism of nonsingular varieties over an algebraically closed field $\Bbbk$ such that $K(X)\hookrightarrow K(Y)$ is Galois. Let $H\subset X$ be the ...
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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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### 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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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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### Comparing heights of rational points on curves through covers

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 ...
Let $L_1$, $\cdots$, $L_k$ be homogenous linear forms in three variables $z_0$, $z_1$, $z_2$ defining $k$ lines in $\mathbb{P}_{2}$. Consider the abelian extension $K ((L_2/L_1)^{1/n}, \cdots, (L_k/... 0answers 190 views ### branch locus of the discriminant map$\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$curves ... 1answer 549 views ### Is this function field extension a Galois extension ? Setting and question Let$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'$the normalization of$...
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 ...