7
votes
1answer
202 views

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 ...
2
votes
0answers
142 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 ...
8
votes
3answers
508 views

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 ...
4
votes
2answers
134 views

Dimension of the space of invariant quadratic differentials in Galois covers

Let $f: X \rightarrow Y $ be a Galois cover of with $X$ and $Y$ algebraic curves over $\mathbb{C}$. I want to compute the dimension of the subspace of $G$-invariants in $H^{0}(X,\omega^{\otimes2})$ ...
1
vote
1answer
164 views

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 ...
4
votes
0answers
167 views

Do regular noetherian schemes of dimension one only have finitely many etale covers of bounded degree

Let $X$ be a regular noetherian scheme of dimension one. Let $d$ be an integer. Question. Are there only finitely many finite etale morphisms $Y\to X$ of degree $d$? I want to exclude finite etale ...
3
votes
0answers
227 views

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 ...
1
vote
1answer
236 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 ...
2
votes
1answer
220 views

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 ...
28
votes
4answers
2k views

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 ...