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 ...
4
votes
1answer
203 views

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 ...
2
votes
0answers
141 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 ...
3
votes
2answers
193 views

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 ...
2
votes
2answers
200 views

Confusion about two statements about cohomology of curves with automorphisms

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 ...
3
votes
2answers
323 views

How to explicitly see the ramification over infinity

Take the equation $y^{d}=\Pi_{1}^{n}(x-t_{i})^{m_{i}}$ over $\mathbb{C}$. This affine equation gives a cyclic cover of $\mathbb{P}^{1}$. Now it is usually said without explanation that if the sum ...
8
votes
3answers
507 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 $H^{0}(X,\omega^{\otimes2})$ ...
2
votes
1answer
174 views

regularity of finite flat branched covers

Let $D$ and $S$ be two regular schemes and let $D$ be a divisor of $S$. Let $C \to S$ be a finite flat morphism, branched along $D$. Is $C$ regular as well?
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 ...
8
votes
1answer
519 views

Question about local description of the branch locus

Let $\pi:Y\to X$ be a dominant, finite morphism of nonsingular varieties over an algebraically closed field $\Bbbk$. Assume furthermore that for all $Q\in Y$, with $P=\pi(Q)$, we have $$\mathcal ...
4
votes
3answers
257 views

Automorphism of finite groups and Hurwitz spaces

If $G$ is a finite group, embedded as a transitive subgroup of $S_n$ for some $n$, will every automorphism of $G$ extend to an inner automorphism of $S_n$? I'm trying to connect the language that's ...
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 ...
2
votes
0answers
267 views

Galois group decomposition of non-cyclic covers

If $\pi: C \rightarrow \mathbb{P}^{1}$ is a cyclic cover of $\mathbb{P}^{1}$ with Galois group $\mathbb{Z}/m \mathbb{Z}$ and thus with the (affine) formula $y^{m}= ...
2
votes
3answers
291 views

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 ...
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
486 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 ...
4
votes
2answers
222 views

Number of connected components of the Hurwitz space $H_n^o$ and subgroups of the fundamental group

A cover (of $\mathbf{P}^1_{\overline{\mathbf{Q}}}$) is a finite morphism $X\to \mathbf{P}^1_{\overline{\mathbf{Q}}}$, where $X$ is a smooth projective connected curve over $\overline{\mathbf{Q}}$. ...
3
votes
2answers
478 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 ...
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 ...
10
votes
2answers
913 views

Finite, Étale Morphism Of Varieties

I have a, probably very simple, question: My intuition tells me that the following statement should be true, but I couldn't find it anywhere and I wanted to make sure I am not missing something. Let ...
2
votes
0answers
225 views

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 ...
6
votes
4answers
1k views

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 ...
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 ...