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The formula is true if the cover is branching is "total", that is, if all sheets of the cover join into a single point. This is a straightforward application of the Riemann-Hurwitz formula (algebraic …

The moduli space of curves of degree $d$ in $\mathbb P^2$ has dimension $\left( \begin{array}{c} d + 2 \\ 2 \end{array}\right)$. The most typical type of singularity is a simple cusp. Having a cusp at …

There is a subtle problem with this idea, that causes serious problems. You observed that $\Lambda_\ell \otimes \mathbb Z_\ell = T_\ell$ but didn't find any other information for it. There is a reason …

If $E'$ is isotrivial, then the fibers $E'\to B$ are all isomorphic to a single fiber, so the Jacobians of every fiber are isomorphic to a single Jacobian abelian variety $A$. Then the Jacobian of eve …

(1) Consider the case where C is a sphere. For any such manifold, we can just add a 3-dimensional ball to get a closed compact 3-fold. So in this case it's just the classification of closed compact 3- …

Consider a regular hyperbolic $4n+2$-gon with vertex angle of genus $3$. I claim we can glue $3$ copies of this polygon together to form an oriented hyperbolic surface of genus $n$, without gluing any …

The reason for this phenomenon is that you are afflicted with a serious mathematical condition, that being: Your monodromy has monodromy. To be less cryptic, the key thing is that the fundamental gro …

Let $F$ be a polynomial of degree at most $n$. For a point $x$ where $f$ has a pole of order $a$ and $g$ has a pole of order $b$, $F(f,g)$ has a pole of order at most $n\max(a,b)$. Locally near $x$, w …

This is not true. For a simple counterexample, take any two $G$-bundles that are not isomorphic. Their fiber product is then a $G \times G$ bundle in two different ways, depending on which copy of $G$ …

For dessin d'enfants, I believe the orientation is superfluous - each edge goes between a black vertex and a white vertex, so picking an orientation is just picking one of those, whcih doesn't help un …

iI think it's best to turn this question around a bit. If such a coincidence occurs, we have two maps $X \to E$. Thus, we can consider the space of maps $X \to E$. If someone hands you a space of cov …

Consider an automorphism of a hyperelliptic curve. It is sufficient to ask what the quotient by this automorphism can be. Because there is a unique hyperelliptic projection to $\mathbb P^1$, it is can …

There are counterexamples as soon as $g > 1$. Let $n$ be the number of surjective homomorphisms $\pi_1(S) \to A_5$, up to $S_5$-conjugacy. (We can see that $n \geq 1$ using the fact that $A_5$ can b …

I think the best interpretation of the paper you link to is that they mean a Riemann surface is a nonsingular projective algebraic curve and they have simply forget to include the condition that $p(t) …

There are many automorphisms of $Y$ over $Z$. One can pull back the cover $X$ along any of these automorphisms of $Y$, producing a new cover of $Y$. If the composition is Galois, then these new covers …