Here's an alternative way to think about it: You can easily deduce from Riemann-Hurwitz that the genus of X is 0, i.e. it is just the projective line. Look at the affine patch: t is not infinity. ...

Yes. The fundamental group of this Riemann surface minus those branch points is $< a_1, b_1, ..., a_g, b_g, c_1, ..., c_r| [a_1,b_1]...[a_g,b_g]c_1...c_r=1>$ (where $g$ is the genus, and $r$ is ...

I see that this question is from a while back, but I figured I add this little morsel: Manish Kumar proved for his thesis that the commutator subgroup of the algebraic fundamental group of A^1 (and, ...

This certainly isn't something I thought a lot about, but there has definitely been interest about "Generalized Fermat Equations" (like the one you listed). Here's a quick link that I found googling ...

A paper I'm reading now is a PERFECT reference for this: "Deformation of tame admissible covers of curves" by Stefan Wewers is written in an expository style. (corollary 3.1.3 is exactly the theorem ...

Hopefully I'll have some time later to ellaborate, but for now - here is a great reference I wish somebody had shown me when I started out, for how to attack questions of this type: http://books....

Here is a possible statement: If f:X->Y and g:X->Y are two finite morphisms of schemes that agree topologically, and they give the same homomorphisms on all the residue fields (if x goes to y, f and ...

Alright, I think I should write my 2 cents here: Obviously $Spec(\mathbb{Q})$ and $\mathbb{A}_K$ are not directly analogous, but they do appear to be in relation to this problem. It seems that they ...

I don't know if this was the original motivation, but this has some interesting motivating ideas: Abstract nonsense versions of "combinatorial" group theory questions