Hi guys,(adsbygoogle = window.adsbygoogle || []).push({});

I new to Algebraic Geometry and was wondering if someone could help me with this problem:

1. The problem statement, all variables and given/known data

Given the algebraic curve [itex]w(z)[/itex] represented by [itex]w^5+w^2+z^2=0[/itex], show that the genus is one by employing the Riemann-Hurwitz formula.

2. Relevant equations

For the mapping [itex]f:X\to S[/itex], the Riemann-Hurwitz formula is given by:

[tex]2g(X)-2=\text{deg}(f)\left[2g(S)-2\right)+\sum (e_w-1)[/tex]

which I assume the expression [itex]f:X\to S[/itex] as applied to my problem can be written as [itex]w(z):\mathbb{C}\to S[/itex], that is, the function w(z) maps the (five-sheeted) complex plane to the Riemann surface given by [itex]S[/itex] and it is this surface that is topologically equivalent to a Riemann sphere with one handle thus having genus one. If that's not the correct way to state that, could someone correct that for me please?

3. The attempt at a solution

1. The problem statement, all variables and given/known data

I believe I understand somewhat well, direct construction of the Riemann surface given by the expression [itex]w^2-p(z)=0[/itex] and the subsequent determination of its genus directly from the union of two Riemann spheres. However I'm not able to deduce that via Riemann-Hurwitz.

But back to the problem above:

I'm taking [itex]g(X)[/itex] to be the genus of the complex plane which is zero.

The degree of [itex]f[/itex] I'm taking to be the highest degree of [itex]f(z,w)[/itex] in [itex]w[/itex] which is five.

The sum is where I'm having the problem. This I belive can be computed from the monodromy group which I assume represents the ramifications around each branch-point which I take to mean the branching around each branch-point including infinity. Here's my analysis of that:

The function has six finite branch points and the surface [itex]S[/itex] ramifies around each one as a two-valued branch and three single-valued branches. I think each monodromy group about these branch-points can be written in terms of [itex]\left\{(n,m)\right\}[/itex] with [itex]n[/itex] being the sheet number and [itex]m[/itex], the branch order as:

[tex]((1,2),(3,1),(4,1),(5,1))[/tex]

Maybe though that's not a valid way to write that. However, around the branch-point at infinity, the function ramifies into a single five-valued branch so the monodromy group is [itex]((1,5))[/itex]

So my (incorrect) analysis of the sum would be [itex]\sum (e_w-1)=(2-1)+(1-1)+(1-1)+(1-1)[/itex] for each finite branch-point and [itex]\sum (e_w-1)=5-1[/itex] for the point at infinity. Plugging all this into the Riemann-Hurwitz formula I obtain:

[tex]-2=5\left[2g-2\right]+10[/tex]

and solving for g I obtain [itex]g=-1/5[/itex] which is obviously not correct.

I don't know, maybe I'm not even qualified to even ask the question. Still I would like to see how it's solved at the very least if someone could help me with that and as I study more the subject, I'm sure it will come together eventually for me.

Thanks,

Jack

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Compute genus via Riemann-Hurwitz and monodromy

**Physics Forums | Science Articles, Homework Help, Discussion**