My question:
Compute genus of compact Riemann surface using Riemann-Hurwitz formula:
$X=\{[x_0,x_1,x_2]\in\Bbb P^2\mid x_2^2x_0=\prod_{i=1}^3(x_1-\lambda_ix_0),\lambda_i\in\Bbb C\}, \lambda_i$ are distinct numbers.
An example:
For homogeneous polynomial $P(z_0,z_1,z_2)=z_0^n+z_1^n+z_2^n$, $P=0$
defines a compact Riemann surface.
$M=\{[z_0,z_1,z_2]\in\Bbb CP^2|P(z_0,z_1,z_2)=0\}$.
To compute its genus, comsider $f:M\to \Bbb CP^1, [z_0,z_1,z_2] \mapsto [z_0,z_1]$, it's a well-defined meromorphic function, $f^{-1}([0,1])=\{[0,1,z_2]|z_2^n=-1\}$.
$f$ is $n$-sheeted, has $n$ branch points, each point with ramification index $n-1$. Total ramification number is $n(n-1).$
From Riemann-Hurwitz formula, genus of $N$ is $\frac 12(n-1)(n-2)$
But for $X=\{[x_0,x_1,x_2]\in\Bbb P^2|x_2^2x_0=\prod_{i=1}^3(x_1-\lambda_ix_0),\lambda_i\in\Bbb C\}$,
how can we find such a $f: X\to \Bbb CP^1$ whose sheet number and branch points can be easily find out?
Thanks for your time and patience.
Best Answer
Look at $\phi\colon X\to\mathbb{P}^1; [x_0,x_1,x_2]\mapsto[x_0,x_1]$. It has degree 2 and ramified only when $x_2^2x_0=0$, so this is ramified over 4 points $[x_0,x_1]=[1,\lambda_i]$ or "$[0,0]$", the limit $x_0,x_1\to 0$, and Riemann-Hurwitz gives $g=1$.