# Compute genus of compact Riemann surface $\{[x_0,x_1,x_2]\in\Bbb P^2|x_2^2x_0=\prod_{i=1}^3(x_1-\lambda_ix_0),\lambda_i\in\Bbb C\}$

Compute genus of compact Riemann surface $$X=\{[x_0,x_1,x_2]\in\Bbb P^2|x_2^2x_0=\prod_{i=1}^3(x_1-\lambda_ix_0),\lambda_i\in\Bbb C\}, \lambda_i$$ are distince numbers.

I know the degree-genus formula, but I want to use Riemann-Hurwitz formula.

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 $$n(n-1).$$

For 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.

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 $$g=1$$.
• Thank you, and how did you find degree and ramification point of $\phi$? I tried to find preimage of $\phi$ but failed. – Andrews Jun 11 at 15:35
• For $x_0\neq 0$ and generic $x_1$ the RHS is nonzero and so only two values of $x_2$ works, hence degree 2. You don't have distinct square-roots only when $x_2=0$ or when $x_0=x_1=0$, so that gives the four points mentioned. – user10354138 Jun 11 at 15:40
• Thank you, and why do you choose function $[x_0,x_1,x_2] \to [x_0,x_1]$ rather than $[x_0,x_1,x_2] \to [x_1,x_2]$ or $[x_0,x_1,x_2] \to [x_1,x_2]$ or other functions? Is there a criterion or technique to choose such a function from given algebraic surface to $\Bbb P^1$? – Andrews Jun 11 at 15:46
• You want the degree to be low, so ramification points and their indices are easy to analyse. Note the $x_2^2$ is the only appearance of $x_2$ in the defining equation, so it makes sense to consider this map. – user10354138 Jun 11 at 15:53