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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.

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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 $g=1$.

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  • $\begingroup$ Thank you, and how did you find degree and ramification point of $\phi$? I tried to find preimage of $\phi$ but failed. $\endgroup$ – Andrews Jun 11 at 15:35
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    $\begingroup$ 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. $\endgroup$ – user10354138 Jun 11 at 15:40
  • $\begingroup$ 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 $\Bbb P^1$? $\endgroup$ – Andrews Jun 11 at 15:46
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    $\begingroup$ 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. $\endgroup$ – user10354138 Jun 11 at 15:53

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