I've got this exercise I can't solve. May someone help me? Thank you.

Let $X, Y, Z$ be homogeneous coordinates on complex projective plan and let $C=\{[X:Y:Z] |X^{4}+XY^{3}+Z^{4}=0\}$. Consider the meromorphic function $f=\frac{X}{Y}$ defined on $C$. 1) Calculate zeroes and poles of $f$ with their orders;

2) Calculate ramification points of $f$ with their indexes and the genus of $C$;

3) Find on $C$ three linearily independent holomorphic differentials

  • $\begingroup$ Hi. It might be good to put where the exercise is from if it is from a book, in case someone else is using it. Also, sharing anything you have tried would be useful :) $\endgroup$ – Joe Tait Jun 28 '13 at 13:06

To answer point 1) The zeroes occur when $X$ is zero - in this case we are forced by the defining equation to have $Z=0$ and hence the only point at which $f$ has a zero is $[0:1:0]$. Similarly we have a pole if $Y=0$, and in this case we are forced to have $X^4 = -Z^4$, which has various solutions over $\mathbb C$. Find these and you have your poles. Finally, note that we cannot have $X=Y=0$, since then we would also have $Z=0$, and $[0:0:0]$ is not a valid point.


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