Problem: Let $X,Y,Z$ be homogeneous coordinates in $\mathbb P^2\mathbb C$ and
$$C=\{[X,Y,Z] \in \mathbb P^2\mathbb C \mid X^5 + Y^5 + Z^5=0\}.$$
Calculate the genus of $C$ and find a base for $\Omega^1(C)$.
Thoughts: $C$ is smooth, thus $g=(5-1)(5-2)/2=6$. In the chart $Z \not=0$ the equation becomes $x^5 + y^5 + 1 =0$ and thus $$\frac {dx}{y^4}=\frac {-dy}{x^4}$$ is a holomorphic $1$-form.
Thanks!
$Dim_{\mathbb C} \Omega^1(C)= g$ thanks to Hodge Theorem.
A base is given by $\frac{\phi(x,y)dx}{F_y}$ with $\phi(x,y)$ every polynomial of degree at most $d-3=5-3=2$.