Let us have a surface $X$ which satisfy the equation $\omega^3 = (z-1)(z-2)\cdots (z-8)$
It's stated in the task, that there are exactly $g$ holomorphic differentials on the curve. How can I find $g$ and construct them?
The genus $g$ of this curve (I tried to use Riemann-Hurwitz theorem)
The map $X(z,\omega) \to CP^1(\omega)$ has degree 3. and we have 8 branching points with 3 leaves at each.
$\chi(Domain) = DegreeOfMap * \chi(Target) - \sum_{points}(BranchingLeaves - 1) $
$2-2g = 3 *(2) - 8*2 $
hence $ 12 = 2g$ and $g = 6$
I know that in case of hyperelliptic curve there are $g-2$ (If i'm not mistaken) holomorphic differetials and they are of sort $\frac{z^k dz}{w}$. But the form of the curve equation was heavily used.
UPD: there is a branhing point at $\infty$, because if we go around all points $ 1 \cdots 8$ at once, we get that cubic root of the right side is multiplied by $\exp{4 \pi /3}$
$2-2g = 3 *(2) - 9*2 $ hence $ 12 = 2g$ and $g = 7$ (I hope that this is true)