I was hesitant about posting this question here, but since it deals with a [partially unanswered question already on this site](https://mathoverflow.net/a/137024/73622) I figured that this would be the best place for it. I apologise in advance if not.


The question deals with explicitly calculating the ramification divisor on a curve in weighted projective space.
The example given in the question was that of $C\_7\subset\mathbb{P}(1,2,3)$.


There is a (generally) degree-6 covering map $\pi\colon\mathbb{P}^2\to\mathbb{P}(1,2,3)$ given by $[x\_0:x\_1:x\_2]\mapsto[x\_0:x\_1^2:x\_2^3]\_{(1,2,3)}$.
Write $[y\_0:y\_1:y\_2]\_{(1,2,3)}$ for the coordinates on $\mathbb{P}=\mathbb{P}(1,2,3)$.
Then $\pi$ branches along the hyperplanes $\{y\_1=0\}$ and $\{y\_2=0\}$, where it is
2-1 and 3-1, respectively.


However, by, say, the Griffiths-Dolgachev-Steenbrink formula (see the linked question), we can calculate that $g\_{C\_7}=1$, since the usual degree-genus formula gives us that $g\_C=15$, where $C=\pi^{-1}(C\_7)\subset\mathbb{P}^2$.
So naively plugging things into Riemann-Hurwitz we see that $$2g\_C-2=6(2g\_{C\_7}-2)+\deg(D)$$ where $D$ is the ramification divisor.
Using the facts that we already know then, we see that $\deg(D)=28$.
But how exactly is $D$ defined?


I've searched through bits of Dolgachev's *Weighted Projective Varieties* and Reid's *Young Person's Guide to Canonical Singularities* but can't seem to find an explicit answer.


Naively, it seems like there would be 14 points in total on $C\_7$ that intersect the two hyperplanes, 7 with index 2 and 7 with index 3.
Plugging this into the usual definition without any thought then gives us that $\deg D=7\*(2-1)+7\*(3-1)=21$.


**Edit:** *I've now answered this question in slightly more generality, but only using pretty basic (i.e. accessible to an undergraduate) ideas, in Theorem 4.3.7 [in this link](https://mathoverflow.net/a/137024/73622).*



Ramification divisor on curves in weighted projective space
The curve $C\_7$ passes through the points $P = (0 : 0 : 1)$ and $Q = (0: 1: 0)$. You can see this by looking at what monomials can be used to define $C$. The group of order $6$ acting on $\mathbf{P}^2$ fixes these two points, hence the map $C\_7 \to C$ has ramification order $6$ at $P$ and $Q$. There are $6$ points on $C\_7$ besides $P$ on the line $x\_1 = 0$. There are $6$ points on $C\_7$ besides $Q$ on the line $x\_2 = 0$. Plugging this into the usual definition then gives us that $\deg D = (6 - 1) + (6 - 1) + 6 \* (2 - 1) + 6 \* (3 - 1) = 28$.


