I was hesitant about posting this question here, but since it deals with a partially unanswered question already on this site 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.*