This question has two parts: A calculation that is giving me a lot of troubles, and a theoretical one on weighted projective spaces.

1) I want to find the genus of the curve $C_7 \subset \mathbb{P}(1,2,3)$. Naive adjunction does not work, because this curve is not well-formed in the sense of [1]. To calculate the genus of the curve, it is possible to use the map $$ \mathbb{P}^2 \to \mathbb{P}(1,2,3) $$ defined by $$ [y_0:y_1:y_2] \to [y_0:y_1^2:y_2^3] $$ This map is a 6 to 1 ramified Galois covering with Galois group $\mathbb{Z}_2 \oplus \mathbb{Z}_3$. If the variables of $\mathbb{P}(1,2,3)$ are $[x:y:z]$. Then our map branches along the the hyperplanes $(z=0)$ and $(y=0)$. It induces a cover $D \to C_7$ where $D$ is a smooth plane curve of degree 7 and genus 15. It is possible to use Hurwitz theorem for finding that the genus of $C_7$ is one. However, I don't understand the ramification divisor! So, I cannot recover the calculation! (This is an example on [1, Note I.3.15.i])

2) A weighted projective space $\mathbb{P}(a_0, \ldots a_n)$ is well formed if $gcd(a_0 \ldots \hat{a}_i \ldots a_n)=1$ for every $i$. However, we find non well formed weighted projective spaces all the time. Even, those with $gcd(a_0, \ldots a_n) \neq 1$. What would be the adjunction formula in those cases?

Thanks!

References:

[1] Working with weighted projective spaces. A.R. Fletcher