Let us consider the weighted projective plane $\mathbb{P}(q_0,q_1,q_2)=\mathrm{Proj}(\mathbb{C}[x_0,x_1,x_2])$ where $x_i$ has weight $q_i$ for every $i\in \{0,1,2\}$. Let $f\in \mathbb{C}[x_0,x_1,x_2]$ be a homogeneous polynomial of degree $d$. Then $f$ defines a curve $C$ in $\mathbb{P}(q_0,q_1,q_2)$.
On the other hand, $\mathrm{Pic}(\mathbb{P}(q_0,q_1,q_2))$ is generated by $\mathcal{O}_{\mathbb{P}(q_0,q_1,q_2)}(n)$ for some $n$.
In the case $q_0=q_1=q_2=1$ we have that $n=1$ and $\mathcal{O}_{\mathbb{P}^2}(C)\simeq \mathcal{O}_{\mathbb{P}^2}(d)$. Can we generalize this for different values of $q_0,q_1,q_2$?
The class group of Weil divisors on a weighted projective plane is isomorphic to the group of reflexive sheaves of rank 1 (with operation of tensor product followed by double dualization), and is a free abelian group of rank 1 generated by the reflexive sheaf $\mathcal{O}(1)$. Under this isomorphism $\mathcal{O}(C) \cong \mathcal{O}(d)$.