I am interested in any properties that can be claimed for the following situation: suppose that $(w_0, \cdots, w_n)$ is a weight vector of positive integers, and suppose that $X,Y \subset \mathbb{P}(w_0, \cdots, w_n)$ are two hypersurfaces in the weighted projective space $\mathbb{P}(w_0, \cdots, w_n)$. What can be said about the intersection $X \cap Y$?

The motivation for this question is the analgous question for $\mathbb{P}(1,1,1,1) = \mathbb{P}^3$, the projective space of in three dimensions. Here we know that $X \cap Y$ would generally be a space curve; and it is known that it can be birationally mapped to a plane curve in $\mathbb{P}^2$. In particular, the number of rational points of height at most $B$ on $X \cap Y$ is of the magnitutde $O_{d,\epsilon}\left(B^{2/d + \epsilon}\right)$.

In the weighted projective case the latter property is less clear since it is not clear how a space curve in $\mathbb{P}(w_0, w_1, w_2, w_3)$ can be mapped to a plane curve in a weighted projective space of lower dimension while more or less preserving the number of rational points (up to a constant factor). Any insight would be appreciated.