Let $P(x)$ and $Q(x)$ be two general polynomial of the same degree $d$ (d can be arbitrary large). Consider the surface $S : z^2 = P(x) Q(y)$ in the projective space $\mathbb{P}^3$. I want to prove the following assertion (which I'm almost sure that should be true):

If the coefficients of the polynomials $P$ and $Q$ are chosen generically, then the surface $S$ is a hyperbolic surface in $\mathbb{P}^3$. Equivalently, I wish to show that there is no rational/elliptic curve on the surface $S$.

By hyperbolicity I mean there is no non-constant holomorphic function $f: \mathbb{C} \rightarrow S$.

Note that the surface $S$ is a singular surface with singularities at $(a_i, b_j, 0)$ where $a_i$ and $b_j$ are roots of the polynomials $P$ and $Q$ respectively.\

Jason, thanks, I think in your answer you mean the hyperbolicity of $D(zw)$ rather than $D(y)$. Also what is the generic condition on the polynomials $P$ and $Q$? Is the affine surface D(zw) in the above surface $S$ always hyperbolic?

If we consider the equation $S$ on a number field $K$. Can we deduce that the there are finite number of $K$-rational points on the affine surface $D(zw)$ by Lang conjecture?