Let $S_{d>3}\subset\mathbb{P}^3_{\mathbb{C}}$ be a smooth surface of degree $d$. What is known (where to read?) about the Picard/cohomology lattice for small d?

e.g. for $d=4$ the cohomology lattice is $E_8^{\oplus2}\oplus H^{\oplus3}$, but what is the class of a hyperplane section in this lattice? Probably many things are known for $d=5,6$? (I need the results for any smooth $S_d$, not necessarily generic.)

And what is known for "not-too-singular" $S_d$? (e.g. with nodes only)

upd. I guess any two smooth surfaces of the same degree in $\mathbb{P}^3$ have the same cohomology lattice? I want to know this lattice and in particular what is the plane section in this lattice, i.e. $\mathcal{O}(1)$.

\ In my actual problem I have a curve lying on such a surface, want to identify its equivalence class. (If you wish: the $\mathbb{P}^3$-degree of $C$ is $\frac{d(d-1)}{2}$, the self-intersection on $S_d$ is: $C^2=\frac{d(d-1)(d-2)}{6}$, so the arithmetic genus is: $p_a=\frac{(2d+1)(d-2)(d-3)}{6}$. Such a curve is quite special, lying on the surface it forces the surface to be rather special too.)

So, I'd need as much information as possible about $S_{d>3}\subset\mathbb{P}^3$. Such a very-classical geometry.