I came across the following quartic surface in $\mathbb{P}^3$: $S: xyz(x-2y-z)+ww(xy+2xz-yz) = 0$
(It seems that S has singularities at $[x:y:z:w] = [1:0:0:0],[0:1:0:0],[0:0:1:0],[0:0:0:1]$.)
I'm interested in the rational points on $S$. Is there some explicit way to obtain rational points (e.g. parametrized rational points) on $S$?
I guess that resolution of singularities may be related but I'm not familiar with explicit resolution of singularities.