Let $Xsubset mathbb{P}_{mathbb{C}}^3$ be a projective singular cubic surface with two singular points. Is the cohomology of such objects known? As an example of the type of surfaces I’d be interested in one can take the hypersurface defined by the homogenous cubic $$x^2w+y^2w+z^2w+xyz-zw^2-w^3=0 $$

I thijnk that the cohomology should be known for projective smooth cubic surfaces which should be realized as blowup of $mathbb{P}^2_{mathbb{C}}$ in six points but I’m not sure about singular ones.