ag.algebraic geometry – Cubic surface in $mathbb{P}^3$ with isolated singularities and generic fiber

I was trying to read the paper “Hypersurfaces complements, Alexander Modules and Monodromy” by Dimca and Nemethi. In example 3.4. they mentioned a cubic surface $V$ in $mathbb{P}^3$ having two singularities of type $A_1$ and $A_5$ and they took $H$, a generic plane so that $X=V-H$ becomes singular. In this case, they also mentioned that the associated polynomial $f$ is tame.

Now my question is as follows: Is $X$ the generic fiber of a polynomial? If so how to recover such a polynomial?