I am trying to compute fundamental group of complement to discriminant hypersurface of $f=x^3+y^3$ singularity via Zarisski-van Kampen theorem. So, I need a versal deformation of singularity to proceed. If I am not mistaken, deformation $x^3+y^3+ax^2+by^2+cxy+d$, where $a,b,c,d,x,y \in \mathbb{C}$, is a versal. I reduce it to the following system, representing disriminant hypersurface $\Sigma_f$, but it's too hard to procced computations. \begin{cases} 2x^3+2y^3+cxy-d=0 \\ 3x^2+a+cy=0 \\ 3y^2+b+cx=0 \end{cases}

Is there any way to get full equation in variables $a,b,c,d$, representing discriminant hypersurface?

Maybe, there is no need for it, because to compute $\pi_1(\mathbb{C}^n\setminus \Sigma_f)$ I need to know points of intersection of line $l$ in general position with $\Sigma_f$ and singular points of $\Sigma_f$, that lays in intersection of plane $P$ (that contains $l$ and also in general position with $\Sigma_f$) and discriminant hypersurface.


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