I am looking for examples of specific quartic projective hypersurfaces over $\mathbb{P}^{3}$. So I am going off the fact the famous Kummer surface, under some parameters, have 16 real $A_{1}$ singularities. This will have global Milnor number 16. I was playing around with coding and making random quartic equations over $\mathbb{P}^{3}$. I was wondering if I can get a Du Val only hypersurface with real roots that is a high Milnor number like the Kummer.
Of course one can easily get high Milnor numbers by extending to unimodal singularities in Arnold's list as there are weird stuff like $Q_{10}$ with Milnor number 10 and so forth, but if we restrict to only $A_{n},D_{n},E_{n}$, I was wondering if we can achieve that. For example, is there an example of real singularities adding up to global Milnor Number 16 with say $E_{6}, A_{5},D_{5}$?
The best I can do is the equation given by $xyzw+x^{2}y^{2}+z^{3}x+w^{2}x^{2}+w^{2}y^{2}=0$. This has 3 singular points. If we denote the coordinates as $[w:x:y:z]$, we have the singularities at $y=1, w=1, x=1$. The singularities with $y=1$ and $w=1$ are of type $A_{5}$ and the singularity with $x=1$ is of type $A_{2}$. This will have global Milnor number 12. I have been trying to go higher but have been stuck.
To conclude, if you know any cool examples of quartic hypersurfaces with real Du Val singularities over $\mathbb{P}^{3}$ that have high global Milnor number, please share. If we exclude Kummer, the max I can do for now is 12.