I posted this in MathStackexchange and was advised to come here. As I am only looking for examples, I didn't feel MathOverflow was necessary.
I am looking for examples of specific quartic projective hypersurfaces over $\mathbb{P}^{3}$. So I am going off the fact of 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 codes 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, preferably say 15 and up?
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 an equation with 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 from just randomly cooking up equations is 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.
There are other slightly lower ones such as $w^{2}y^{2}-xy^{3}+zy^{3}+xwz^{2}+y^{4}+w^{2}x^{2}=0$. This has 3 singularities of type $E_{6},A_{3},A_{2}$. Others are $w^{2}z^{2}+x^{3}z+xy^{2}z+w^{2}y^{2}+x^{2}y^{2}$ with $D_{4},A_{5},A_{1}$ and $z^{4}+xz^{3}+w^{2}z^{2}+xy^{2}w+x^{2}yw+x^{2}y^{2}$ with $D_{4}, A_{3}, A_{2}$.
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. Honestly, even a slight change to Kummer would be cool. Maybe $14 A_{1 }$ and $1 A_{2}$. All the examples I see are type $A_{1}$/ordinary double points such as Kummer surface. It would be nice to see other Du Val singularities with high global Milnor number.
