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A 3-fold compound Du Val singularity of type $A_{k} \ (D_{k}, E_{k})$ is defined as a singular point on a 3-fold which is locally given by $$ F(x,y,z,w)=f(x,y,z)+wg(x,y,z,w)=0, $$ where $g(x,y,z,w)$ is an arbitrary polynomial and $f(x,y,z)$ defines a 2-fold singularity of type $A_{k} \ (D_{k}, E_{k})$. In other words, a 3-fold cDV singularity is a singularity whose hyperplane section is a 2-dols A-D-E singularity.

My question is, is the $g(x,y,z,w)$ above necessary to be generic (or the hyperplane in the latter)? For example can $x^2+y^3+z^5+zw^2$ be a $cE_{8}$ and $cD_{6}$ at the same time (cutting by $w=0$ and $y=0$ respectively)? Or should one think of it as $cD_{4}$, cutting it by a general hyperplane $y=z$?

Thank you for your help.

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