  An intriguing correspondence between four-qubit systems and simple
singularity of type $D_4$ is established. We first consider an algebraic
variety $X$ of separable states within the projective Hilbert space
$\mathbb{P}(\mathcal{H})=\mathbb{P}^{15}$. Then, cutting $X$ with a specific
hyperplane $H$, we prove that the $X$-hypersurface, defined from the section
$X\cap H\subset X$, has an isolated singularity of type $D_4$; it is also shown
that this is the "worst-possible" isolated singularity one can obtain by this
construction. Moreover, it is demonstrated that this correspondence admits a
dual version by proving that the equation of the dual variety of $X$, which is
nothing but the Cayley hyperdeterminant of type $2\times 2\times 2\times 2$,
can be expressed in terms of the SLOCC invariant polynomials as the
discriminant of the miniversal deformation of the $D_4$-singularity.
