  In this paper, we use singularity theory to study the entanglement nature of
pure three-qutrit systems. We first consider the algebraic variety $X$ of
separable three-qutrit states within the projective Hilbert space
$\mathbb{P}(\mathcal{H}) = \mathbb{P}^{26}$. Given a quantum pure state
$|\varphi\rangle\in \mathbb{P}(\mathcal{H})$ we define the
$X_\varphi$-hypersuface by cutting $X$ with a hyperplane $H_\varphi$ defined by
the linear form $\langle\varphi|$ (the $X_\varphi$-hypersurface of $X$ is
$X\cap H_\varphi \subset X$).
  We prove that when $|\varphi\rangle$ ranges over the SLOCC entanglement
classes, the "worst" possible singular $X_\varphi$-hypersuface with isolated
singularities, has a unique singular point of type $D_4$.
