  To characterize entanglement of tripartite
$\mathbb{C}^d\otimes\mathbb{C}^d\otimes\mathbb{C}^d$ systems, we employ
algebraic-geometric tools that are invariants under Stochastic Local Operation
and Classical Communication (SLOCC), namely $k$-secant varieties and
one-multilinear ranks. Indeed, by means of them, we present a classification of
tripartite pure states in terms of a finite number of families and subfamilies.
At the core of it stands out a fine-structure grouping of three-qutrit
entanglement.
