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To characterize entanglement of tripartite$\mathbb{C}^d\otimes\mathbb{C}^d\otimes\mathbb{C}^d$ systems, we employalgebraic-geometric tools that are invariants under Stochastic Local Operationand Classical Communication (SLOCC), namely $k$-secant varieties andone-multilinear ranks. Indeed, by means of them, we present a classification oftripartite 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-qutritentanglement.

An Algebraic-Geometric Characterization of Tripartite Entanglement