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We propose a new approach to the geometry of the four-qubit entanglementclasses depending on parameters. More precisely, we use invariant theory andalgebraic geometry to describe various stratifications of the Hilbert space bySLOCC invariant algebraic varieties. The normal forms of the four-qubitclassification of Verstraete {\em et al.} are interpreted as dense subsets ofcomponents of the dual variety of the set of separable states and an algorithmbased on the invariants/covariants of the four-qubit quantum states is proposedto identify a state with a SLOCC equivalent normal form (up to qubitspermutation).

Entanglement of four-qubit systems: a geometric atlas with polynomial compass II (the tame world)