The aim of the paper is to propose geometric descriptions of multipartite entangled states using algebraic geometry. In the context of this paper, geometric means each stratum of the Hilbert space, corresponding to an entangled state, is an open subset of an algebraic variety built by classical geometric constructions (tangent lines, secant lines) from the set of separable states. In this setting we describe well-known classifications of multipartite entanglement such as 2 × 2 × (n + 1), for n ≥ 1, quantum systems and a new description with the 2 × 3 × 3 quantum system. Our results complete the approach of A. Miyake and makes stronger connections with recent work of algebraic geometers. Moreover for the quantum systems detailed in this paper we propose an algorithm, based on the classical theory of invariants, to decide to which subvariety of the Hilbert space a given state belongs.Let x and y be two points of P N , the secant line P 1 xy is the unique line in P N containing x and y. We define the join of two varieties X and Y to be the (Zariski) closure of the union of the secant lines with x ∈ X and y ∈ Y :Suppose Y ⊂ X and let T ⋆ X,Y,y0 denote the union of P 1 * 's where P 1 * is the limit of P 1 xy with x ∈ X, y ∈ Y and x, y → y 0 ∈ Y . The union of the T ⋆ X,Y,y0 is defined as the variety of relative tangent stars of X with respect to P m × P n ⊂ σ 2 (P m × P n ) ⊂ · · · ⊂ σ min(m,n)−1 (P m × P n ) ⊂ P(H) Remark 2.2. As noticed in [2], a projective line P 1 xy in the Hilbert space P(H) represents all possible superpositions of the statesx,ŷ ∈ H.Definition 2.1. Let X ⊂ P(V ) be an irreducible variety of dimension n. The s-secant variety σ s (X) is said to be nondefective if either dim(σ s (X)) = sn + s − 1 or σ s (X) = P(V ).A direct consequence of Theorem 1 is the following proposition :

We point out an explicit connection between graphs drawn on compact Riemann surfaces defined over the fieldQ of algebraic numbers -so-called Grothendieck's dessins d'enfants -and a wealth of distinguished point-line configurations. These include simplices, cross-polytopes, several notable projective configurations, a number of multipartite graphs and some 'exotic' geometries. Among them, remarkably, we find not only those underlying Mermin's magic square and magic pentagram, but also those related to the geometry of two-and three-qubit Pauli groups. Of particular interest is the occurrence of all the three types of slim generalized quadrangles, namely GQ(2,1), GQ(2,2) and GQ(2,4), and a couple of closely related graphs, namely the Schläfli and Clebsch ones. These findings seem to indicate that dessins d'enfants may provide us with a new powerful tool for gaining deeper insight into the nature of finite-dimensional Hilbert spaces and their associated groups, with a special emphasis on contextuality.

We investigate the geometry of the four qubit systems by means of algebraic geometry and invariant theory, which allows us to interpret certain entangled states as algebraic varieties. More precisely we describe the nullcone, i.e., the set of states annihilated by all invariant polynomials, and also the so called third secant variety, which can be interpreted as the generalization of GHZ-states for more than three qubits. All our geometric descriptions go along with algorithms which allow us to identify any given state in the nullcone or in the third secant variety as a point of one of the 47 varieties described in the paper. These 47 varieties correspond to 47 non-equivalent entanglement patterns, which reduce to 15 different classes if we allow permutations of the qubits.

We propose a new approach to the geometry of the four-qubit entanglement classes depending on parameters. More precisely, we use invariant theory and algebraic geometry to describe various stratifications of the Hilbert space by SLOCC invariant algebraic varieties. The normal forms of the four-qubit classification of Verstraete et al. are interpreted as dense subsets of components of the dual variety of the set of separable states and an algorithm based on the invariants/covariants of the four-qubit quantum states is proposed to identify a state with a SLOCC equivalent normal form (up to qubits permutation).