Complex multi-projective variety and entanglement

In this paper, we will show that a vanishing generalized concurrence of a separable state can be seen as an algebraic variety called the Segre variety. This variety define a quadric space which gives a geometric picture of separable states. For pure, bi- and three-partite states the variety equals the generalized concurrence. Moreover, we generalize the Segre variety to a general multipartite state by relating to a quadric space defined by two-by-two subdeterminants.