The XJC-correspondence

For any n >= 3, we prove that there are equivalences betweenirreducible n-dimensional non-degenerate complex projective varieties X subset of P2n+1 different from rational normal scrolls and 3-covered by cubic curves, up to projective equivalence,n-dimensional complex Jordan algebras J of rank 3, up to isotopy,quadro-quadric Cremona transformations C : Pn-1 -> Pn-1 of the complex projective space of dimension n 1, up to linear equivalence.These three equivalences form what we call the XJC-correspondence.We also provide some applications to the classification of particular types of varieties in the class defined above and of quadro-quadric Cremona transformations.