The XJC-correspondence

For any n ≥ 3, we prove that there are equivalences between irreducible n-dimensional non-degenerate complex projective varieties X ⊂ 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 W P^n-1 P^n-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.