The XJC-correspondence

For any n ≥ 3, we prove that there are equivalences between • irreducible n-dimensional non degenerate projective varieties X ⊂ P2n+1 different from rational normal scrolls and 3-covered by cubic curves, up to projective equivalence; • quadro-quadricCremonatransformationsofPn−1,uptolinearequivalence; • n-dimensional complex Jordan algebras of rank 3, up to isotopy. We also provide some applications to the classification of particular types of varieties in the class defined above and of quadro-quadric Cremona transformations.