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Extremal varieties 3-rationally connected by cubics, quadro-quadric Cremona transformations and rank 3 Jordan algebras

For any $n\geq 3$, we prove that there exist equivalences between these apparently unrelated objects: irreducible $n$-dimensional non degenerate projective varieties $X\subset \mathbb P^{2n+1}$ different from rational normal scrolls and 3-covered by twisted cubic curves, up to projective equivalence; quadro-quadric Cremona transformations of $ \mathbb P^{n-1}$, up to linear equivalence; $n$-dimensional complex Jordan algebras of rank three, up to isotopy. We also provide some applications to the classification of particular classes of varieties in the class defined above and of quadro-quadric Cremona transformations, proving also a structure theorem for these birational maps and for varieties 3-covered by twisted cubics by reinterpreting for these objects the solvability of the radical of a Jordan algebra.