The XJC-correspondence

Abstract

Abstract For any n ≥ 3 ${n\geq 3}$ , we prove that there are equivalences between • irreducible n-dimensional non-degenerate complex projective varieties 𝑿 ⊂ ℙ 2 ⁢ n + 1 ${{\boldsymbol{X}}\subset\mathbb{P}^{2n+1}}$ different from rational normal scrolls and 3-covered by cubic curves, up to projective equivalence, • n-dimensional complex Jordan algebras 𝑱 ${{\boldsymbol{J}}}$ of rank 3, up to isotopy, • quadro-quadric Cremona transformations 𝑪 : ℙ n - 1 ⇢ ℙ n - 1 ${{\boldsymbol{C}}:\mathbb{P}^{n-1}\dashrightarrow\mathbb{P}^{n-1}}$ of the complex projective space of dimension n - 1 ${n-1}$ , up to linear equivalence. These three equivalences form what we call the 𝑋𝐽𝐶 ${\mathit{XJC}}$ -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.