On homaloidal polynomial functions of degree 3 and prehomogeneous vector spaces

Chaput, Pierre-Emmanuel; Sabatino, Pietro

doi:10.48550/arxiv.1011.5975

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“…In [15] it is proved the surprising and nice result that there are only four examples of quadroquadric Cremona transformations with smooth irreducible base locus. These four examples are related to the so called Severi varieties and are linked to the four simple complex Jordan algebras of hermitian 3 × 3 matrices with coefficients in the complexification of the four real division algebras R, C, H and O, see [15], [40], and also [17], [9] and Corollary 5.9 here.…”

Section: Introductionmentioning

confidence: 99%

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

Pirio,

Russo

2011

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For any n ≥ 3, we prove that there are equivalences between • irreducible n-dimensional non degenerate projective varieties X ⊂ P 2n+1 different from rational normal scrolls and 3-covered by rational cubic curves, up to projective equivalence; • quadro-quadric Cremona transformations of 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. cubic over a rank three complex Jordan algebra (Theorem 3.7). Some particular versions of the XJC-correspondence are the following: cartesian products of varieties 3-covered by twisted cubics correspond to direct product Jordan algebras of rank three and to the so called elementary quadratic transformations (Proposition 4.3); smooth varieties 3-covered by twisted cubics, modulo projective equivalence, are in bijection with semi-simple rank three Jordan algebras, modulo isotopy, and with semi-special quadro-quadric Cremona transformations, modulo linear equivalence (Theorem 4.4).The XJC-correspondence is extended in Section 4.3 to cover some degenerated cases: rational normal scrolls, Jordan algebras with a cubic norm and 'fake' quadro-quadric Cremona transformations, respectively. Moreover, the XJC-correspondence leads us to some new constructions and definitions. The theory of the radical and the semisimple part of a Jordan algebra suggested the definitions of semi-simple part, semi-simple rank and semi-simple dimension of a quadro-quadric Cremona transformation, or of an extremal variety 3-covered by twisted cubics, providing for instance a general Structure Theorem for these maps, see Theorem 5.16. As an application we prove in Corollary 5.9 that every homaloidal polynomial f of degree 3 defining a quadro-quadric Cremona transformation whose ramification locus scheme is cut out by f is, modulo linear equivalence, the norm of a rank 3 semi-simple Jordan algebra, providing a new short proof of [17, Theorem 3.10] and of [9, Theorem 2, Corollary 4].The paper is organized as follows. In Section 1 we introduce some notation which is not standard. In Section 2 we define precisely the objects studied giving some examples: the X-world consisting of extremal varieties 3-covered by twisted cubics; the C-world consisting of quadro-quadric Cremona transformations and the J-world consisting of rank three complex Jordan algebras. Moreover the natural equivalence relations: projective equivalence, linear equivalence, respectively isotopy are introduced as well the notion of cubic Jordan pair. In Section 3 we define the correspondences between the three sets modulo equivalences. First from the J-world to the X and C worlds. Then from the C-world to the X-world. We prove the equivalence between...

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