Pré-Publication, Document De Travail Année : 2011

On projective varieties n-covered by curves of degree δ

Résumé

As proved recently in [PT], for varieties Xr+1PN such that through n2 general points there passes an irreducible curve C of degree δn1 we have Nπ(r,n,δ+r(n1)+2), where π(r,n,d) is the Castelnuovo-Harris bound function for the geometric genus of an irreducible non-degenerate variety YrPn+r1 of degree d. A lot of examples of varieties as in the title and attaining the previous bound for the embedding dimension are constructed from Castelnuovo varieties and were thus dubbed {\it of Castelnuovo type} in [PT], where it is also proved that all extremal varieties as above are of this kind, except possibly when n>2, r>1 and δ=2n3. One of the main results of the paper is the classification of extremal varieties Xr+1P2r+3 3-covered by twisted cubics and not of Castelnuovo type. Interesting examples are provided by the so called {\it twisted cubics over complex Jordan algebras of rank 3}, as pointed out by Mukai. By relating to an extremal variety 3-covered by twisted cubics, via tangential projection, a quadro-quadric Cremona transformation in Pr we are able to classify all these object either for r4 or under the smoothness assumption. In the last case we obtain that they are either smooth rational normal scrolls (hence of Castelnuovo type) or the Segre embeddings of \p1×Qr or one of the four Lagrangian Grassmannians. We end by discussing some open problems pointing towards the equivalence of these apparently unrelated objects: extremal varieties 3-covered by twisted cubics, quadro-quadric Cremona transformations of Pr and complex Jordan algebras of dimension r+1 and of rank three.

Dates et versions

hal-00709670 , version 1 (19-06-2012)

Identifiants

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Luc Pirio, Francesco Russo. On projective varieties n-covered by curves of degree δ. 2011. ⟨hal-00709670⟩
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