On projective varieties n-covered by curves of degree δ
Résumé
As proved recently in [PT], for varieties Xr+1⊂PN such that through n≥2 general points there passes an irreducible curve C of degree δ≥n−1 we have N≤π(r,n,δ+r(n−1)+2), where π(r,n,d) is the Castelnuovo-Harris bound function for the geometric genus of an irreducible non-degenerate variety Yr⊂Pn+r−1 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=2n−3. One of the main results of the paper is the classification of extremal varieties Xr+1⊂P2r+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 r≤4 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.