2011/09/16 by Luc Pirio, Pirio, Luc, Francesco Russo +1
Computer Science · Mathematics · Physics and Astronomy · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Nonlinear Waves and Solitons #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.1109.3566
openalex publication_date 2011/09/16 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
As proved recently in [PT], for varieties Xr+1⊂ \mathbb 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⊂\mathbb 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⊂ \mathbb 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 \mathbb 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 \mathbb Pr and complex Jordan algebras of dimension r+1 and of rank three.