On projective varieties $n$-covered by curves of degree $δ$

As proved recently in [PT], for varieties $X^{r+1}\subset \mathbb P^N$ such that through $n\geq 2$ general points there passes an irreducible curve $C$ of degree $δ\geq n-1$ we have $N\leq π(r,n,δ+r(n-1)+2)$, where $π(r,n,d)$ is the Castelnuovo-Harris bound function for the geometric genus of an irreducible non-degenerate variety $Y^r\subset\mathbb P^{n+r-1}$ 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 $X^{r+1}\subset \mathbb P^{2r+3}$ 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 $\mathbb P^r$ we are able to classify all these object either for $r\leq 4$ 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 $\p^1\times Q^r$ 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 $\mathbb P^r$ and complex Jordan algebras of dimension $r+1$ and of rank three.

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