Explicit n-connectedness by irreducible curves and the XJC-correspondence

When an irreducible projective variety $X$ is embedded in some projective space $\mathbb P^N $ (or more generally when an arbitrary Cartier divisor $D$ is fixed on $X$), one can consider the property of being generically $n$–connected by irreducible curves of a fixed degree $\delta$ (with respect to $D$ in the abstract case), that is through $n\geq 2$ general points of $X\subset\mathbb P^N$ there passes an irreducible curve of degree $\delta$ contained in $X$.
Natural constraints for the existence of such varieties immediately appear. We shall present results of joint work with Luc Pirio showing that there is a natural bound on $N$ (or on $\dim(|D|)$) depending on $\dim(X)$, $n$ and $\delta\geq n-1$ such that the boundary examples are rational varieties which are $n$–connected by smooth rational curves of degree $\delta$ in such a way that there exists a unique curve of the family passing though $n$ general points.
Natural constraints for the existence of such varieties immediately appear. We shall present results of joint work with Luc Pirio showing that there is a natural bound on $N$ (or on $\dim(|D|)$) depending on $\dim(X)$, $n$ and $\delta\geq n-1$ such that the boundary examples are rational varieties which are $n$–connected by smooth rational curves of degree $\delta$ in such a way that there exists a unique curve of the family passing though $n$ general points.
We shall also present an application of the previous bound to the top intersection of nef divisors on a variety X as above, generalizing a result of Fano. Moreover we shall illustrate the proof for the classification of the boundary cases for $n=2$ to show a projective incarnation Mori's famous characterization of projective spaces as the unique projective manifolds with ample tangent bundle.
The classification of varieties attaining the bound of the embedding dimension is related to that of Castelnuovo varieties except for $\delta= 2n-3$, $n\geq 3$. The first exceptional case being that of extremal varieties with $n=3$ and $\delta=3$, that is m-dimensional varieties $X\subset P^{2m+1}$ such that through 3 general points there passes a twisted cubic contained in it. We shall explain and outline the proof of the so called $XJC$ correspondence asserting that for every $m\geq 3$ there are equivalences between: the previous class of algebraic varieties different from rational normal scrolls; quadro-quadric Cremona transformations of $P^{m-1}$, modulo projective equivalence; $m$ dimensional complex Jordan algebras of rank 3, modulo isotopy.