Models of rationally connected manifolds

Abstract

We study rationally connected (projective) manifolds $X$ via the concept of a model $(X,Y)$, where $Y$ is a smooth rational curve on $X$ having ample normal bundle. Models are regarded from the view point of Zariski equivalence, birational on $X$ and biregular around $Y$. Several numerical invariants of these objects are introduced and a notion of minimality is proposed for them. The important special case of models Zariski equivalent to $(P^n,line)$ is investigated more thoroughly. When the (ample) normal bundle of $Y$ in $X$ has minimal degree, new such models are constructed via special vector bundles on $X$. Moreover, the formal geometry of the embedding of $Y$ in $X$ is analysed for some non-trivial examples.