Models of rationally connected manifolds. (Q1396408)

A model (of rationally connected manifolds) is a pair \((X,Y)\) where \(X\) is a projective manifold and \(Y\subset X\) is a smooth rational curve with ample normal bundle in \(X\). Two models \((X,Y)\) and \((X',Y')\) are said to be Zariski equivalent if there exists a birational isomorphism \(\phi: X \to X'\) which is biregular on open subsets containing \(Y\) and \(Y'\) and such that \(\phi(Y)=Y'\). In this paper the authors study the models from the point of view of Zariski equivalence. In particular they analyze models Zariski equivalent to \((X,Y)\) where either \(X\) is a projective space and \(Y\) is a line or \(X ={\mathbb P}(E)\) is a projective bundle over a smooth curve and \(Y\) is a quasi-line.