Ample subvarieties and rationally connected fibrations (Q930546)

Let \(Y\) be a smooth projective variety admitting a family of rational curves \(V_Y\) such that through a general point of \(Y\) there is a curve parametrized by \(V_Y\). To such a family is possible to associate an equivalence relation on \(Y\) and a proper fibration defined on an open set whose very general fibers are equivalence classes of this relation. Assume now that \(Y \subset X\) is a subvariety with ample normal bundle; the paper under review deals mainly with the problem of extending fibrations as above to the ambient manifold. The first result is obtained assuming just that the restriction map \(N^1(X) \to N^1(Y)\) is surjective; in this case, if \(V_X\) is a family of rational curves on \(X\) which restricts to \(V_Y\) then there is a commutative diagram \[ \begin{matrix} Y & \begin{matrix} \pi \\ \dashrightarrow \end{matrix} & Y /\!\!/_{V_Y} \\ i\downarrow & & \delta \downarrow \\ X & \begin{matrix} \Phi \\ \dashrightarrow \end{matrix} & X /\!\!/_{V_X}\end{matrix} \] where \(\Phi\) and \(\pi\) are the fibrations associated to the families and \(\delta\) is a surjective morphism. One of the main results of the paper then establishes conditions which ensure that the morphism \(\delta\) appearing in the above diagram is generically finite. Two sufficient conditions are found, namely (i) \(V_X\) is an unsplit family; (ii) codim\(_YX < \dim Y - \dim Y /\!\!/_{V_Y}\). This results is then applied to the extension problem for Mori elementary contraction of fiber type; the best results are obtained for projective bundles and quadric fibration. For instance, among other results, the author prove that if \(Y\) is the zero scheme of a regular section of an ample vector bundle on \(X\), has a projective bundle structure \(\pi:Y \to Z\) and codim\(_YX < \dim Y - \dim Z\) then \(X\) admits a projective bundle structure \(\tilde \pi:X \to Z\) extending that of \(Y\).