 
Summary: Lifting of extremal rays of the Mori cone
from an ample section
Marco Andreatta and Gianluca Occhetta
Abstract
Let E be an ample vector bundle of rank r on a complex projective
manifold X such that there exists a section s 2 (E) whose zero locus
Z = (s = 0) is a smooth submanifold of the expected dimension dim X
r := n r. We study the problem of lifting birational contractions of Z
to the ambient variety, proving an extension property for blowups and
we apply our results to classify X as above such that Z is a Pbundle on
a surface with nonnegative Kodaira dimension.
1 Introduction
Let X be a smooth complex projective variety of dimension n and E an ample
vector bundle of rank r on X such that there exists a section s 2 (E) whose
zero locus Z = (s = 0) is a smooth submanifold of the expected dimension
dim Z = dim X r = n r.
A classical and natural problem is to ascend the geometric properties of Z
to get informations on the geometry of X ; for an account of the results in case
r = 1, i.e. when Z is an ample divisor see [5, Chapter 5]. In [1] we considered
the problem from the point of view of Mori theory, posing the following que
