Applications of the Ein-Lazarsfeld criterion for spannedness of adjoint bundles (Q1323429)

In this paper using Ein-Lazarsfeld's criteria for the spannedness of the adjoint bundle \(K_ X \otimes L\), of a nef and big line bundle, \(L\), on a complex projective 3-fold \(X\) and some inequalities for the Chern classes of ample and spanned vector bundles given by the authors in \S1, they prove: If \(E\) is an ample and spanned vector bundle on a smooth 3-fold \(X\) then \(K_ X \otimes \text{det} (E)\) is spanned if \(\text{rank} (E) \geq 3\) and \(L^ 3 \geq 850\) or if \(\text{rank} (E) \geq 4\) and \(L^ 3 \geq 162\) or if \(\text{rank} (E) \geq 13\). In the last section, using again Ein-Lazarsfeld's criteria the authors study projective \(n\)-folds without rational curves.