On the Generation of Certain Bundles of the Projective Space

Extending a result of Manivel, we prove the following: THEOREM. Suppose $\sum\limits_i {b_i \geqslant } \sum\limits_i {a_i } + n$ and $$\sum\limits_i {b_i } n + d_i d_i - 1] \geqslant \sum\limits_i {a_i } n + l_i l_i - 1] + n.$$ Then the kernel E(d) of the general morphism: $$\mathop \oplus \limits_{i = 1}^v (Bi \otimes O_{P^n } (d_i )) \to \mathop \oplus \limits_{j = 1}^v (A_j \otimes O_{P^n } (l_i ))$$ (l ₁>...>l _s>d ₁>...>d _v) is a globally generated vector bundle, except for at most finitely many sets $\left\{ {b_i ,a_i } \right\}$ .