Some remarks on the cohomology of an invertible sheaf

AbstractThis note is mainly devoted to three-folds. In the first part we give an example of an invertible sheaf. ℒ with Kodaira dimension three such that the base loous of the complete linear system |ℒ| has codimension two and H2 (ℒ−1)≠(0). This example gives a negative answer to a question posed by C. P. Bamanujam in [R]. Moreover we obtain a result on the vanishing of the second cohomology group.In the second part we extend to the case of dimension greater than two same results proved by Zariski in [Z2] for surfaces. These results concern the cohomology of high multiples of an effective divisor