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 sheafLwith Kodaira dimension three such that the base locus 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. Ramanujam 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