On a plurigenus of a canonical threefold

摘要

For a canonical threefold X, we know that h(0)(X, O-x, (nK(x))) greater than or equal to 1 for a sufficiently large n. When chi(O-x) > 0, it is not easy to get such an integer n. Fletcher showed that h(0)(X, O-x(12K(x))) greater than or equal to 1 and h(0)(X, O-x(24K(x))) greater than or equal to 2 when chi(O-x) = 1. He inquired about existence of a canonical threefold with given conditions which shows the result sharp. We show that such an example does not exist. Using a different technique, we prove h(0)(X, O-x (12K(x))) greater than or equal to 2. [References: 4]