Abstract
For a canonical threefold X, we know that h 0(X, 𝒪 X (nK X )) ≥ 1 for a sufficiently large n. When χ(𝒪 X ) > 0, it is not easy to get such an integer n. Fletcher showed that h 0(X, 𝒪 X (12K X )) ≥ 1 and h 0(X, 𝒪 X (24K X )) ≥ 2 when χ(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 )) ≥ 2.
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Acknowledgment
This paper was supported by KOSEF R01-1999-00004.