• Communications of the Korean Mathematical Society
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• v.25 no.2
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• pp.161-165
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• 2010

Even though there is a formula for $h^0$(X, $\cal{O}_X(nK_X)$) for a canonical threefold X, it is not easy to compute $h^0$(X, $\cal{O}_X(nK_X)$) because the formula has a term due to singularities. In this paper, we find a way to control the term due to singularities. We show nonvanishing of plurigenus for the case when the index r in the singularity type $\frac{1}{r}$(1, -1, b) is sufficiently large.

• Communications of the Korean Mathematical Society
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• v.27 no.1
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• pp.37-46
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• 2012

It is well known that plurigenus does not vanish for a sufficiently large multiple on a canonical threefold over $\mathbb{C}$. There is Reid Fletcher formula for plurigenus. But, unlike in the case of surface of general type, it is not easy to compute plurigenus. In this paper, we in-duce a different version of Reid-Fletcher formula and show that the constant term in the induced formula has periodic properties. Using these properties we have an application to nonvanishing of plurigenus.