Effective non-vanishing of global sections of multiple adjoint bundles for polarized 3-folds

Details Effective non-vanishing of global sections of multiple adjoint bundles for polarized 3-folds Effective non-vanishing of global sections of multiple adjoint bundles for polarized 3-folds

2009-10-15

arxiv.org/abs/0910.2785v1

Mathematics

Algebraic Geometry

25 pages

Scientific paper

Let $X$ be a smooth complex projective variety of dimension three and let $L$ be an ample line bundle on $X$. In this paper, we provide a lower bound of the dimension of the global sections of $m(K_{X}+L)$ under the assumption that $\kappa(K_{X}+L)$ is non-negative. In particular, we get the following: (1) if $\kappa(K_{X}+L)$ is greater than or equal to zero and less than or equal to two, then $h^{0}(K_{X}+L)$ is positive. (2) If $\kappa(K_{X}+L)$ is equal to three, then $h^{0}(2(K_{X}+L))$ is greater than or equal to three. Moreover we get a classification of $(X,L)$ such that $\kappa(K_{X}+L)$ is equal to three and $h^{0}(2(K_{X}+L))$ is equal to three or four.

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