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

Abstract

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.

25 pages

References in corpus (1)

• Remarks on the second sectional geometric genus of quasi-polarized manifolds and their applications