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

TL;DR

This paper establishes lower bounds on the global sections of multiple adjoint bundles on polarized 3-folds, providing positivity results and classifying cases with maximal Kodaira dimension.

Contribution

It offers new bounds for sections of adjoint bundles on threefolds and classifies cases with maximal Kodaira dimension and small section counts.

Findings

If $ abla(K_{X}+L)$ is between 0 and 2, then $h^{0}(K_{X}+L)$ is positive.

If $ abla(K_{X}+L)=3$, then $h^{0}(2(K_{X}+L)) \\geq 3.

Classification of $(X,L)$ with $ abla(K_{X}+L)=3$ and small $h^{0}(2(K_{X}+L))$.

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.