On sectional genus of quasi-polarized 3-folds

Let $X$ be a smooth projective variety over $\mathbb{C}$ and $L$ a nef-big (resp. ample) divisor on $X$. Then $(X,L)$ is called a quasi-polarized (resp. polarized) manifold. Then we conjecture that $g(L)\geq q(X)$, where $g(L)$ is the sectional genus of $L$ and $q(X)=\operatorname{dim}H^{1}(\mathcal{O}_{X})$ is the irregularity of $X$. In general it is unknown whether this conjecture is true or not, even in the case of $\operatorname{dim}X=2$. For example, this conjecture is true if $\operatorname{dim}X=2$ and $\operatorname{dim}H^{0}(L)>0$. But it is unknown if $\operatorname{dim}X\geq 3$ and $\operatorname{dim}H^{0}(L)>0$. In this paper, we prove $g(L)\geq q(X)$ if $\operatorname{dim}X=3$ and $\operatorname{dim}H^{0}(L)\geq 2$. Furthermore we classify polarized manifolds $(X,L)$ with $\operatorname{dim}X=3$, $\operatorname{dim}H^{0}(L)\geq 3$, and $g(L)=q(X)$.