On polarized surfaces $(X,L)$ with $h^{0}(L)>0$, $\kappa (X)=2$, and $g(L)=q(X)$

Let $X$ be a smooth projective surface over $\mathbb {C}$ and $L$ an ample Cartier divisor on $X$. If the Kodaira dimension $\kappa (X)\leq 1$ or $\operatorname {dim}H^{0}(L)>0$, the author proved $g(L)\geq q(X)$, where $q(X)=\operatorname {dim}H^{1}(\mathcal {O}_{X})$. If $\kappa (X)\leq 1$, then the author studied $(X,L)$ with $g(L)=q(X)$. In this paper, we study the polarized surface $(X,L)$ with $\kappa (X)=2$, $g(L)=q(X)$, and $\operatorname {dim}H^{0}(L)>0$.