Published online by Cambridge University Press: 20 November 2018
Let $X$ be a smooth projective surface over the complex number field and let
$L$ be a nef-big divisor on
$X$. Here we consider the following conjecture; If the Kodaira dimension
$\kappa (X)\ge 0$, then
${{K}_{X}}L\,\ge \,2q(X)\,-\,4$, where
$q\left( X \right)$ is the irregularity of
$X$. In this paper, we prove that this conjecture is true if (1) the case in which
$\kappa (X)=0$ or 1, (2) the case in which
$\kappa (X)=2$ and
${{h}^{0}}(L)\,\ge \,2$
, or (3) the case in which
$\kappa (X)=2$,
$X$ is minimal,
${{h}^{0}}(L)\,=\,1$
, and
$L$ satisfies some conditions.