A lower bound for $K_X L$ of quasi-polarized surfaces $(X,L)$ with non-negative Kodaira dimension

http://dx.doi.org/10.4153/CJM-1998-059-x
Canad. J. Math. 50(1998), 1209-1235

  Published:1998-12-01
 Printed: Dec 1998

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 $\kappa(X)\geq 0$, then $K_{X}L\geq 2q(X)-4$, where $q(X)$ 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 $h^{0}(L)\geq 2$, or (3) the case in which $\kappa(X)=2$, $X$ is minimal, $h^{0}(L)=1$, and $L$ satisfies some conditions.