# About the nef threshold of the divisor L

I have this definition:

Let $S$ a complex algebraic projective surface then we can consider this set $A=\{t\in \mathbb{R} |H+tK_S\in \overline{\mathcal{A(S)}}\}$ where $H$ is an ample divisor of $S$, $K_S$ is the canonical divisor of $S$ and $\overline{\mathcal{A(S)}}$ is the nef cone of $S$.
Set $t_0=\sup A$. It's easy to see that $0<t_0<+\infty$.

Suppose that $S$ is a minimal surface with $K_S$ not nef. So by Kawamata Rationality Theorem we know that $t_0=\frac{u}{v}$ with $u,v$ positive integers.
Take $H$ an ample divisor on $S$ and define $L=vH+uK_S$. Obviously $L$ is not an ample divisor but can i say that $L$ is nef? My idea is to prove that $H+t_0K_S$ lies on the boundary of the ample cone.

• yes sorry i mean kawamata rationality theorem – dario Oct 4 '14 at 13:12
• There are several mistakes in the question --- in particular the definition of $t_0$ doesn't make sense as written. But anyway, yes $L$ is nef, because of the definition of $t_0$ and the fact that $\overline{\mathcal A (S)}$ is closed. – user64687 Oct 4 '14 at 13:26