Rational Curves and Strictly nef Divisors on Calabi-Yau Threefolds

We give a criterion for a nef divisor $D$ to be semi-ample on a Calabi-Yau threefold $X$ when $D^3=0=c_2(X)\cdot D$ and $c_3(X)\neq 0$. As a direct consequence, we show that on such a variety $X$, if $D$ is strictly nef and $\nu(D)\neq 1$, then $D$ is ample; we also show that if there exists a Cariter divisor $D\not\equiv 0$ in the boundary of the nef cone of $X$, then $X$ contains a rational curve when its topological Euler characteristic is not $0$.