Rational curves and strictly nef divisors on Calabi--Yau threefolds

Abstract

We give a criterion for a nef divisor $D$ to be semiample on a Calabi--Yau threefold $X$ when $D^3=0=c_2(X)\cdot D$ and $c_3(X)\ne 0$. As a direct consequence, we show that on such a variety $X$, if $D$ is strictly nef and $\hat{I}½(D)\ne 1$, then $D$ is ample; we also show that if there exists a nef non-ample divisor $D$ with $D\not\equiv 0$, then $X$ contains a rational curve when its topological Euler characteristic is not $0$.

18 pages, comments are welcome!