Calabi-Yau-threefolds with Picard number $\rho (X)=2$ and their Kaehler cone II

We prove the rationality of the K"ahler cone and the positivity of $c_2(X)$, if $X$ is a Calabi-Yau-threefold with $\rho (X)=2$ and has an embedding into a ${\text{bb}P}^n$-bundle over ${\text{bb}P}^m$ in the cases $(n,m)=(1,3),(3,1)$. The case $(n,m)=(2,2)$ has been done in the first part of this paper. Moreover, if $(n,m)=(3,1)$, we describe the ‘other’ contraction different from the projection.