Geometry of the mirror models dual to the complete intersection of two cubics

In many cases involving mirror symmetry, it is important to have concrete mirror dual families. In this paper, we construct a natural crepant resolutions of the Batyrev-Borisov mirror dual family to the complete intersection of two cubic hypersurfaces in ${\mathbb{P}}^5$. It is, similarly to the mirrors of quintic threefolds, a family over ${\mathbb{P}}^1$ with singular fibers over the set $\{0,\mathrm{\infty }\}\cup {\mu }_6$. We compute the Picard-Fuchs equation and limiting mixed Hodge structures of the singular fibers. We find that the singular fiber over $\mathrm{\infty }$ has maximal unipotent monodromy, and $0$ is of a new type compared to the quintic case.