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

Abstract: In many cases involving mirror symmetry, it is important to have concrete mirror dual families. In this paper, we construct 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 ${0, \infty} \cup \mu_6$. We have the Picard-Fuchs equation and limiting mixed Hodge structures of the singular fibers. We find that the singular fiber over $\infty$ has maximal unipotent monodromy, and $0$ is of a new type compared to the quintic case.