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

Abstract

We construct a natural crepant resolution 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 compute an explicit height function producing the MPCP desingularization of the $B$-model toric ambient space. We compute the limiting mixed Hodge structures of the singular fibers. We show that the singular fiber over $\infty$ has maximal unipotent monodromy, whereas the singular fiber over $0$ is of a new type compared to the quintic case.