  We introduce the notion of cracked polytope, and - making use of joint work
with Coates and Kasprzyk - construct the associated toric variety $X$ as a
subvariety of a non-singular toric variety $Y$ under certain conditions.
Restricting to the case in which this subvariety is a complete intersection, we
present a sufficient condition for a smoothing of $X$ to exist inside $Y$. We
exhibit a relative anti-canonical divisor for this smoothing of $X$, and show
that the general member is simple normal crossings.
