Cracked polytopes and Fano toric complete intersections

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 smooth 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.