Cracked Polytopes and Fano Toric Complete Intersections

TL;DR

This paper introduces cracked polytopes and constructs associated Fano toric complete intersections, providing conditions for smoothings and analyzing their geometric properties within toric varieties.

Contribution

It defines cracked polytopes and constructs Fano toric complete intersections with conditions for smoothings and normal crossing divisors.

Findings

Constructed toric varieties from cracked polytopes.

Provided sufficient conditions for smoothings of these varieties.

Identified a relative anti-canonical divisor with simple normal crossings.

Abstract

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.