# From Cracked Polytopes to Fano Threefolds

## TL;DR

This paper introduces a method to construct Fano threefolds with specific properties from cracked polytopes using Laurent inversion, and explores classification problems related to these polytopes.

## Contribution

It develops a new construction technique for Fano threefolds from cracked polytopes and classifies possible unimodular polytope pieces in three dimensions.

## Key findings

- Constructed new examples of Fano threefolds with high Picard rank.
- Classified unimodular polytopes that can appear as parts of cracked polytopes.
- Extended understanding of the combinatorial structure of cracked polytopes.

## Abstract

We construct Fano threefolds with very ample anti-canonical bundle and Picard rank greater than one from cracked polytopes - polytopes whose intersection with a complete fan forms a set of unimodular polytopes - using Laurent inversion; a method developed jointly with Coates-Kasprzyk. We also give constructions of rank one Fano threefolds from cracked polytopes, following work of Christophersen-Ilten and Galkin. We explore the problem of classifying polytopes cracked along a given fan in three dimensions, and classify the unimodular polytopes which can occur as 'pieces' of a cracked polytope.