From cracked polytopes to Fano threefolds

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.