Equivalence classes for smooth Fano polytopes

TL;DR

This paper classifies equivalence classes of smooth Fano polytopes, introduces notions of F- and I-isolation, and provides complete classifications for certain dimensions and vertex counts.

Contribution

It introduces F- and I-equivalence classes and characterizes these classes for 5-dimensional polytopes and those with n+3 vertices, also constructing I-isolated examples.

Findings

Classified all F- and I-equivalence classes for 5-dimensional smooth Fano polytopes.

Provided a complete characterization of equivalence classes for polytopes with n+3 vertices.

Constructed a family of I-isolated smooth Fano polytopes.

Abstract

Let $\mathcal{F}(n)$ be the set of smooth Fano $n$-polytopes up to unimocular equivalence. In this paper, we consider the F-equivalence or I-equivalence classes for $\mathcal{F}(n)$ and introduce F-isolated or I-isolated smooth Fano $n$-polytopes. First, we describe all of F-equivalence classes and I-equivalence classes for $\mathcal{F}(5)$. We also give a complete characterization of F-equivalence classes (I-equivalence classes) for smooth Fano $n$-polytopes with $n+3$ vertices and construct a family of I-isolated smooth Fano polytopes.