A classification of smooth convex 3-polytopes with at most 16 lattice points

TL;DR

This paper classifies all smooth convex lattice 3-polytopes with up to 16 lattice points, revealing 103 distinct types, mostly Cayley polytopes and their subdivisions, and describes related toric threefold embeddings.

Contribution

It provides a complete classification of smooth convex lattice 3-polytopes with at most 16 lattice points, including their isomorphism types and associated toric embeddings.

Findings

103 distinct polytopes meeting criteria

99 are strict Cayley polytopes

4 are subdivisions or blow-ups of Cayley polytopes

Abstract

We provide a complete classification up to isomorphism of all smooth convex lattice 3-polytopes with at most 16 lattice points. There exist in total 103 different polytopes meeting these criteria. Of these, 99 are strict Cayley polytopes and the remaining 4 are obtained as inverse stellar subdivisions of such polytopes. We derive a classification, up to isomorphism, of all smooth embeddings of toric threefolds in $\mathbb{P}^N$ where $N\le 15$. Again we have in total 103 such embeddings. Of these, 99 are projective bundles embedded in $\mathbb{P}^N$ and the remaining 4 are blow-ups of such toric threefolds.