Few smooth d-polytopes with n lattice points

TL;DR

This paper establishes finiteness results for smooth d-polytopes with a fixed number of lattice points and applies these results to classify certain embeddings of toric varieties, providing a comprehensive enumeration for specific cases.

Contribution

It proves finiteness of smooth d-polytopes with a given number of lattice points and enumerates all smooth 3-polytopes with up to 12 lattice points.

Findings

Finitely many smooth d-polytopes exist for fixed d and lattice points.

Complete enumeration of smooth 3-polytopes with up to 12 lattice points.

Finiteness of embeddings of Q-factorial toric varieties into projective space.

Abstract

We prove that, for fixed n there exist only finitely many embeddings of Q-factorial toric varieties X into P^n that are induced by a complete linear system. The proof is based on a combinatorial result that for fixed nonnegative integers d and n, there are only finitely many smooth d-polytopes with n lattice points. We also enumerate all smooth 3-polytopes with at most 12 lattice points. In fact, it is sufficient to bound the singularities and the number of lattice points on edges to prove finiteness.