On empty lattice simplices in dimension 4

TL;DR

This paper classifies all empty lattice simplices in four dimensions, showing they relate to cyclic quotient singularities and most have small width, advancing understanding of lattice geometry.

Contribution

Provides an almost complete classification of 4D empty lattice simplices, linking them to cyclic quotient singularities and bounding their width.

Findings

All 4D empty lattice simplices correspond to cyclic quotient singularities.

Most simplices have width at most 2.

The classification relies on conjectural and proven results in algebraic geometry.

Abstract

We give an almost complete classification of empty lattice simplices in dimension 4 using the conjectural results of Mori-Morrison-Morrison, later proved by Sankaran and Bober. In particular, all of these simplices correspond to cyclic quotient singularities, and all but finitely many of them have width bounded by 2.