3-Dimensional Lattice Polytopes Without Interior Lattice Points

Abstract

A theorem of Howe states that every 3-dimensional lattice polytope $P$ whose only lattice points are its vertices, is a Cayley polytope, i.e. $P$ is the convex hull of two lattice polygons with distance one. We want to generalize this result by classifying 3-dimensional lattice polytopes without interior lattice points. The main result will be, that they are up to finite many exceptions either Cayley polytopes or there is a projection, which maps the polytope to the double unimodular 2-simplex. To every such polytope we associate a smooth projective surface of genus 0.