Singularities of Discrete Improper Indefinite Affine Spheres

TL;DR

This paper studies the singularities of discrete improper affine spheres, classifying certain edges and vertices as discrete cuspidal edges and swallowtails, and provides characterizations of ruled discrete improper affine spheres.

Contribution

It introduces a classification of singularities in discrete improper affine spheres and characterizes ruled cases, advancing understanding of discrete affine differential geometry.

Findings

Discrete cuspidal edges identified as singular edges

Discrete swallowtails identified as singular vertices

Characterizations of ruled discrete improper affine spheres

Abstract

We consider in this paper discrete improper affine spheres based on asymptotic nets. In this context, we distinguish the discrete edges and vertices that must be considered singular. The singular edges can be considered as discrete cuspidal edges, while some of the singular vertices can be considered as discrete swallowtails. The classification of singularities of discrete nets is a quite difficult task, and our results can be considered as a fisrt step in this direction. We also prove some characterizations of ruled discrete improper affine spheres which are analogous to the smooth case.