Tropical surface singularities

TL;DR

This paper classifies and analyzes singular tropical surfaces in toric threefolds, detailing the locations and nature of their singular points, and providing a comprehensive understanding of their geometric structure.

Contribution

It offers a complete classification of maximal-dimensional singular tropical surfaces and describes the possible locations of their singular points, including metric restrictions.

Findings

Singular points are located at vertices, midpoints, or baricenters of faces.

Maximal-dimensional singular tropical surfaces have finitely many singular points.

Certain faces of the surfaces may have metric restrictions affecting singularities.

Abstract

In this paper, we study tropicalisations of singular surfaces in toric threefolds. We completely classify singular tropical surfaces of maximal-dimensional type, show that they can generically have only finitely many singular points, and describe all possible locations of singular points. More precisely, we show that singular points must be either vertices, or generalized midpoints and baricenters of certain faces of singular tropical surfaces, and, in some cases, there may be additional metric restrictions to faces of singular tropical surfaces.