Tropical curves with a singularity in a fixed point

TL;DR

This paper analyzes tropicalizations of algebraic curves with fixed singularities, classifying their tropical types and exploring algebraic preimages, thereby advancing understanding of tropical singularities and their algebraic counterparts.

Contribution

It provides a classification of maximal dimensional tropical singularities and links them to algebraic preimages, extending tropical geometry theory.

Findings

Singularities tropicalize to vertices of higher valence or multiplicity, or to weighted edges.

Classified maximal types of tropical singular curves: crossings, 3-valent vertices, and weighted edges.

Connected tropical singularities to algebraic preimages, enriching the tropical-algebraic correspondence.

Abstract

In this paper, we study tropicalisations of families of curves with a singularity in a fixed point. The tropicalisation of such a family is a linear tropical variety. We describe its maximal dimensional cones using results about linear tropical varieties from Ardila and Klivans and from Feichtner and Sturmfels. We show that a singularity tropicalises either to a vertex of higher valence or of higher multiplicity, or to an edge of higher weight. We then classify maximal dimensional types of singular tropical curves. For those, the singularity is either a crossing of two edges, or a 3-valent vertex of multiplicity 3, or a point on an edge of weight 2 whose distances to the neighbouring vertices satisfy a certain metric condition. We also study algebraic preimages of our singular tropical curves.