Tropical curves with a singularity in a fixed point

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.

34 pages, 26 figrues; replaced by the published version

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• Matroid polytopes, nested sets and Bergman fans
• A Field of Generalised Puiseux Series for Tropical Geometry
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Cited by in corpus (9)

• Lifting tropical bitangents
• Generic Tropical Varieties
• Tropicalized quartics and canonical embeddings for tropical curves of genus 3
• Enumeration of Complex and Real Surfaces via Tropical Geometry
• The Newton polygon of a planar singular curve and its subdivision
• The tropical discriminant in positive characteristic
• Secondary fans and Tropical Severi varieties
• On real tropical bases and real tropical discriminants
• The tropical discriminant of a polynomial map on a plane