Tropical curves with a singularity in a fixed point

Hannah Markwig, Thomas Markwig, Eugenii Shustin

Research output: Contribution to journalArticlepeer-review

Abstract

In this paper, we study tropicalisations of families of plane 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. 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 generic singular tropical curves enhanced with refined tropical limits and construct canonical simple parameterisations for them, explaining the above metric condition.

Original languageEnglish
Pages (from-to)383-418
Number of pages36
JournalManuscripta Mathematica
Volume137
Issue number3
DOIs
StatePublished - Mar 2012

Keywords

  • 05B35
  • 14T05
  • 51M20

Fingerprint

Dive into the research topics of 'Tropical curves with a singularity in a fixed point'. Together they form a unique fingerprint.

Cite this