How to repair tropicalizations of plane curves using modifications

TL;DR

This paper introduces tropical modifications as a method to improve embeddings of plane curves in tropical geometry, enabling better reflection of the curve's geometry, especially for elliptic cubics, through an effective algorithm and theoretical insights.

Contribution

It develops a novel approach using tropical modifications to repair embeddings of plane curves, with a focus on elliptic cubics and their j-invariants, supported by an algorithm and elementary proof.

Findings

Effective algorithm for re-embedding elliptic cubics in dimension 4

Tropical modifications improve the reflection of the j-invariant in tropicalizations

Elementary proof using local discriminants of the A-discriminant

Abstract

Tropical geometry is sensitive to embeddings of algebraic varieties inside toric varieties. The purpose of this paper is to advertise tropical modifications as a tool to locally repair bad embeddings of plane curves, allowing the re-embedded tropical curve to better reflect the geometry of the input curve. Our approach is based on the close connection between analytic curves (in the sense of Berkovich) and tropical curves. We investigate the effect of these tropical modifications on the tropicalization map defined on the analytification of the given curve. Our study is motivated by the case of plane elliptic cubics, where good embeddings are characterized in terms of the j-invariant. Given a plane elliptic cubic whose tropicalization contains a cycle, we present an effective algorithm, based on non-Archimedean methods, to linearly re-embed the curve in dimension 4 so that its tropicalization reflects the j-invariant. We give an alternative elementary proof of this result by interpreting the initial terms of the A-discriminant of the defining equation as a local discriminant in the Newton subdivision.