Lifting tropical self intersections

TL;DR

This paper investigates the tropicalization of plane curve intersections with identical tropicalizations, characterizing the structure of the resulting tropical divisors and their realizability, especially for genus at most 1.

Contribution

It provides a detailed description of the space of tropical divisors from intersecting plane curves and introduces a combinatorial method for constructing realizable divisors.

Findings

The set of tropical divisors forms a pure dimensional balanced polyhedral complex.

All tropical divisors in the expected dimension are realizable when genus ≤ 1.

A new combinatorial tool for constructing realizable tropical divisors is introduced.

Abstract

We study the tropicalization of intersections of plane curves, under the assumption that they have the same tropicalization. We show that the set of tropical divisors that arise in this manner is a pure dimensional balanced polyhedral complex and compute its dimension. When the genus is at most 1, we show that all the tropical divisors that move in the expected dimension are realizable. As part of the proof, we introduce a combinatorial tool for explicitly constructing large families of realizable tropical divisors.