Realizability of Tropical Curves in a Plane in the Non-Constant Coefficient Case

TL;DR

This paper develops an algorithm to determine if a tropical curve in a plane over a non-constant coefficient field is realizable as an algebraic curve, and characterizes the space of such realizable curves.

Contribution

It introduces a complete algorithm for relative realizability of tropical curves in a plane and provides conditions for realizability in specific cases.

Findings

The space of realizable curves forms an abstract polyhedral set.

Necessary and sufficient conditions are established for certain tropical complexes.

A non-algorithmic solution is provided for complexes with one bounded edge.

Abstract

Let X be a plane in a torus over an algebraically closed field K, with tropicalization the matroidal fan Sigma. In this paper we present an algorithm which completely solves the question whether a given one-dimensional balanced polyhedral complex in Sigma is relatively realizable, i.e. whether it is the tropicalization of an algebraic curve, over the field of Puiseux series over K, in X. The algorithm implies that the space of all such relatively realizable curves of fixed degree is an abstract polyhedral set. In the case when X is a general plane in 3-space, we use the idea of this algorithm to prove some necessary and some sufficient conditions for relative realizability. For 1-dimensional polyhedral complexes in Sigma that have exactly one bounded edge, passing through the origin, these necessary and sufficient conditions coincide, so that they give a complete non-algorithmic solution of the relative realizability problem.