Embeddings and immersions of tropical curves

TL;DR

This paper develops methods for immersing and embedding tropical curves in Euclidean spaces, introduces the tropical crossing number, and applies these results to construct algebraic curves with specific properties.

Contribution

It introduces the tropical crossing number, establishes bounds for it, and demonstrates how to construct algebraic curves from tropical curves using these immersions.

Findings

Tropical crossing number is at most quadratic in the number of edges.

Explicit computations of crossing numbers for genus up to two.

Construction of algebraic curves from tropical immersions.

Abstract

We construct immersions of trivalent abstract tropical curves in the Euclidean plane and embeddings of all abstract tropical curves in higher dimensional Euclidean space. Since not all curves have an embedding in the plane, we define the tropical crossing number of an abstract tropical curve to be the minimum number of self-intersections, counted with multiplicity, over all its immersions in the plane. We show that the tropical crossing number is at most quadratic in the number of edges and this bound is sharp. For curves of genus up to two, we systematically compute the crossing number. Finally, we use our immersed tropical curves to construct totally faithful nodal algebraic curves via lifting results of Mikhalkin and Shustin.