Projections of tropical varieties and their self-intersections

TL;DR

This paper explores how tropical varieties behave under projections, focusing on algebraic properties, dual subdivisions, and the occurrence of self-intersections, especially for curves, with bounds and constructions provided.

Contribution

It characterizes algebraic properties of projected tropical varieties and introduces bounds and constructions for self-intersections in projections.

Findings

Characterization of algebraic properties of projected tropical varieties.

Bounds established for self-intersections of tropical curves.

Constructions demonstrating many self-intersections.

Abstract

We study algebraic and combinatorial aspects of (classical) projections of $m$-dimensional tropical varieties onto $(m+1)$-dimensional planes. Building upon the work of Sturmfels, Tevelev, and Yu on tropical elimination as well as the work of the authors on projection-based tropical bases, we characterize algebraic properties of the relevant ideals and provide a characterization of the dual subdivision (as a subdivision of a fiber polytope). This dual subdivision naturally leads to the issue of self-intersections of a tropical variety under projections. For the case of curves, we provide some bounds for the (unweighted) number of self-intersections of projections onto the plane and give constructions with many self-intersections.