Arithmetic Counts of Tropical Plane Curves and Their Properties

TL;DR

This paper explores the properties of arithmetic counts of tropical plane curves, establishing their invariance, deriving a Caporaso-Harris formula, and demonstrating polynomiality through floor diagrams.

Contribution

It introduces new invariance and polynomiality results for arithmetic counts of tropical curves, and develops a Caporaso-Harris formula in this context.

Findings

Arithmetic count is independent of point configuration.

A Caporaso-Harris formula for arithmetic counts is established.

Polynomiality of arithmetic counts is proven using floor diagrams.

Abstract

Recently, the first and third author proved a correspondence theorem which recovers the Levine-Welschinger invariants of toric del Pezzo surfaces as a count of tropical curves weighted with arithmetic multiplicities. In this paper, we study properties of the arithmetic count of plane tropical curves satisfying point conditions. We prove that this count is independent of the configuration of point conditions. Moreover, a Caporaso-Harris formula for the arithmetic count of plane tropical curves is obtained by moving one point to the very left. Repeating this process until all point conditions are stretched, we obtain an enriched count of floor diagrams which coincides with the tropical count. Finally, we prove polynomiality properties for the arithmetic counts using floor diagrams.