On moduli spaces of roots in algebraic and tropical geometry

TL;DR

This paper constructs a tropical moduli space for roots of divisors on tropical curves, explores its relation to algebraic moduli spaces, and reveals combinatorial structures governed by graph flows.

Contribution

It introduces a new tropical moduli space for roots, linking tropical and algebraic geometry through the skeleton of Jarvis's moduli space and graph flow posets.

Findings

The tropical moduli space parametrizes roots of divisors on tropical curves.

The space relates to the skeleton of Jarvis's moduli space of limit roots.

The combinatorics are governed by the poset of flows on graphs.

Abstract

In this paper we construct a tropical moduli space parametrizing roots of divisors on tropical curves. We study the relation between this space and the skeleton of Jarvis moduli space of nets of limit roots on stable curves. We show that the combinatorics of the moduli space of tropical roots is governed by the poset of flows, a poset parametrizing certain flows on graphs.