Rubber tori in the boundary of expanded stable maps

TL;DR

This paper provides a canonical, coordinate-free description of the rubber torus in the boundary of expanded stable maps, using tropical geometry, facilitating recursive boundary analysis and future localization applications.

Contribution

It introduces an intrinsic, tropical-based construction of the higher-rank rubber torus, advancing the understanding of boundary structures in expanded stable maps.

Findings

Canonical construction of the rubber torus from tropical moduli space

Encoding of torus action via linear tropical position map

Identification of maps differing by rubber action through 2-morphisms

Abstract

We investigate torus actions on logarithmic expansions in the context of enumerative geometry. Our main result is an intrinsic and coordinate-free description of the higher-rank rubber torus appearing in the boundary of the space of expanded stable maps. The rubber torus is constructed canonically from the tropical moduli space, and its action on each stratum of the expanded target is encoded in a linear tropical position map. The presence of 2-morphisms in the universal target forces expanded stable maps differing by the rubber action to be identified. This provides the first step towards a recursive description of the boundary of the expanded moduli space, with future applications including localisation and rubber calculus.