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Rubber tori in the boundary of expanded stable maps

Francesca Carocci, Navid Nabijou

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.

Rubber tori in the boundary of expanded stable maps

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.

Paper Structure

This paper contains 8 sections, 6 theorems, 49 equations.

Table of Contents

  1. Rubber actions on logarithmic expansions
  2. Subdivision preliminaries
  3. Tropical expansions
  4. Rubber action (toric case)
  5. Rubber action on locally-closed strata (toric case)
  6. Rubber action (general case)
  7. Pullback of the universal target
  8. Static target and the rubber action

Key Result

Theorem A

The rubber torus is the torus associated to the tropical moduli space: The action of the rubber torus on the irreducible component $Y_v$ of the expanded target is given by composing the homomorphism of tori associated to the tropical position map with the natural action $T_{\sigma_v} \curvearrowright Y_v$.

Theorems & Definitions (20)

  • Theorem A: Theorem \ref{['prop: rubber action general']} for busy people
  • Definition 1.1: MR20
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Theorem 1.5
  • proof
  • Theorem 1.6
  • proof
  • Remark 1.7
  • ...and 10 more