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Balancing properties of tropical moduli maps

Karl Christ, Xiang He, Ilya Tyomkin

TL;DR

This work develops a higher-dimensional tropicalization framework for families of curves by tropicalizing the base along the skeleton of a strictly semistable pair and producing a family of parameterized tropical curves over this skeleton. The main contribution is a balancing (harmonic/quasi-harmonic) property for the induced tropical moduli map into M^{trop}_{g,n,nabla}, together with a local surjectivity condition on codimension-one strata. This yields a new relative realizability criterion for parameterized tropical curves and supports applications such as irreducibility results for Severi and Hurwitz spaces in arbitrary characteristic. The approach connects degenerations, tropicalization, and moduli theory without relying on deformation theory, enabling combinatorial and geometric insights into liftability and moduli questions in tropical geometry.

Abstract

Given a family of parameterized algebraic curves over a strictly semistable pair, we show that the simultaneous tropicalization of the curves in the family forms a family of parameterized tropical curves over the skeleton of the strictly semistable pair. We show that the induced tropical moduli map satisfies a certain balancing condition, which allows us to describe properties of its image and deduce a new liftability criterion.

Balancing properties of tropical moduli maps

TL;DR

This work develops a higher-dimensional tropicalization framework for families of curves by tropicalizing the base along the skeleton of a strictly semistable pair and producing a family of parameterized tropical curves over this skeleton. The main contribution is a balancing (harmonic/quasi-harmonic) property for the induced tropical moduli map into M^{trop}_{g,n,nabla}, together with a local surjectivity condition on codimension-one strata. This yields a new relative realizability criterion for parameterized tropical curves and supports applications such as irreducibility results for Severi and Hurwitz spaces in arbitrary characteristic. The approach connects degenerations, tropicalization, and moduli theory without relying on deformation theory, enabling combinatorial and geometric insights into liftability and moduli questions in tropical geometry.

Abstract

Given a family of parameterized algebraic curves over a strictly semistable pair, we show that the simultaneous tropicalization of the curves in the family forms a family of parameterized tropical curves over the skeleton of the strictly semistable pair. We show that the induced tropical moduli map satisfies a certain balancing condition, which allows us to describe properties of its image and deduce a new liftability criterion.
Paper Structure (14 sections, 3 theorems, 26 equations)

This paper contains 14 sections, 3 theorems, 26 equations.

Table of Contents

  1. Introduction
  2. Notation and terminology
  3. Polyhedral complexes
  4. Tropical curves
  5. Abstract and parameterized tropical curves
  6. Moduli of parameterized tropical curves
  7. Families of parameterized tropical curves over polyhedral complexes
  8. Families of curves
  9. Toric varieties and parameterized curves
  10. Tropicalization of parameterized curves
  11. Strictly semistable pairs and their tropicalization
  12. Main theorem
  13. Proof of part (1) Theorem \ref{['thm:main thm']}
  14. Proof of Part (2) of Theorem \ref{['thm:main thm']}

Key Result

Corollary 1

Let $X$ be a toric variety and $f\colon {\mathcal{C}} \rightarrow X$ a family of parameterized curves defined over the field $K$. Assume that the base $B$ of the family is quasi-projective, and consider the set of tropicalizations Let $M_{[\Theta]}\subset M^{\mathrm{trop}}_{g,n,\nabla}$ be a stratum. Then,

Theorems & Definitions (21)

  • Corollary
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Remark 2.6
  • Definition 2.7
  • Remark 2.8
  • Definition 2.9
  • ...and 11 more