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On the piecewise quasipolynomiality of double tropical Welschinger invariants

Vincenzo Reda

TL;DR

The paper addresses the problem of counting real curves on toric surfaces via double tropical Welschinger invariants associated with $h$-transverse polygons, proving that these counts are piecewise quasipolynomial across chamber structures. It develops a robust framework based on real floor diagrams, a correspondence theorem to tropical curves, and weighted Ehrhart theory to convert weighted lattice sums into higher-dimensional unweighted lattice counts, enabling precise chamberwise descriptions. The main contributions include proving quasipolynomiality for the counting maps $G_{(\mathbf{d}^r;\mathbf{d}^l),\mathbf{c},g}^{n_1,n_2}$ (and its $\mathbf{c}$-dependent extension) with degree $g$, introducing $s$-real multiplicities and new combinatorial Welschinger-type numbers, and providing explicit examples to illustrate the piecewise structure. The results enhance understanding of tropical enumerative invariants, their chamber structure, and potential deformation behavior, with implications for real curve counts on a broad class of toric surfaces, beyond Hirzebruch surfaces.

Abstract

Ardila and Brugallé conjectured that double tropical Welschinger invariants of Hirzebruch surfaces are piecewise quasipolynomial. In this work, we prove the conjecture holds in full generality, i.e. for toric surfaces corresponding to h-transverse polygons. Furthermore, we define new combinatorial Welschinger-type numbers for h-transverse polygons and show that they are likewise piecewise quasipolynomial.

On the piecewise quasipolynomiality of double tropical Welschinger invariants

TL;DR

The paper addresses the problem of counting real curves on toric surfaces via double tropical Welschinger invariants associated with -transverse polygons, proving that these counts are piecewise quasipolynomial across chamber structures. It develops a robust framework based on real floor diagrams, a correspondence theorem to tropical curves, and weighted Ehrhart theory to convert weighted lattice sums into higher-dimensional unweighted lattice counts, enabling precise chamberwise descriptions. The main contributions include proving quasipolynomiality for the counting maps (and its -dependent extension) with degree , introducing -real multiplicities and new combinatorial Welschinger-type numbers, and providing explicit examples to illustrate the piecewise structure. The results enhance understanding of tropical enumerative invariants, their chamber structure, and potential deformation behavior, with implications for real curve counts on a broad class of toric surfaces, beyond Hirzebruch surfaces.

Abstract

Ardila and Brugallé conjectured that double tropical Welschinger invariants of Hirzebruch surfaces are piecewise quasipolynomial. In this work, we prove the conjecture holds in full generality, i.e. for toric surfaces corresponding to h-transverse polygons. Furthermore, we define new combinatorial Welschinger-type numbers for h-transverse polygons and show that they are likewise piecewise quasipolynomial.

Paper Structure

This paper contains 15 sections, 9 theorems, 68 equations, 8 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Double tropical Welschinger invariants
  3. Techniques
  4. Results
  5. Organization
  6. Acknowledgements
  7. Preliminaries
  8. Real floor diagrams
  9. Weighted Ehrhart theory
  10. Piecewise quasipolynomiality of double tropical Welschinger invariants
  11. Notation
  12. Proof of Theorem \ref{['thm-quasipolynomiality']}
  13. Example
  14. Combinatorial game
  15. Example

Key Result

Theorem 1.4

Let $(\textbf{d}^r;\textbf{d}^l)\in\mathbb Z^{n+m}$ be a vector with positive integer coordinates and $g\geq0,n_1,n_2>0$ fixed integers and $\textbf{c}=(\textbf{c}^r,\textbf{c}^l)\in\mathbb Z^{n+m}$ such that $c_1^r>\dots>c_n^r$ and $c_1^l<\dots<c_m^l$. The map $G_{(\textbf{d}^r,\textbf{d}^l),\textb

Figures (8)

  • Figure 1:
  • Figure 2:
  • Figure 3: Hyperplane arrangement in $\Lambda$
  • Figure 4:
  • Figure 5:
  • ...and 3 more figures

Theorems & Definitions (36)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3: $s$-pair
  • Remark 2.4
  • Definition 2.5: $s$-real floor diagram
  • Definition 2.6: $s$-real multiplicity
  • ...and 26 more