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Tropical vertex and real enumerative geometry

Eugenii Shustin

Abstract

We show that the commutator relations in the refined tropical vertex group can be expressed via the enumeration of suitable real rational curves in toric surfaces.

Tropical vertex and real enumerative geometry

Abstract

We show that the commutator relations in the refined tropical vertex group can be expressed via the enumeration of suitable real rational curves in toric surfaces.
Paper Structure (7 sections, 4 theorems, 79 equations, 1 figure)

This paper contains 7 sections, 4 theorems, 79 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Tropical vertex and its deformation
  3. Commutators in the tropical vertex group
  4. Commutators in the deformed tropical vertex group
  5. Relative Block-Göttsche invariants and Mikhalkin's quantization
  6. The case of $q=-1$
  7. Relative Welschinger invariants

Key Result

Lemma 3.1

If $\boldsymbol{z}$ and $\boldsymbol{z}'$ are in general position on $D_a$ and $D_b$, respectively, then ${\mathcal{M}}_{0,m+m'+1}(X_{a,b},\beta,\beta',\boldsymbol{z},\boldsymbol{z}')$ is finite, and for each of its elements, the map $\boldsymbol{n}:{\mathbb P}^1\to X_{a,b}$ is a birational immersio

Figures (1)

  • Figure 1: Proof of Proposition \ref{['lrtv3']}, Part 7.5

Theorems & Definitions (8)

  • Lemma 3.1
  • Proposition 3.2
  • proof
  • Proposition 4.1
  • proof
  • Remark 4.2
  • Proposition 5.1
  • proof