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Computation of refined toric invariants II

Thomas Blomme

TL;DR

This work extends refined toric enumerative geometry by linking real refined curve counts to tropical refined counts beyond the purely real boundary case. It defines and computes a quantum index for general oriented real curves, and develops a tropicalization framework plus a real realization (correspondence) theorem that lifts first-order tropical solutions to genuine real curves. The main result expresses the refined invariants R_{Δ,s} in terms of tropical refined counts N^{∂,trop}_{Δ(s)} with explicit q-analogs, unifying classical and tropical techniques via vertex-by-vertex local enumerations. The approach enables computation of refined toric invariants for complex configurations where complex points lie on multiple toric divisors, providing a robust bridge between real algebraic geometry and tropical refined invariants with potential connections to broader refinements in enumerative geometry.

Abstract

In 2015, G.~Mikhalkin introduced a refined count for real rational curves in toric surfaces. The counted curves have to pass through some real and complex points located on the toric boundary of the surface, and the count is refined according to the value of a so called quantum index. This count happens only to depend on the number of complex points on each toric divisors, leading to an invariant. First, we give a way to compute the quantum index of any oriented real rational curve, getting rid of the previously needed "purely imaginary" assumption on the complex points. Then, we use the tropical geometry approach to relate these classical refined invariants to tropical refined invariants, defined using Block-Göttsche multiplicity. This generalizes the result of Mikhalkin relating both invariants in the case where all the points are real, and the result of the author where complex points are located on a single toric divisor.

Computation of refined toric invariants II

TL;DR

This work extends refined toric enumerative geometry by linking real refined curve counts to tropical refined counts beyond the purely real boundary case. It defines and computes a quantum index for general oriented real curves, and develops a tropicalization framework plus a real realization (correspondence) theorem that lifts first-order tropical solutions to genuine real curves. The main result expresses the refined invariants R_{Δ,s} in terms of tropical refined counts N^{∂,trop}_{Δ(s)} with explicit q-analogs, unifying classical and tropical techniques via vertex-by-vertex local enumerations. The approach enables computation of refined toric invariants for complex configurations where complex points lie on multiple toric divisors, providing a robust bridge between real algebraic geometry and tropical refined invariants with potential connections to broader refinements in enumerative geometry.

Abstract

In 2015, G.~Mikhalkin introduced a refined count for real rational curves in toric surfaces. The counted curves have to pass through some real and complex points located on the toric boundary of the surface, and the count is refined according to the value of a so called quantum index. This count happens only to depend on the number of complex points on each toric divisors, leading to an invariant. First, we give a way to compute the quantum index of any oriented real rational curve, getting rid of the previously needed "purely imaginary" assumption on the complex points. Then, we use the tropical geometry approach to relate these classical refined invariants to tropical refined invariants, defined using Block-Göttsche multiplicity. This generalizes the result of Mikhalkin relating both invariants in the case where all the points are real, and the result of the author where complex points are located on a single toric divisor.

Paper Structure

This paper contains 44 sections, 45 theorems, 159 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Enumeration of rational curves in toric surfaces
  3. Quantum indices of real curves
  4. Refined enumerative geometry
  5. Tropical curves and real tropical curves
  6. Real abstract tropical curves
  7. Real parametrized tropical curves
  8. Plane tropical curves
  9. Real parametrizations of a plane tropical curve
  10. Moment of an edge
  11. Moduli space of tropical curves and refined multiplicity of a simple tropical curve
  12. Tropicalization
  13. Tropicalization of a marked curve
  14. Tropicalization of a parametrized curve
  15. Tropicalization of a plane curve
  16. ...and 29 more sections

Key Result

Theorem 1

(Mikhalkin, prop existence quantum index) Let $\varphi:\mathbb{C} P^1\rightarrow\mathbb{C}\Delta$ be a oriented real rational curve. Let $\varepsilon_j\theta_j$, with $\varepsilon_j=\pm 1$ and $0<\theta_j<\pi$, be the arguments of the complex intersection points of $\mathbb{H}$ with the toric bounda The half-integer $k(\mathbb{H},\varphi)$ is called the quantum index of the oriented curve.

Figures (8)

  • Figure 1: Abstract real tropical curve with its real structure depicted by doubling the exchanged edges.
  • Figure 2: Co-amoeba of a line.
  • Figure 3: Co-amoeba of a parabola with order map: $-1$ for red (triangles with left vertical side), $+1$ for blue (triangles with right vertical side).
  • Figure 4: Coemoeba of an half ellipse tangent to the infinite axis. On $(a)$ the coamoeba in $N_{2\pi}$, on $(b)$ in $N_\pi$
  • Figure 5: Depiction of the change of coordinate in the tangent ellipse case
  • ...and 3 more figures

Theorems & Definitions (114)

  • Theorem
  • Remark 1.1
  • Theorem 1.2: Mikhalkinmikhalkin2017quantum
  • Remark 1.3
  • Theorem 1.4: Mikhalkinmikhalkin2017quantum
  • Remark 1.5
  • Theorem : \ref{['theorem paper']}
  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • ...and 104 more