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Tropical enumeration of curves in blowups of the projective plane

Brett Parker

TL;DR

The paper develops a uniform, tropical-gluing approach to Gromov-Witten invariants of all blowups of CP^2 by embedding the problem into the exploded-manifold framework and employing relative invariants. It constructs a recursive formula for the relative potential ${F}$, then uses a tropical gluing formula to assemble absolute invariants $G_n$ from relative data, with a termination criterion that yields a simplified recursion. The method recovers and extends prior zero-genus and small-n point recursions, connects to classical degeneration techniques, and provides computational tools (e.g., a Mathematica implementation) to verify results up to degree 12 and beyond. The framework yields a uniform, genus-agnostic description of curve counts across arbitrary numbers of blowups, establishing a powerful bridge between tropical geometry, log geometry, and Gromov-Witten theory. This approach has potential to reprove known formulas (e.g., Bryan–Leung) and to produce new invariants for large families of blown-up toric surfaces.

Abstract

We describe a method for recursively calculating Gromov-Witten invariants of all blowups of the projective plane. This recursive formula is different from the recursive formulas due to Göttsche and Pandharipande in the zero genus case, and Caporaso and Harris in the case of no blowups. We use tropical curves and a recursive computation of Gromov-Witten invariants relative a normal crossing divisor.

Tropical enumeration of curves in blowups of the projective plane

TL;DR

The paper develops a uniform, tropical-gluing approach to Gromov-Witten invariants of all blowups of CP^2 by embedding the problem into the exploded-manifold framework and employing relative invariants. It constructs a recursive formula for the relative potential , then uses a tropical gluing formula to assemble absolute invariants from relative data, with a termination criterion that yields a simplified recursion. The method recovers and extends prior zero-genus and small-n point recursions, connects to classical degeneration techniques, and provides computational tools (e.g., a Mathematica implementation) to verify results up to degree 12 and beyond. The framework yields a uniform, genus-agnostic description of curve counts across arbitrary numbers of blowups, establishing a powerful bridge between tropical geometry, log geometry, and Gromov-Witten theory. This approach has potential to reprove known formulas (e.g., Bryan–Leung) and to produce new invariants for large families of blown-up toric surfaces.

Abstract

We describe a method for recursively calculating Gromov-Witten invariants of all blowups of the projective plane. This recursive formula is different from the recursive formulas due to Göttsche and Pandharipande in the zero genus case, and Caporaso and Harris in the case of no blowups. We use tropical curves and a recursive computation of Gromov-Witten invariants relative a normal crossing divisor.

Paper Structure

This paper contains 24 sections, 28 theorems, 112 equations, 18 figures.

Table of Contents

  1. Background
  2. Stratified $\mathbb Z$--affine spaces
  3. Complex manifolds with simple normal crossing divisors
  4. Toric manifolds and toric varieties
  5. Log geometry
  6. Exploded manifolds
  7. The explosion functor
  8. The smooth part and tropical part functors
  9. The moduli stack of holomorphic curves
  10. Contact data, and counting curves with constrained ends.
  11. Refinements
  12. Decomposition of counts of curves with constrained ends
  13. Tropical completion
  14. Balancing condition of tropical curves
  15. Tropical gluing formula
  16. ...and 9 more sections

Key Result

Theorem 1.10

Let $\gamma$ be a tropical curve in $\underline{ \bold B}$ with constrained contact data $\Gamma$. If Algorithm alg terminates with all edges of $\gamma$ labeled rigid, where $n_{\gamma}$ appears in equation (decomposition), and $n_{\gamma_{v}}$ is the virtual number of curves in $\bold B\check\rvert_{v}$ with constrained contact data specified by $\gamma_{v}$ with incoming edges constrained and

Figures (18)

  • Figure 1: The moment polytope, and dual fan of $M$
  • Figure 2: A rigid tropical curve in the tropical part of $\mathop{\mathrm{Expl}}\nolimits(M,N)$, or $\underline{ \bold M}$.
  • Figure 3: Tropical curves contributing to $n_{(-1,0)}$
  • Figure 4: Tropical curves contributing to $n_{(-1,g)}$
  • Figure 5: Tropical curves contributing to $n_{(-2,1)}$
  • ...and 13 more figures

Theorems & Definitions (48)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Remark 1.5
  • Definition 1.6
  • Definition 1.7
  • Remark 1.8
  • Theorem 1.10: Simplified gluing formula
  • Claim 1.11
  • ...and 38 more