Quadratically Enriched Plane Curve Counting via Tropical Geometry

TL;DR

This work advances a quadratically enriched framework for enumerating rational curves on smooth toric del Pezzo surfaces by translating the problem into tropical geometry with a refined multiplicity $ ext{mult}^{ ext{A}^1}$. A key result is a correspondence theorem equating the Grothendieck–Witt-valued algebraic count $N^{ ext{A}^1}_{ riangle}(oldsymbol{r},(d_1,ullet))$ with a tropical count $N^{ ext{A}^1, ext{trop}}_{ riangle}(oldsymbol{r},(d_1,ullet))$ computed via vertically stretched point configurations, floor diagrams, and twin-tree structures. The framework unifies complex GW invariants, real Welschinger invariants, and multiquadratic A^1-enriched counts within a single tropical paradigm, and it provides an algorithmic pathway for actual computations across several del Pezzo surfaces. The results include concrete computational data for $oldsymbol{P}^2$ and other toric del Pezzo surfaces, illustrating both the method and its predictive power, as well as potential universal formulas for multiquadratic σ anchored in the Grothendieck–Witt calculus. Overall, the paper delivers a robust tropical toolkit for quadratically enriched plane curve counting with broad implications for arithmetic and real enumerative geometry.

Abstract

We prove that the quadratically enriched count of rational curves in a smooth toric del Pezzo surface passing through $k$-rational points and pairs of conjugate points in quadratic field extensions $k\subset k(\sqrt{d_i})$ can be determined by counting certain tropical stable maps through vertically stretched point conditions with a suitable multiplicity. Building on the floor diagram technique in tropical geometry, we provide an algorithm to compute these numbers. Our tropical algorithm computes not only these new quadratically enriched enumerative invariants, but simultaneously also the complex Gromov-Witten invariant, the real Welschinger invariant counting curves satisfying real point conditions only, the real Welschinger invariant of curves satisfying pairs of complex conjugate and real point conditions, and the quadratically enriched count of curves satisfying $k$-rational point conditions.

Figures (12)

• Figure 1: The dual polygons of degrees of smooth toric del Pezzo surfaces.
• Figure 2: A local picture of a marked end of a tropical stable map and its image in ${\mathbb R}^2$. We use the convention to draw marked ends as dashed lines.
• Figure 3: Possibilities for double point conditions of a tropical stable map. We depict the image $f(\Gamma)$ but indicate the parametrization by the abstract graph. We call these local building blocks the vertex types.
• Figure 4: If an end of a component of double edges was not fixed by the double point conditions, we could deform the double edges into two edges.
• Figure 5: Examples for twin trees and their multiplicities as in \ref{['def-multtwintree']}. The color coding from \ref{['not:colors']} applies. In the picture on the right, the encircled vertices are possible choices for $v_0$ in the proof of \ref{['lem-algebratwintree']}.

Theorems & Definitions (147)

• Theorem 1.2: Main Theorem
• Theorem 1.3
• Theorem 1.4: \ref{['thm-flooreqtrop']}
• Definition 2.1
• Remark 2.2
• Definition 2.3: Degree of a tropical stable map
• Definition 2.4: Simple tropical stable maps
• Proposition 2.5
• Example 2.6
• Remark 2.7