Quadratically Enriched Plane Curve Counting via Tropical Geometry

TL;DR

This paper introduces a tropical geometry-based algorithm to compute quadratically enriched counts of rational curves in toric del Pezzo surfaces, unifying various enumerative invariants including Gromov-Witten and Welschinger invariants.

Contribution

It develops a novel tropical geometric method and algorithm for calculating quadratically enriched curve counts, extending existing techniques to new enumerative invariants.

Findings

Algorithm computes quadratically enriched invariants and classical Gromov-Witten invariants.

Unifies computation of real, complex, and enriched curve counts.

Provides explicit methods for enumerating rational curves in toric surfaces.

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.