Year: 2025
Type: article
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 divisor, 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.14H99, 14M25, 14N10, 14T90 1. Introduction 3129 2. Tropical curves and real tropical curves 3135 3. Quantum indices of real curves 3137 4. Tropical enumerative problem and refined curve counting 3146 5. Realization and correspondence theorem in the real case 3149 6. Proof of Theorem 4.8 3162
Source: Geometry & Topology
Author: Thomas Blomme
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
OpenAlex ID: W4414462017