A Caporaso-Harris type Formula for relative refined invariants

TL;DR

This paper introduces a recursive formula and algorithm for computing refined counts of real rational curves in toric surfaces, enhancing the understanding of their invariants through tropical geometry and quantum indices.

Contribution

It provides the first recursive formula for refined invariants of real rational curves, enabling efficient computation via tropical geometry methods.

Findings

Derived a recursive formula for refined invariants

Developed an algorithm for practical computation

Confirmed invariance of counts under point choices

Abstract

G. Mikhalkin introduced a refined count for real rational curves in a toric surface which pass through some points on the toric boundary of the surface. The refinement is provided by the value of a so-called quantum index. Moreover, he proved that the result of this refined count does not depend on the choice of the points. The correspondence theorem allows one to compute these invariants using the tropical geometry approach and the refined Block-G\"ottsche multiplicities. In this paper we give a recursive formula for these invariants, that leads to an algorithm to compute them.