A Quadratically Enriched Correspondence Theorem

TL;DR

This paper extends Mikhalkin's correspondence theorem by incorporating quadratic enrichments into both algebraic and tropical curve counts, linking algebraic geometry with tropical geometry through enriched invariants.

Contribution

It introduces a quadratic enrichment of Mikhalkin's multiplicity and Levine's quadratic Welschinger sign, establishing a new correspondence theorem between algebraic and tropical curves.

Findings

Established a quadratic enrichment of Mikhalkin's multiplicity.

Proved a correspondence between algebraic and tropical curves with quadratic enrichments.

Extended Mikhalkin's theorem to include Levine's quadratic Welschinger sign.

Abstract

We quadratically enrich Mikhalkin's correspondence theorem. That is, we prove a correspondence between algebraic curves on a toric surface counted with Levine's quadratic enrichment of the Welschinger sign, and tropical curves counted with a quadratic enrichment of Mikhalkin's multiplicity for tropical curves.