Tropical refined curve counting via motivic integration

TL;DR

This paper offers a geometric interpretation of refined tropical curve counting invariants using motivic measures, validating the approach for genus 1 and arbitrary genus after specialization.

Contribution

It introduces a new geometric perspective on refined tropical curve counting via motivic integration, extending the understanding to higher genus cases.

Findings

Validates the interpretation for genus 1 linear series

Extends the interpretation to arbitrary genus after specialization

Connects tropical invariants with motivic measures in algebraic geometry

Abstract

We propose a geometric interpretation of Block and G\"ottsche's refined tropical curve counting invariants in terms of virtual $\chi_{-y}$-specializations of motivic measures of semialgebraic sets in relative Hilbert schemes. We prove that this interpretation is correct for linear series of genus 1, and in arbitrary genus after specializing from $\chi_{-y}$ to Euler characteristic.