Tropical refined curve counting via motivic integration

Abstract

We propose a geometric interpretation of Block and Göttsche'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.

44 pages; 2 figures. v2: minor errors and typos corrected; Section 6 added (proof of the δ=1 case)

References in corpus (2)

• Refined curve counting with tropical geometry
• Motivic Serre invariants, ramification, and the analytic Milnor fiber

Cited by in corpus (7)

• Scattering diagrams, theta functions, and refined tropical curve counts
• A tropical motivic Fubini theorem with applications to Donaldson-Thomas theory
• Motivic integration and Milnor fiber
• Bounded integral and motivic Milnor fiber
• Computation of refined toric invariants II
• Motivic volume of families of polarized rigid-analytic tori
• Polynomiality properties of tropical refined invariants