Polynomiality properties of tropical refined invariants

Abstract

Tropical refined invariants of toric surfaces constitute a fascinating interpolation between real and complex enumerative geometries via tropical geometry. They were originally introduced by Block and Göttsche, and further extended by Göttsche and Schroeter in the case of rational curves.In this paper, we study the polynomial behavior of coefficients of these tropical refined invariants. We prove that coefficients of small codegree are polynomials in the Newton polygon of the curves under enumeration, when one fixes the genus of the latter. This provides a surprising reappearance, in a dual setting, of the so-called node polynomials and the Göttsche conjecture. Our methods, based on floor diagrams introduced by Mikhalkin and the first author, are entirely combinatorial. Although the combinatorial treatment needed here is different, we follow the overall strategy designed by Fomin and Mikhalkin and further developed by Ardila and Block. Hence our results may suggest phenomena in complex enumerative geometry that have not been studied yet.In the particular case of rational curves, we extend our polynomiality results by including the extra parameter s recording the number of ψ classes. Contrary to the polynomiality with respect to ∆, the one with respect to s may be expected from considerations on Welschinger invariants in real enumerative geometry. This pleads in particular in favor of a geometric definition of Göttsche-Schroeter invariants.

Floor diagrams and enumerative invariants of line bundles over an elliptic curve

“…Recently, Brugallé and Puentes [BJ22] proved a polynomiality statement for the coefficients of a fixed codegree in the families of refined invariance of toric surfaces. Such a behavior could also be studied for the hereby introduced refined invariants using the floor diagrams.…”