Combinatorial Theory

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 \(\psi\) classes. Contrary to the polynomiality with respect to \( \Delta\), 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.

Mathematics Subject Classifications: Primary 14T15, 14T90, 05A15; Secondary 14N10, 52B20

Keywords: Tropical refined invariants, enumerative geometry, Welschinger invariants, Gromov-Witten invariants, floor diagrams