Severi degrees on toric surfaces

Abstract Ardila and Block used tropical results of Brugallé and Mikhalkin to count nodal curves on a certain family of toric surfaces. Building on a linearity result of the first author, we revisit their work in the context of the Göttsche–Yau–Zaslow formula for counting nodal curves on arbitrary smooth surfaces, addressing several questions they raised by proving stronger versions of their main theorems. In the process, we give new combinatorial formulas for the coefficients arising in the Göttsche–Yau–Zaslow formulas, and give correction terms arising from rational double points in the relevant family of toric surfaces.