Counting Curves on Toric Surfaces

A few years ago, Tzeng settled a remarkable conjecture of Goettsche on counting nodal curves on smooth surfaces, proving that the formulas are given by certain universal polynomials. At the same time, Ardila and Block used the tropical approach of Brugalle, Mihalkin and Fomin to count nodal curves on a certain class of (not necessarily smooth) toric surfaces, and obtained similar polynomiality behavior. In light of recent work of Block, Colley and Kennedy, we revisit the work of Ardila and Block, giving simpler proofs of stronger results, and in particular expressing the formulas in a manner compatible with Tzeng’s theorem for smooth surfaces. In this way, we obtain explicit combinatorial formulas for the coefficients arising in Tzeng’s theorem, and also give correction terms arising from singularities. Although the motivation and main theorems are expressed in terms of algebraic geometry, the work itself is largely combinatorial. This is joint work with Fu Liu.