Multiplicativity of the Generating Functions for the Refined Node Polynomials on Toric Surfaces

Abstract

The advent of tropical enumerative geometry [Mik05, BM07, FM10], led to emergence of combinatorial strategies for tackling enumerative problems on complex algebraic varieties. In this thesis, we use long-edge graphs, which are combinatorial abstractions of tropical plane curves to prove that the generating function for the refined node polynomials on a subclass of toric surfaces exhibits a multiplicative structure. Long edge graphs were originally introduced by Block, Colley and Kennedy [BCK14] and Liu [Liu16] to study similar questions for the standard Severi degrees