Computing Severi Degrees with Long-edge Graphs

TL;DR

This paper explores a class of finite-edge graphs related to floor diagrams and tropical curves to better understand the generating series for Severi degrees through combinatorial methods.

Contribution

It introduces a new combinatorial framework using long-edge graphs to analyze Severi degrees and connects these graphs to existing tropical geometry tools.

Findings

Established a link between long-edge graphs and Severi degrees

Provided a combinatorial approach to compute Severi degrees

Enhanced understanding of the structure of generating series for Severi degrees

Abstract

We study a class of graphs with finitely many edges in order to understand the nature of the formal logarithm of the generating series for Severi degrees in elementary combinatorial terms. These graphs are related to floor diagrams associated to plane tropical curves originally developed by Brugalle and Mikhalkin, and used by Block, Fomin, and Mikhalkin to calculate Severi degrees of the projective plane and node polynomials of plane curves.