Refined node polynomials via long edge graphs

TL;DR

This paper extends the understanding of refined Severi degrees for algebraic surfaces, utilizing long-edge graphs and tropical geometry, and explores potential generalizations to singular surfaces and curves with multiple points.

Contribution

It generalizes existing results on Severi degrees to refined invariants on P^2 and rational ruled surfaces, and proposes conjectural extensions to more complex surface and curve types.

Findings

Refined Severi degrees are expressed via long-edge graphs.

Generalizations to singular surfaces are proposed, with some conjectural.

The multiplicative nature of generating functions is supported for new cases.

Abstract

The generating functions of the Severi degrees for sufficiently ample line bundles on algebraic surfaces are multiplicative in the topological invariants of the surface and the line bundle. Recently new proofs of this fact were given for toric surfaces by Block, Colley, Kennedy and Liu, Osserman, using tropical geometry and in particular the combinatorial tool of long-edged graphs. In the first part of this paper these results are for P^2 and rational ruled surfaces generalized to refined Severi degrees. In the second part of the paper we give a number of mostly conjectural generalizations of this result to singular surfaces, and curves with prescribed multiple points.