Communications in Number Theory and Physics

Volume 10 (2016)

Number 2

Refined node polynomials via long edge graphs

Pages: 193 – 234



Lothar Göttsche (International Center for Theoretical Physics (ICTP), Trieste, Italy)

Benjamin Kikwai (International Center for Theoretical Physics (ICTP), Trieste, Italy)


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 $\mathbb{P}^2$ and rational ruled surfaces generalised to refined Severi degrees. In the second part of the paper we give a number of mostly conjectural generalisations of this result to singular surfaces, and curves with prescribed multiple points. The formulas involve modular forms and theta functions.

Published 19 July 2016