01201nas a2200145   4500008004100000245005000041210005000091260003400141300001400175490000700189520075600196100002200952700003100974856005001005       2016                        eng d00aRefined node polynomials via long edge graphs0 aRefined node polynomials via long edge graphs  bInternational Press of Boston  a193–2340 v103 a<p>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.</p>1 aGöttsche, Lothar1 aKikwai, Benjamin, Kipkirui  uhttp://dx.doi.org/10.4310/CNTP.2016.v10.n2.a2