Structure of node polynomials for curves on surfaces (Q2922209)

For a smooth projective surface \(S\) and a linear system \(|L|\) let \(N_r\) be the degree of the locus of \(r\)-nodal curves. It is known from the Göttsche conjecture that \(N_r\) is a polynomial of the four relevant numbers of the situation. An earlier paper of the author proves that \(r!N_r=P_r(a_1,\ldots,a_r)\) where \(P_r\) is the \(r\)th complete exponential Bell polynomial and \(a_i\) are universal integer coefficient linear polynomials of the four numbers.NEWLINENEWLINEIn this paper, the author gives an intersection theoretic interpretation of the \(a_i\) linear forms. In addition the relation of the results with \textit{M. È. Kazaryan}'s [Russ. Math. Surv. 58, No. 4, 665--724 (2003); translation from Usp. Mat. Nauk 58, No. 4, 29--88 (2003; Zbl 1062.58039)] multi-singularity formulas is discussed.