Computing Node Polynomials for Plane Curves

TL;DR

This paper develops an explicit algorithm to compute node polynomials for plane curves, verifies G"ottsche's conjecture up to delta 14, and extends previous results on polynomial coefficients.

Contribution

It introduces a new algorithm for computing all node polynomials and verifies the conjecture's threshold and coefficients for higher delta values.

Findings

Computed node polynomials for delta <= 14

Verified G"ottsche's conjecture up to delta 14

Determined the first 9 coefficients of N_delta(d) for general delta

Abstract

According to the G\"ottsche conjecture (now a theorem), the degree N^{d, delta} of the Severi variety of plane curves of degree d with delta nodes is given by a polynomial in d, provided d is large enough. These "node polynomials" N_delta(d) were determined by Vainsencher and Kleiman-Piene for delta <= 6 and delta <= 8, respectively. Building on ideas of Fomin and Mikhalkin, we develop an explicit algorithm for computing all node polynomials, and use it to compute N_delta(d) for delta <= 14. Furthermore, we improve the threshold of polynomiality and verify G\"ottsche's conjecture on the optimal threshold up to delta <= 14. We also determine the first 9 coefficients of N_delta(d), for general delta, settling and extending a 1994 conjecture of Di Francesco and Itzykson.