Computing Node Polynomials for Plane Curves

Abstract

According to the Gttsche conjecture (now a theorem), the degree N d, of the Severi variety of plane curves of degree d with nodes is given by a polynomial in d, provided d is large enough. These "node polynomials" N (d) were determined by Vainsencher and Kleiman-Piene for 6 and 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 (d) for 14. Furthermore, we improve the threshold of polynomiality and verify Gttsche's conjecture on the optimal threshold up to 14. We also determine the first nine coefficients of N (d), for general , settling and extending a 1994 conjecture of Di Francesco and Itzykson.