Computing Node Polynomials for Plane Curves

Abstract

According to the G¨ ottsche conjecture (now a theorem), the degree N d; of the Severi variety of plane curves of degreed with nodes is given by a polynomial ind, providedd is large enough. These 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 Gconjecture on the optimal threshold up to 14. We also determine the first 9 coefficients of N (d), for general , settling and extending a 1994 conjecture of Di Francesco and Itzykson. R´ esum´ e. Selon la Conjecture de G¨ ottsche (maintenant un Th´` eme), le degr´