Computing Node Polynomials for Plane Curves
نویسنده
چکیده
According to the Göttsche conjecture (now a theorem), the degree N of the Severi variety of plane curves of degree d with δ nodes is given by a polynomial in d, provided d is large enough. These “node polynomials” Nδ(d) were determined by Vainsencher and Kleiman–Piene for δ ≤ 6 and δ ≤ 8, respectively. Building on ideas of Fomin and Mikhalkin, we develop an explicit algorithm for computing all node polynomials, and use it to compute Nδ(d) for δ ≤ 14. Furthermore, we improve the threshold of polynomiality and verify Göttsche’s conjecture on the optimal threshold up to δ ≤ 14. We also determine the first 9 coefficients of Nδ(d), for general δ, settling and extending a 1994 conjecture of Di Francesco and Itzykson. Résumé. Selon la Conjecture de Göttsche (maintenant un Théorème), le degréN de la variété de Severi des courbes planes de degré d avec δ noeuds est donné par un polynôme en d, pour d assez grand. Ces polynômes de noeudsNδ(d) ont été déterminés par Vainsencher et Kleiman–Piene pour δ ≤ 6 et δ ≤ 8, respectivement. S’appuyant sur les idées de Fomin et Mikhalkin, nous développons un algorithme explicite permettant de calculer tous les polynômes de noeuds, et l’utilisons pour calculer Nδ(d), pour δ ≤ 14. De plus, nous améliorons le seuil de polynomialité et vérifions la Conjecture de Göttsche sur le seuil optimal jusqu’à δ ≤ 14. Nous déterminons aussi les 9 premiers coéfficients de Nδ(d), pour un δ quelconque, confirmant et étendant la Conjecture de Di Francesco et Itzykson de 1994.
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