Computing Node Polynomials for Plane Curves

Block, Florian

Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010), DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010) (2010).

Résumé

According to the Göttsche 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 $\delta \leq 6$ and $\delta \leq 8$, 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 $\delta \leq 14$. Furthermore, we improve the threshold of polynomiality and verify Göttsche's conjecture on the optimal threshold up to $\delta \leq 14$. 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.

Détail

DOI : 10.46298/dmtcs.2861