Computing Node Polynomials for Plane Curves

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 $\delta \le 6$ and $\delta \le 8$, 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 $\delta \le 14$. Furthermore, we improve the threshold of polynomiality and verify Göttsche’s conjecture on the optimal threshold up to $\delta \le 14$. We also determine the first nine coefficients of $N_\delta(d)$, for general $\delta$, settling and extending a 1994 conjecture of Di Francesco and Itzykson.

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Published 19 August 2011