Florian Block - Computing Node Polynomials for Plane Curves

dmtcs:2861 - Discrete Mathematics & Theoretical Computer Science, January 1, 2010, DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010) - https://doi.org/10.46298/dmtcs.2861
Computing Node Polynomials for Plane CurvesArticle

Authors: Florian Block 1

  • 1 Department of Mathematics - University of Michigan

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.


Volume: DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010)
Section: Proceedings
Published on: January 1, 2010
Imported on: January 31, 2017
Keywords: labeled floor diagram.,Severi degree,curve enumeration,plane curve,node polynomial,[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO],[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]
Funding:
    Source : OpenAIRE Graph
  • Deep Drug Discovery and Deployment; Code: PTDC/CCI-BIO/29266/2017

14 Documents citing this article

Consultation statistics

This page has been seen 175 times.
This article's PDF has been downloaded 240 times.