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Abstract: According to the G\-ottsche conjecture now a theorem, the degree N^{d,delta} of the Severi variety of plane curves of degree d with delta nodes isgiven by a polynomial in d, provided d is large enough. These -nodepolynomials- N deltad were determined by Vainsencher and Kleiman-Piene fordelta <= 6 and delta <= 8, respectively. Building on ideas of Fomin andMikhalkin, we develop an explicit algorithm for computing all node polynomials,and use it to compute N deltad for delta <= 14. Furthermore, we improve thethreshold of polynomiality and verify G\-ottsche-s conjecture on the optimalthreshold up to delta <= 14. We also determine the first 9 coefficients ofN deltad, for general delta, settling and extending a 1994 conjecture of DiFrancesco and Itzykson.



Autor: Florian Block

Fuente: https://arxiv.org/







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