2010/06/01 by Block, Florian
#05A99 #14N10 #14N35 #14T05 #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1006.0218
According to the Göttsche conjecture (now a theorem), the degree Nd, 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" Ndelta(d) were determined by Vainsencher and Kleiman-Piene for delta <= 6 and delta <= 8, respectively. Building on ideas of Fomin and Mikhalkin, we develop an explicit algorithm for computing all node polynomials, and use it to compute Ndelta(d) for delta <= 14. Furthermore, we improve the threshold of polynomiality and verify Göttsche's conjecture on the optimal threshold up to delta <= 14. We also determine the first 9 coefficients of Ndelta(d), for general delta, settling and extending a 1994 conjecture of Di Francesco and Itzykson.