Computing Node Polynomials for Plane Curves
Florian Block · Mathematical Research Letters · 2011
According to the Göttsche conjecture (now a theorem), the degree N d,δ of the Severi variety of plane curves of degree d with δ nodes is given by a polynomial in d, provided d is large enough.These "node polynomials" N δ (d) were determined by Vainsencher and Kleiman-Piene for δ ≤ 6 and δ ≤ 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 δ (d) for δ ≤ 14.Furthermore, we improve the threshold of polynomiality and verify Göttsche's conjecture on the optimal threshold up to δ ≤ 14.We also determine the first nine coefficients of N δ (d), for general δ, settling and extending a 1994 conjecture of Di Francesco and Itzykson.2 δ generic points in P 2 ?The answer to this question is the Severi degree N d,δ , the degree of the corresponding Severi variety.In 1994, Di Francesco and Itzykson [6] conjectured that N d,δ is given by a polynomial in d (assuming δ is fixed and d is sufficiently large).It is not hard to see that, if such a polynomial exists, it has to be of degree 2δ.Recently, Fomin and Mikhalkin [7, Theorem 5.1] established the polynomiality of N d,δ using tropical geometry and floor decompositions.More precisely, they showed that there exists, for every δ ≥ 1, a node polynomial N δ (d) which satisfies N d,δ = N δ (d) for all d ≥ 2δ.(The δ = 0 case is trivial asFor δ = 1, 2, 3, the polynomiality of the Severi degrees and the formulas for N δ (d) were determined in the 19th century.For δ = 4, 5, 6, this was only achieved by Vainsencher [13] in 1995.In 2001, Kleiman and Piene [9] settled the cases δ = 7, 8. Earlier, Göttsche [8] conjectured a more detailed (still not entirely explicit) description of these polynomials for counting nodal curves on smooth projective algebraic surfaces. Main results.In this paper we develop, building on ideas of Fomin and Mikhalkin [7], an explicit algorithm (see Algorithm 1) for computing the node polynomials N δ (d) for arbitrary δ.This algorithm is used to calculate N δ (d) for all δ ≤ 14.