Relative node polynomials for plane curves

Block, Florian

Journal of Algebraic Combinatorics, Tome 36 (2012) no. 2, pp. 279-308.

Résumé

Summary: We generalize the recent work of S. Fomin and G. Mikhalkin on polynomial formulas for Severi degrees. The degree of the Severi variety of plane curves of degree d and $\delta $ nodes is given by a polynomial in d, provided $\delta $ is fixed and d is large enough. We extend this result to generalized Severi varieties parametrizing plane curves that, in addition, satisfy tangency conditions of given orders with respect to a given line. We show that the degrees of these varieties, appropriately rescaled, are given by a combinatorially defined "relative node polynomial" in the tangency orders, provided the latter are large enough. We describe a method to compute these polynomials for arbitrary $\delta $, and use it to present explicit formulas for $\delta \leq 6$. We also give a threshold for polynomiality, and compute the first few leading terms for any $\delta $.

Détail

Keywords: enumerative geometry, floor diagram, Gromov-Witten theory, node polynomial, tangency conditions