Abstract
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 δ nodes is given by a polynomial in d, provided δ 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 δ, and use it to present explicit formulas for δ≤6. We also give a threshold for polynomiality, and compute the first few leading terms for any δ.
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Block, F. Relative node polynomials for plane curves. J Algebr Comb 36, 279–308 (2012). https://doi.org/10.1007/s10801-011-0337-x
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DOI: https://doi.org/10.1007/s10801-011-0337-x