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Relative Node Polynomials for Plane Curves

Abstract : We generalize the recent work of Fomin and 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 which, 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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Florian Block. Relative Node Polynomials for Plane Curves. 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011), 2011, Reykjavik, Iceland. pp.199-210. ⟨hal-01215084⟩

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