Relative node polynomials for plane curves

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.

Computing Node Polynomials for Plane Curves

“…By extending ideas of S. Fomin and G. Mikhalkin [7] and of the present paper, we can obtain polynomiality results for relative Severi degrees, the degrees of generalized Severi varieties (see [5,14]). This is discussed in the separate paper [1]; see Remark 3.9.…”

Enumeration of rational curves in a moving family ofP2

“…This is analogous to the case of counting δ-nodal degree d curves in P 2 ; let N δ d denote that number. A formula for this number can be explicitly found in [2] for instance. On the other hand, let N P 2 d denote the number of rational degree d curves in P 2 through 3d− 1 generic points.…”

“…One of the most fundamental and studied problems in enumerative geometry is the following: what is N δ d , the number of degree d curves in P 2 that have δ distinct nodes and pass through d(d+3) 2 − δ generic points? The question was studied more than a hundred years ago by Zeuthen ([21]) and has been studied extensively in the last thirty years by Ran ([16], [15]), Vainsencher ([20]), Caporaso-Harris ( [4]), Kazarian ([8]), Kleiman and Piene ([9]), Florian Block ( [2]), Tzeng and Li ([18], [19]), Kool, Shende and Thomas ( [13]), Berczi ( [1]) and Fomin and Mikhalkin ( [6]), amongst others. This question has been investigated from several perspectives and is very well understood.…”

A proof of the Göttsche-Yau-Zaslow formula

“…Recently, Fomin and Mikhalkin [FM09] proved the polynomiality with tropical geometry and find many interesting properties of node polynomials. Block [Bl10] generalized it to relative node polynomials and proved that there is a formal power series which specializes to all relative node polynomials. These results suggest that enumerating curves with broader conditions may possess a generalized structure, which could be used to provide answers to open problems and interpretations for known results.…”