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JOURNAL OF
ALGEBRAIC
COMBINATORICS
  Editors-in-chief: C. A. Athanasiadis, T. Lam, A. Munemasa, H. Van Maldeghem
ISSN 0925-9899 (print) • ISSN 1572-9192 (electronic)
 

Relative node polynomials for plane curves

Florian Block

DOI: 10.1007/s10801-011-0337-x

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\leq 6. We also give a threshold for polynomiality, and compute the first few leading terms for any δ.

Pages: 279–308

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

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References

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